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passage: ## Granular gas
Researchers from the University of Twente, the University of Patras in Greece, and the Foundation for Fundamental Research on Matter have constructed a Feynman–Smoluchowski engine which, when not in thermal equilibrium, converts pseudo-Brownian motion into work by means of a granular gas, which is a conglomeration of solid particles vibrated with such vigour that the system assumes a gas-like state. The constructed engine consisted of four vanes which were allowed to rotate freely in a vibrofluidized granular gas. Because the ratchet's gear and pawl mechanism, as described above, permitted the axle to rotate only in one direction, random collisions with the moving beads caused the vane to rotate. This seems to contradict Feynman's hypothesis. However, this system is not in perfect thermal equilibrium: energy is constantly being supplied to maintain the fluid motion of the beads. Vigorous vibrations on top of a shaking device mimic the nature of a molecular gas. Unlike an ideal gas, though, in which tiny particles move constantly, stopping the shaking would simply cause the beads to drop. In the experiment, this necessary out-of-equilibrium environment was thus maintained. Work was not immediately being done, though; the ratchet effect only commenced beyond a critical shaking strength. For very strong shaking, the vanes of the paddle wheel interacted with the gas, forming a convection roll, sustaining their rotation.
|
https://en.wikipedia.org/wiki/Brownian_ratchet
|
passage: The correlation just scales the covariance by the product of the standard deviation of each variable. Consequently, the correlation is a dimensionless quantity that can be used to compare the linear relationships between pairs of variables in different units. If the points in the joint probability distribution of X and Y that receive positive probability tend to fall along a line of positive (or negative) slope, ρXY is near +1 (or −1). If ρXY equals +1 or −1, it can be shown that the points in the joint probability distribution that receive positive probability fall exactly along a straight line. Two random variables with nonzero correlation are said to be correlated. Similar to covariance, the correlation is a measure of the linear relationship between random variables.
The correlation coefficient between the random variables
$$
X
$$
and
$$
Y
$$
is
$$
\rho_{XY}=\frac{\operatorname{cov}(X,Y)}{\sqrt{V(X)V(Y)}}=\frac{\sigma_{XY}}{\sigma_X\sigma_Y}.
$$
## Important named distributions
Named joint distributions that arise frequently in statistics include the multivariate normal distribution, the multivariate stable distribution, the multinomial distribution, the negative multinomial distribution, the multivariate hypergeometric distribution, and the elliptical distribution.
|
https://en.wikipedia.org/wiki/Joint_probability_distribution
|
passage: In 8086 CPUs the instruction is used for , with being a pseudo-opcode to encode the instruction . Some disassemblers recognize this and will decode the instruction as . Similarly, IBM assemblers for System/360 and System/370 use the extended mnemonics and for and with zero masks. For the SPARC architecture, these are known as synthetic instructions.
Some assemblers also support simple built-in macro-instructions that generate two or more machine instructions. For instance, with some Z80 assemblers the instruction is recognized to generate followed by . These are sometimes known as pseudo-opcodes.
Mnemonics are arbitrary symbols; in 1985 the IEEE published Standard 694 for a uniform set of mnemonics to be used by all assemblers. The standard has since been withdrawn.
#### Data directives
There are instructions used to define data elements to hold data and variables. They define the type of data, the length and the alignment of data. These instructions can also define whether the data is available to outside programs (programs assembled separately) or only to the program in which the data section is defined. Some assemblers classify these as pseudo-ops.
Assembly directives
Assembly directives, also called pseudo-opcodes, pseudo-operations or pseudo-ops, are commands given to an assembler "directing it to perform operations other than assembling instructions". Directives affect how the assembler operates and "may affect the object code, the symbol table, the listing file, and the values of internal assembler parameters".
|
https://en.wikipedia.org/wiki/Assembly_language%23Assembler
|
passage: Then the map
$$
f
$$
is said to be differentiable at a point
$$
x
$$
in
$$
X
$$
if there exists a (necessarily unique) linear transformation
$$
f'(x) : \mathbb{R}^n \to \mathbb{R}^m
$$
, called the derivative of
$$
f
$$
at
$$
x
$$
, such that
$$
\lim_{ h \to 0 } \frac{1}{|h|} |f(x + h) - f(x) - f'(x)h| = 0
$$
where
$$
f'(x)h
$$
is the application of the linear transformation
$$
f'(x)
$$
to
$$
h
$$
. If
$$
f
$$
is differentiable at
$$
x
$$
, then it is continuous at
$$
x
$$
since
$$
|f(x + h) - f(x)| \le (|h|^{-1}|f(x + h) - f(x) - f'(x)h|) |h| + |f'(x)h| \to 0
$$
as
$$
h \to 0
$$
.
As in the one-variable case, there is
This is proved exactly as for functions in one variable.
|
https://en.wikipedia.org/wiki/Calculus_on_Euclidean_space
|
passage: A precessional ram was responsible for rotating the gyroscope to change the direction of the precessional force to counteract any forces causing the vehicle imbalance. The one-of-a-kind prototype is now at the Lane Motor Museum in Nashville, Tennessee.
## Consumer electronics
In addition to being used in compasses, aircraft, computer pointing devices, etc., gyroscopes have been introduced into consumer electronics.
Since the gyroscope allows the calculation of orientation and rotation, designers have incorporated them into modern technology. The integration of the gyroscope has allowed for more accurate recognition of movement within a 3D space than the previous lone accelerometer within a number of smartphones. Gyroscopes in consumer electronics are frequently combined with accelerometers for more robust direction- and motion-sensing. Examples of such applications include smartphones such as the Samsung Galaxy Note 4, HTC Titan, Nexus 5, iPhone 5s, Nokia 808 PureView and Sony Xperia, game console peripherals such as the PlayStation 3 controller and the Wii Remote, and virtual reality headsets such as the Oculus Rift. Some features of Android phones like PhotoSphere or 360 Camera and to use VR gadget do not work without a gyroscope sensor in the phone.
Nintendo has integrated a gyroscope into the Wii console's Wii Remote controller by an additional piece of hardware called "Wii MotionPlus". It is also included in the 3DS, Wii U GamePad, and Nintendo Switch Joy-Con and Pro controllers, which detect movement when turning and shaking.
Cruise ships use gyroscopes to level motion-sensitive devices such as self-leveling pool tables.
|
https://en.wikipedia.org/wiki/Gyroscope
|
passage: Many of the early commercial machines carried on from the one-off machines and were designed for rapid mathematical calculations needed for scientific, engineering and military purposes. But some were designed for data-processing workloads generated by the large, existing punch card ecosystem. IBM in particular divided its computers into scientific and commercial lines, which shared electronic technology and peripherals but had completely incompatible instruction set architectures and software. This practice continued into its second-generation (transistorized) machines, until reunification by the IBM System/360 project. See IBM 700/7000 series
Below is a list of these first generation commercial computers.
Computer Date Units Notes IBM 604 1948 5,600 First all-electronic calculator for use with unit record equipment. Could multiply and divide data from punched cards. Had 1,250 tubes. IBM CPC 1949700 Combined an IBM 604 with other unit record machines to carry out a sequence of calculations defined by instructions on a deck of punched cards. Ferranti Mark 1 1951 9 First commercially available stored program computer, based on Manchester Mark 1. UNIVAC I 1951 46 First mass-produced stored-program computer. Used delay-line memory. LEO I 19511First computer for commercial applications. Built and used by J. Lyons and Co., a restaurant and bakery chain. Based on EDSAC design. IBM 701 1952 19Built by IBM, also known as the Defense Calculator, based on the IAS computer, with Williams tube memory. The head of IBM famously expected to sell 5 units, but got orders for 18 on the first sales trip.
|
https://en.wikipedia.org/wiki/Vacuum-tube_computer
|
passage: Eliminating bond pads thus permits a more compact integrated circuit, on a smaller die; this increases the number of dies that may be fabricated on a wafer, and thus reduces the cost per die.
- Reducing the number of external pins also reduces assembly and packaging costs. A serial device may be packaged in a smaller and simpler package than a parallel device.
- Smaller and lower pin-count packages occupy less PCB area.
- Lower pin-count devices simplify PCB routing.
There are two major SPI flash types. The first type is characterized by small blocks and one internal SRAM block buffer allowing a complete block to be read to the buffer, partially modified, and then written back (for example, the Atmel AT45 DataFlash or the Micron Technology Page Erase NOR Flash). The second type has larger sectors where the smallest sectors typically found in this type of SPI flash are 4 KB, but they can be as large as 64 KB. Since this type of SPI flash lacks an internal SRAM buffer, the complete block must be read out and modified before being written back, making it slow to manage. However, the second type is cheaper than the first and is therefore a good choice when the application is code shadowing.
The two types are not easily exchangeable, since they do not have the same pinout, and the command sets are incompatible.
Most FPGAs are based on SRAM configuration cells and require an external configuration device, often a serial flash chip, to reload the configuration bitstream every power cycle.
|
https://en.wikipedia.org/wiki/Flash_memory
|
passage: For the sequel, use the sample mean:
$$
\overline{X} = (X_1+\cdots+X_n)/n
$$
and the (unbiased) sample variance:
$$
s^2 = {1 \over n-1}\sum_{i=1}^n (X_i-\overline{X})^2
$$
#### Unknown mean, known variance
Given a normal distribution with unknown mean μ but known variance
$$
\sigma^2
$$
, the sample mean
$$
\overline{X}
$$
of the observations
$$
X_1,\dots,X_n
$$
has distribution
$$
N(\mu,\sigma^2/n),
$$
while the future observation
$$
X_{n+1}
$$
has distribution
$$
N(\mu,\sigma^2).
$$
Taking the difference of these cancels the μ and yields a normal distribution of variance
$$
\sigma^2+(\sigma^2/n),
$$
thus
$$
\frac{X_{n+1}-\overline{X}}{\sqrt{\sigma^2+(\sigma^2/n)}} \sim N(0,1).
$$
Solving for
$$
X_{n+1}
$$
gives the prediction distribution
$$
N(\overline{X},\sigma^2+(\sigma^2/n)),
$$
from which one can compute intervals as before. This is a predictive confidence interval in the sense that if one uses a quantile range of 100p%, then on repeated applications of this computation, the future observation
$$
X_{n+1}
$$
will fall in the predicted interval 100p% of the time.
|
https://en.wikipedia.org/wiki/Prediction_interval
|
passage: Fritz is substituted for X in rule #3 to see if its consequent matches the goal, so rule #3 becomes:
If Fritz is a frog – Then Fritz is green
Since the consequent matches the goal ("Fritz is green"), the rules engine now needs to see if the antecedent ("Fritz is a frog") can be proven. The antecedent, therefore, becomes the new goal:
Fritz is a frog
2. Again substituting Fritz for X, rule #1 becomes:
If Fritz croaks and Fritz eats flies – Then Fritz is a frog
Since the consequent matches the current goal ("Fritz is a frog"), the inference engine now needs to see if the antecedent ("Fritz croaks and eats flies") can be proven. The antecedent, therefore, becomes the new goal:
Fritz croaks and Fritz eats flies
3. Since this goal is a conjunction of two statements, the inference engine breaks it into two sub-goals, both of which must be proven:
Fritz croaks
Fritz eats flies
4. To prove both of these sub-goals, the inference engine sees that both of these sub-goals were given as initial facts. Therefore, the conjunction is true:
Fritz croaks and Fritz eats flies
therefore the antecedent of rule #1 is true and the consequent must be true:
Fritz is a frog
therefore the antecedent of rule #3 is true and the consequent must be true:
Fritz is green
This derivation, therefore, allows the inference engine to prove that Fritz is green. Rules #2 and #4 were not used.
|
https://en.wikipedia.org/wiki/Backward_chaining
|
passage: So a sentence can be defined recursively (very roughly) as something with a structure that includes a noun phrase, a verb, and optionally another sentence. This is really just a special case of the mathematical definition of recursion.
This provides a way of understanding the creativity of language—the unbounded number of grammatical sentences—because it immediately predicts that sentences can be of arbitrary length: Dorothy thinks that Toto suspects that Tin Man said that.... There are many structures apart from sentences that can be defined recursively, and therefore many ways in which a sentence can embed instances of one category inside another. Over the years, languages in general have proved amenable to this kind of analysis.
The generally accepted idea that recursion is an essential property of human language has been challenged by Daniel Everett on the basis of his claims about the Pirahã language. Andrew Nevins, David Pesetsky and Cilene Rodrigues are among many who have argued against this. Literary self-reference can in any case be argued to be different in kind from mathematical or logical recursion.
Recursion plays a crucial role not only in syntax, but also in natural language semantics. The word and, for example, can be construed as a function that can apply to sentence meanings to create new sentences, and likewise for noun phrase meanings, verb phrase meanings, and others. It can also apply to intransitive verbs, transitive verbs, or ditransitive verbs.
|
https://en.wikipedia.org/wiki/Recursion
|
passage: It can occur as a result of a pre-existing infection or one acquired during pregnancy.
- Iatrogenic transmission, due to medical procedures such as injection or transplantation of infected material.
- Vector-borne transmission, transmitted by a vector, which is an organism that does not cause disease itself but that transmits infection by conveying pathogens from one host to another.
The relationship between virulence versus transmissibility is complex; with studies have shown that there were no clear relationship between the two. There is still a small number of evidence that partially suggests a link between virulence and transmissibility.
## Diagnosis
Diagnosis of infectious disease sometimes involves identifying an infectious agent either directly or indirectly. In practice most minor infectious diseases such as warts, cutaneous abscesses, respiratory system infections and diarrheal diseases are diagnosed by their clinical presentation and treated without knowledge of the specific causative agent. Conclusions about the cause of the disease are based upon the likelihood that a patient came in contact with a particular agent, the presence of a microbe in a community, and other epidemiological considerations. Given sufficient effort, all known infectious agents can be specifically identified.
Diagnosis of infectious disease is nearly always initiated by medical history and physical examination. More detailed identification techniques involve the culture of infectious agents isolated from a patient. Culture allows identification of infectious organisms by examining their microscopic features, by detecting the presence of substances produced by pathogens, and by directly identifying an organism by its genotype.
|
https://en.wikipedia.org/wiki/Infection
|
passage: De Dion suspensions are also in this category, as they rigidly connect the wheels together.
Independent suspension allows wheels to rise and fall on their own without affecting the opposite wheel. Suspensions with other devices, such as sway bars that link the wheels in some way, are still classed as independent.
Semi-dependent suspension is a third type. In this case, the motion of one wheel does affect the position of the other, but they are not rigidly attached to each other. Twist-beam rear suspension is such a system.
### Dependent suspensions
Dependent systems may be differentiated by the system of linkages used to locate them, both longitudinally and transversely. Often, both functions are combined in a set of linkages.
Examples of location linkages include:
- Satchell link
- Panhard rod
- Watt's linkage
- WOBLink
- Mumford linkage
- Leaf springs used for location (transverse or longitudinal)
- Fully elliptical springs usually need supplementary location links, and are no longer in common use
- Longitudinal semi-elliptical springs used to be common, and are still used in heavy-duty trucks and aircraft. They have the advantage, that the spring rate can easily be made progressive (non-linear).
- A single transverse leaf spring for both front wheels and/or both back wheels, supporting solid axles, was used by Ford Motor Company, before and soon after World War II, even on expensive models. It had the advantages of simplicity and low unsprung weight (compared to other solid-axle designs).
|
https://en.wikipedia.org/wiki/Car_suspension
|
passage: #### Post-Apple silicon transition
At WWDC 2022, Apple announced an updated MacBook Air based on a new M2 chip. It incorporates several changes from the 14-inch MacBook Pro, such as a flat, slab-shaped design, full-sized function keys, MagSafe charging, and a Liquid Retina display, with rounded corners and a display cutout incorporating a 1080p webcam.
The Mac Studio with M2 Max and M2 Ultra chips and the Mac Pro with M2 Ultra chip was unveiled at WWDC 2023, and the Intel-based Mac Pro was discontinued on the same day, completing the Mac transition to Apple silicon chips. The Mac Studio was received positively as a modest upgrade over the previous generation, albeit similarly priced PCs could be equipped with faster GPUs. However, the Apple silicon-based Mac Pro was criticized for several regressions, including memory capacity and a complete lack of CPU or GPU expansion options. A 15-inch MacBook Air was also introduced, and is the largest display included on a consumer-level Apple laptop.
The MacBook Pro was updated on October 30, 2023, with updated M3 Pro and M3 Max chips using a 3 nm process node, as well as the standard M3 chip in a refreshed iMac and a new base model MacBook Pro. Reviewers lamented the base memory configuration of 8 GB on the standard M3 MacBook Pro. In March 2024, the MacBook Air was also updated to include the M3 chip. In October 2024, several Macs were announced with the M4 series of chips, including the iMac, a redesigned Mac Mini, and the MacBook Pro; all of which included 16 GB of memory as standard.
|
https://en.wikipedia.org/wiki/Mac_%28computer%29
|
passage: If there are
$$
k
$$
ordered pairs, the degree of
$$
L(x)
$$
is at most
$$
k - 1
$$
. The polynomial
$$
L(x)
$$
can be computed explicitly in terms of the input data
$$
(x_j, y_j)
$$
.
#### Polynomial decomposition
A decomposition of a polynomial is a way of expressing it as a composition of other polynomials of degree larger than 1. A polynomial that cannot be decomposed is indecomposable. Ritt's polynomial decomposition theorem asserts that if
$$
f = g_1 \circ g_2 \circ \cdots \circ g_m = h_1 \circ h_2 \circ \cdots\circ h_n
$$
are two different decompositions of the polynomial
$$
f
$$
, then
$$
m = n
$$
and the degrees of the indecomposables in one decomposition are the same as the degrees of the indecomposables in the other decomposition (though not necessarily in the same order).
### Factorization
Except for factorization, all previous properties of are effective, since their proofs, as sketched above, are associated with algorithms for testing the property and computing the polynomials whose existence are asserted. Moreover these algorithms are efficient, as their computational complexity is a quadratic function of the input size.
The situation is completely different for factorization: the proof of the unique factorization does not give any hint for a method for factorizing.
|
https://en.wikipedia.org/wiki/Polynomial_ring
|
passage: The human eye responds to this wavelength combination as if it were a combination of blue and white light.
Some of the scattering can also be from sulfate particles. For years after large Plinian eruptions, the blue cast of the sky is notably brightened by the persistent sulfate load of the stratospheric gases. Some works of the artist J. M. W. Turner may owe their vivid red colours to the eruption of Mount Tambora in his lifetime.
In locations with little light pollution, the moonlit night sky is also blue, because moonlight is reflected sunlight, with a slightly lower color temperature due to the brownish color of the Moon. The moonlit sky is not perceived as blue, however, because at low light levels human vision comes mainly from rod cells that do not produce any color perception (Purkinje effect).
## Of sound in amorphous solids
Rayleigh scattering is also an important mechanism of wave scattering in amorphous solids such as glass, and is responsible for acoustic wave damping and phonon damping in glasses and granular matter at low or not too high temperatures. This is because in glasses at higher temperatures the Rayleigh-type scattering regime is obscured by the anharmonic damping (typically with a ~λ−2 dependence on wavelength), which becomes increasingly more important as the temperature rises.
## In amorphous solids – glasses – optical fibers
Rayleigh scattering is an important component of the scattering of optical signals in optical fibers.
|
https://en.wikipedia.org/wiki/Rayleigh_scattering
|
passage: Next, Bohr was told by his friend, Hans Hansen, that the Balmer series is calculated using the Balmer formula, an empirical equation discovered by Johann Balmer in 1885 that described wavelengths of some spectral lines of hydrogen. This was further generalized by Johannes Rydberg in 1888, resulting in what is now known as the Rydberg formula.
After this, Bohr declared, "everything became clear".
In 1913 Niels Bohr put forth three postulates to provide an electron model consistent with Rutherford's nuclear model:
1. The electron is able to revolve in certain stable orbits around the nucleus without radiating any energy, contrary to what classical electromagnetism suggests. These stable orbits are called stationary orbits and are attained at certain discrete distances from the nucleus. The electron cannot have any other orbit in between the discrete ones.
1. The stationary orbits are attained at distances for which the angular momentum of the revolving electron is an integer multiple of the reduced Planck constant:
$$
m_\mathrm{e} v r = n \hbar
$$
, where
$$
n= 1, 2, 3, ...
$$
is called the principal quantum number, and
$$
\hbar = h/2\pi
$$
. The lowest value of
$$
n
$$
is 1; this gives the smallest possible orbital radius, known as the Bohr radius, of 0.0529 nm for hydrogen. Once an electron is in this lowest orbit, it can get no closer to the nucleus.
|
https://en.wikipedia.org/wiki/Bohr_model
|
passage: As a concrete example, consider the factorial function , recursively defined by
.
In the lambda expression which is to represent this function, a parameter (typically the first one) will be assumed to receive the lambda expression itself as its value, so that calling it with itself as its first argument will amount to the recursive call. Thus to achieve recursion, the intended-as-self-referencing argument (called here, reminiscent of "self", or "self-applying") must always be passed to itself within the function body at a recursive call point:
with to hold, so and
and we have
Here becomes the same inside the result of the application , and using the same function for a call is the definition of what recursion is. The self-application achieves replication here, passing the function's lambda expression on to the next invocation as an argument value, making it available to be referenced there by the parameter name to be called via the self-application , again and again as needed, each time re-creating the lambda-term .
The application is an additional step just as the name lookup would be. It has the same delaying effect. Instead of having inside itself as a whole up-front, delaying its re-creation until the next call makes its existence possible by having two finite lambda-terms inside it re-create it on the fly later as needed.
This self-applicational approach solves it, but requires re-writing each recursive call as a self-application.
|
https://en.wikipedia.org/wiki/Lambda_calculus
|
passage:
$$
\mathcal{L}_X (dx^b) = d i_X (dx^b) = d X^b = \partial_a X^b dx^a
$$
.
Hence for a covector field, i.e., a differential form,
$$
A = A_a(x^b)dx^a
$$
we have:
$$
\mathcal{L}_X A = X (A_a) dx^a + A_b \mathcal{L}_X (dx^b) = (X^b \partial_b A_a + A_b\partial_a (X^b))dx^a
$$
The coefficient of the last expression is the local coordinate expression of the Lie derivative.
For a covariant rank 2 tensor field
$$
T = T_{ab}(x^c)dx^a \otimes dx^b
$$
we have:
$$
\begin{align}
(\mathcal {L}_X T) &= (\mathcal {L}_X T)_{ab} dx^a\otimes dx^b\\
BLOCK1\end{align}
$$
If
$$
T = g
$$
is the symmetric metric tensor, it is parallel with respect to the Levi-Civita connection (aka covariant derivative), and it becomes fruitful to use the connection.
|
https://en.wikipedia.org/wiki/Lie_derivative
|
passage: Deterministic method solution is based on a grid-based numerical method such as the spherical harmonics approach, whereas the Monte Carlo is the stochastic approach used to solve the BTE.
Monte Carlo method
The semiclassical Monte Carlo method is a statistical method used to yield exact solution to the Boltzmann transport equation which includes complex band structure and scattering processes. This approach is semiclassical for the reason that scattering mechanisms are treated quantum mechanically using the Fermi's Golden Rule, whereas the transport between scattering events is treated using the classical particle notion. The Monte Carlo model in essence tracks the particle trajectory at each free flight and chooses a corresponding scattering mechanism stochastically. Two of the great advantages of semiclassical Monte Carlo are its capability to provide accurate quantum mechanical treatment of various distinct scattering mechanisms within the scattering terms, and the absence of assumption about the form of carrier distribution in energy or k-space. The semiclassical equation describing the motion of an electron is
$$
\frac{dr}{dt} = \frac{1}{\hbar} \nabla_k E(k)
$$
$$
\frac{dk}{dt} = \frac{qF(r)}{\hbar}
$$
where F is the electric field, E(k) is the energy dispersion relation, and k is the momentum wave vector. To solve the above equation, one needs strong knowledge of the band structure (E(k)).
|
https://en.wikipedia.org/wiki/Monte_Carlo_methods_for_electron_transport
|
passage: A final 3 must be preceded by a 0 or 5; a final 8 must be preceded by a 2 or 7.
In base 10, the digital root of a nonzero triangular number is always 1, 3, 6, or 9. Hence, every triangular number is either divisible by three or has a remainder of 1 when divided by 9:
The digital root pattern for triangular numbers, repeating every nine terms, as shown above, is "1, 3, 6, 1, 6, 3, 1, 9, 9".
The converse of the statement above is, however, not always true. For example, the digital root of 12, which is not a triangular number, is 3 and divisible by three.
If is a triangular number, is an odd square, and , then is also a triangular number. Note that will always be a triangular number, because , which yields all the odd squares are revealed by multiplying a triangular number by 8 and adding 1, and the process for given is an odd square is the inverse of this operation.
The first several pairs of this form (not counting ) are: , , , , , , ... etc. Given is equal to , these formulas yield , , , , and so on.
|
https://en.wikipedia.org/wiki/Triangular_number
|
passage: (11111101111110 mod 100011011)
|-
| = || {3F7E mod 11B} = {01}
|-
| = || 1 (decimal)
|}
The latter can be demonstrated through long division (shown using binary notation, since it lends itself well to the task. Notice that exclusive OR is applied in the example and not arithmetic subtraction, as one might use in grade-school long division. ):
11111101111110 (mod) 100011011
^100011011
01110000011110
^100011011
0110110101110
^100011011
010101110110
^100011011
00100011010
^100011011
000000001
(The elements {53} and {CA} are multiplicative inverses of one another since their product is 1.)
Multiplication in this particular finite field can also be done using a modified version of the "peasant's algorithm". Each polynomial is represented using the same binary notation as above. Eight bits is sufficient because only degrees 0 to 7 are possible in the terms of each (reduced) polynomial.
This algorithm uses three variables (in the computer programming sense), each holding an eight-bit representation. a and b are initialized with the multiplicands; p accumulates the product and must be initialized to 0.
At the start and end of the algorithm, and the start and end of each iteration, this invariant is true: a b + p is the product. This is obviously true when the algorithm starts. When the algorithm terminates, a or b will be zero so p will contain the product.
|
https://en.wikipedia.org/wiki/Finite_field_arithmetic
|
passage: Again, there are many complexity classes between the two, such as
$$
\textsf{NL}
$$
and
$$
\textsf{NC}
$$
, and it is not known if they are distinct or equal classes.
It is suspected that
$$
\textsf{P}
$$
and
$$
\textsf{BPP}
$$
are equal. However, it is currently open if
$$
\textsf{BPP} = \textsf{NEXP}
$$
.
## Intractability
A problem that can theoretically be solved, but requires impractical and infinite resources (e.g., time) to do so, is known as an . Conversely, a problem that can be solved in practice is called a , literally "a problem that can be handled". The term infeasible (literally "cannot be done") is sometimes used interchangeably with intractable, though this risks confusion with a feasible solution in mathematical optimization.
Tractable problems are frequently identified with problems that have polynomial-time solutions (
$$
\textsf{P}
$$
,
$$
\textsf{PTIME}
$$
); this is known as the Cobham–Edmonds thesis. Problems that are known to be intractable in this sense include those that are EXPTIME-hard. If
$$
\textsf{NP}
$$
is not the same as
$$
\textsf{P}
$$
, then NP-hard problems are also intractable in this sense.
|
https://en.wikipedia.org/wiki/Computational_complexity_theory
|
passage: The fast fission factor describes the contribution of fast fissions to the effective neutron multiplication factor
- The bounds of this factor are 1 and infinity, with a value of 1 describing a system for which only thermal neutrons are causing fissions. A value of 2 would denote a system in which thermal and fast neutrons are causing equal amounts of fissions.
-
$$
p
$$
is the ratio of the number of neutrons that begin thermalization to the number of neutrons that reach thermal energies.
- Many isotopes have "resonances" in their capture cross-section curves that occur in energies between fast and thermal.
- If a neutron begins thermalization (i.e. begins to slow down), there is a possibility it will be absorbed by a non-multiplying material before it reaches thermal energy.
- The bounds of this factor are 0 and 1, with a value of 1 describing a system for which all fast neutrons that do not leak out and do not cause fast fissions eventually reach thermal energies.
-
$$
P_{\mathrm{TNL}}
$$
describes the probability that a thermal neutron will not escape the system without interacting.
- The bounds of this factor are 0 and 1, with a value of 1 describing a system for which thermal neutrons will never escape without interacting, i.e. an infinite system.
- Also written as
$$
L_{th}
$$
-
$$
f
$$
is the ratio of number of thermal neutrons absorbed in by fissile nuclei versus the number of neutrons absorbed in all materials in the system.
-
|
https://en.wikipedia.org/wiki/Nuclear_chain_reaction
|
passage: #### Important special cases
The Jacobi polynomials with
$$
\alpha=\beta
$$
are called the Gegenbauer polynomials (with parameter
$$
\gamma = \alpha+1/2
$$
)
For
$$
\alpha=\beta=0
$$
, these are called the Legendre polynomials (for which the interval of orthogonality is [−1, 1] and the weight function is simply 1):
$$
P_0(x) = 1,\, P_1(x) = x,\,P_2(x) = \frac{3x^2-1}{2},\,
P_3(x) = \frac{5x^3-3x}{2},\ldots
$$
For
$$
\alpha=\beta=\pm 1/2
$$
, one obtains the Chebyshev polynomials (of the second and first kind, respectively).
|
https://en.wikipedia.org/wiki/Classical_orthogonal_polynomials
|
passage: In 1887, Hermann Carl Vogel and Julius Scheiner discovered the "annual Doppler effect", the yearly change in the Doppler shift of stars located near the ecliptic, due to the orbital velocity of the Earth. In 1901, Aristarkh Belopolsky verified optical redshift in the laboratory using a system of rotating mirrors.
Beginning with observations in 1912, Vesto Slipher discovered that the Andromeda Galaxy had a blue shift, indicating that it was moving towards the Earth. Slipher first reported his measurement in the inaugural volume of the Lowell Observatory Bulletin. Three years later, he wrote a review in the journal Popular Astronomy. In it he stated that "the early discovery that the great Andromeda spiral had the quite exceptional velocity of –300 km[/s] showed the means then available, capable of investigating not only the spectra of the spirals but their velocities as well." Slipher reported the velocities for 15 spiral nebulae spread across the entire celestial sphere, all but three having observable "positive" (that is recessional) velocities.
Until 1923 the nature of the nebulae was unclear. By that year Edwin Hubble had established that these were galaxies and worked out a procedure to measure distance based on the period-luminosity relation of variable Cepheids stars. This made it possible to test a prediction by Willem de Sitter in 1917 that redshift would be correlated with distance.
|
https://en.wikipedia.org/wiki/Redshift
|
passage: Compared to the direct matrix inverse, inverse solutions using QR decomposition are more numerically stable as evidenced by their reduced condition numbers.
To solve the underdetermined linear problem
$$
A \mathbf x = \mathbf b
$$
where the matrix
$$
A
$$
has dimensions
$$
m \times n
$$
and rank first find the QR factorization of the transpose of where Q is an orthogonal matrix (i.e. and R has a special form:
$$
R = \left[\begin{smallmatrix} R_1 \\ 0 \end{smallmatrix}\right]
$$
. Here
$$
R_1
$$
is a square
$$
m \times m
$$
right triangular matrix, and the zero matrix has dimension After some algebra, it can be shown that a solution to the inverse problem can be expressed as:
$$
\mathbf x = Q \left[\begin{smallmatrix}
\left(R_1^\textsf{T}\right)^{-1} \mathbf b \\
BLOCK0\end{smallmatrix}\right]
$$
where one may either find
$$
R_1^{-1}
$$
by Gaussian elimination or compute
$$
\left(R_1^\textsf{T}\right)^{-1} \mathbf b
$$
directly by forward substitution. The latter technique enjoys greater numerical accuracy and lower computations.
|
https://en.wikipedia.org/wiki/QR_decomposition
|
passage: This implies that both of these are differential algebras.
By definition,
$$
\delta:\mathfrak{g}\to\mathfrak{g}
$$
is a derivation on
$$
\mathfrak{g}
$$
if it obeys Leibniz's law:
$$
\delta([v,w])=[\delta(v),w]+[v,\delta(w)]
$$
(When
$$
\mathfrak{g}
$$
is the space of left invariant vector fields on a group
$$
G
$$
, the Lie bracket is that of vector fields.) The lifting is performed by defining
$$
\begin{align}\delta(v\otimes w \otimes \cdots \otimes u)
=& \, \delta(v) \otimes w \otimes \cdots \otimes u \\
&+ v\otimes \delta(w) \otimes \cdots\otimes u \\
&+ \cdots + v\otimes w \otimes \cdots \otimes \delta(u).
\end{align}
$$
Since
$$
\mbox{ad}_x
$$
is a derivation for any
$$
x\in\mathfrak{g},
$$
the above defines
$$
\mbox{ad}_x
$$
acting on
$$
T(\mathfrak{g})
$$
and
$$
U(\mathfrak{g}).
$$
From the PBW theorem, it is clear that all central elements are linear combinations of symmetric homogeneous polynomials in the basis elements
$$
e_a
$$
of the Lie algebra.
|
https://en.wikipedia.org/wiki/Universal_enveloping_algebra
|
passage: bounded in
$$
(X, \tau(X, Y, b)),
$$
bounded in
$$
(X, \beta(X, Y, b))
$$
).
### Topologies compatible with a pair
If
$$
(X, Y, b)
$$
is a pairing over
$$
\mathbb{K}
$$
and
$$
\mathcal{T}
$$
is a vector topology on
$$
X
$$
then
$$
\mathcal{T}
$$
is a topology of the pairing and that it is compatible (or consistent) with the pairing
$$
(X, Y, b)
$$
if it is locally convex and if the continuous dual space of
$$
\left(X, \mathcal{T}\right) = b(\,\cdot\,, Y).
$$
If
$$
X
$$
distinguishes points of
$$
Y
$$
then by identifying
$$
Y
$$
as a vector subspace of
$$
X
$$
's algebraic dual, the defining condition becomes:
$$
\left(X, \mathcal{T}\right)^{\prime} = Y.
$$
Some authors (e.g. [Trèves 2006] and [Schaefer 1999]) require that a topology of a pair also be Hausdorff, which it would have to be if
$$
Y
$$
distinguishes the points of
$$
X
$$
(which these authors assume).
The weak topology
$$
\sigma(X, Y, b)
$$
is compatible with the pairing
$$
(X, Y, b)
$$
|
https://en.wikipedia.org/wiki/Dual_system
|
passage: b = c; fb = fc;
if (side == -1)
fa /= 2;
side = -1;
} else if (fa * fc > 0) {
/* fc and fa have same sign, copy c to a */
a = c; fa = fc;
if (side == +1)
fb /= 2;
side = +1;
} else {
/* fc * f_ very small (looks like zero) */
break;
}
}
return c;
}
int main(void) {
printf("%0.15f\n", FalsiMethod(&f, 0, 1, 5E-15, 100));
return 0;
}
|
https://en.wikipedia.org/wiki/Regula_falsi
|
passage: Parsing the formats do prove a bit harder, since one more layer must be followed even for some classes plists were supposed to support. Like the binary format which also has an object table, it is possible to create circular references in . Since there is not a UID data type in XML, the integers are stored in a dictionary under the key "CF$UID".
Apple publishes an open-source in Swift Corelibs Foundation; like the closed-source Apple Foundation, it restricts output formats to binary and XML only. It also has some test cases showing the results of serialization. GNUstep also has a compatible implementation, which does not limit output formats.
## Path language
There is not a single, standardized path language for property lists like XPath does for XML, but informal conventions used by various programs exist.
- A dot syntax version is found in the keypath argument of Apple's . It appears to derive from .
- A different format is used by , with a colon syntax for indexing.
Neither format is able to express a key with the separator character in it.
## Other platforms
### Windows
Although best known on Apple or Darwin systems, including iOS and macOS, plist files are also present on Windows computers when Apple software, such as iTunes or Safari are installed. On Windows, the files are typically binary files, although some applications may generate PLIST files in the other formats.
On Windows the Apple plist files are stored in the user's home directory under . These plist files on Windows typically store preferences and other information, rather than using the Windows registry.
|
https://en.wikipedia.org/wiki/Property_list
|
passage: The volume of each infinitesimal disc is therefore . The limit of the Riemann sum of the volumes of the discs between and becomes integral (1).
Assuming the applicability of Fubini's theorem and the multivariate change of variables formula, the disk method may be derived in a straightforward manner by (denoting the solid as D):
$$
V = \iiint_D dV = \int_a^b \int_{g(z)}^{f(z)} \int_0^{2\pi} r\,d\theta\,dr\,dz = 2\pi \int_a^b\int_{g(z)}^{f(z)} r\,dr\,dz = 2\pi \int_a^b \frac{1}{2}r^2\Vert^{f(z)}_{g(z)} \,dz = \pi \int_a^b (f(z)^2 - g(z)^2)\,dz
$$
### Shell Method of Integration
The shell method (sometimes referred to as the "cylinder method") is used when the slice that was drawn is parallel to the axis of revolution; i.e. when integrating perpendicular to the axis of revolution.
|
https://en.wikipedia.org/wiki/Solid_of_revolution
|
passage: Combining the last two equations one gets the seismic wave equation in homogeneous media
$$
\rho \partial_t^2 u_i = \lambda\partial_i \sum_k \partial_k u_k + \mu\sum_j \bigl(\partial_i\partial_j u_j + \partial_j\partial_j u_i\bigr)
$$
Using the nabla operator notation of vector calculus,
$$
\nabla = (\partial_1, \partial_2, \partial_3)
$$
, with some approximations, this equation can be written as
$$
\rho \partial_t^2 \boldsymbol{u} = \left(\lambda + 2\mu \right) \nabla\left(\nabla \cdot \boldsymbol{u}\right) - \mu\nabla \times \left(\nabla \times \boldsymbol{u}\right)
$$
Taking the curl of this equation and applying vector identities, one gets
$$
\partial_t^2(\nabla\times\boldsymbol{u}) = \frac{\mu}{\rho}\nabla^2 \left(\nabla\times\boldsymbol{u}\right)
$$
This formula is the wave equation applied to the vector quantity
$$
\nabla\times \boldsymbol{u}
$$
, which is the material's shear strain.
|
https://en.wikipedia.org/wiki/S_wave
|
passage: In other primates, the thumb is short and unable to touch the little finger. The ulnar opposition facilitates the precision grip and power grip of the human hand, underlying all the skilled manipulations.
### Other changes
A number of other changes have also characterized the evolution of humans, among them an increased reliance on vision rather than smell (highly reduced olfactory bulb); a longer juvenile developmental period and higher infant dependency; a smaller gut and small, misaligned teeth; faster basal metabolism; loss of body hair; an increase in
eccrine sweat gland density that is ten times higher than any other catarrhinian primates, yet humans use 30% to 50% less water per day compared to chimps and gorillas; more REM sleep but less sleep in total; a change in the shape of the dental arcade from u-shaped to parabolic; development of a chin (found in Homo sapiens alone); styloid processes; and a descended larynx. As the human hand and arms adapted to the making of tools and were used less for climbing, the shoulder blades changed too. As a side effect, it allowed human ancestors to throw objects with greater force, speed and accuracy.
## Use of tools
The use of tools has been interpreted as a sign of intelligence, and it has been theorized that tool use may have stimulated certain aspects of human evolution, especially the continued expansion of the human brain. Paleontology has yet to explain the expansion of this organ over millions of years despite being extremely demanding in terms of energy consumption.
|
https://en.wikipedia.org/wiki/Human_evolution
|
passage: Then the reaction proceeds as an avalanche until two radicals meet and recombine.
X. + R-H -> X-H + R.
R. + X2 -> R-X + X.
Reactions during the chain reaction of radical substitution
### Addition and elimination
The addition and its counterpart, the elimination, are reactions that change the number of substituents on the carbon atom, and form or cleave multiple bonds. Double and triple bonds can be produced by eliminating a suitable leaving group. Similar to the nucleophilic substitution, there are several possible reaction mechanisms that are named after the respective reaction order. In the E1 mechanism, the leaving group is ejected first, forming a carbocation. The next step, the formation of the double bond, takes place with the elimination of a proton (deprotonation). The leaving order is reversed in the E1cb mechanism, that is the proton is split off first. This mechanism requires the participation of a base. Because of the similar conditions, both reactions in the E1 or E1cb elimination always compete with the SN1 substitution.
The E2 mechanism also requires a base, but there the attack of the base and the elimination of the leaving group proceed simultaneously and produce no ionic intermediate. In contrast to the E1 eliminations, different stereochemical configurations are possible for the reaction product in the E2 mechanism, because the attack of the base preferentially occurs in the anti-position with respect to the leaving group. Because of the similar conditions and reagents, the E2 elimination is always in competition with the SN2-substitution.
|
https://en.wikipedia.org/wiki/Chemical_reaction
|
passage: $$
and the six possible equations are (with the relevant set shown at right):
$$
\begin{alignat}{5}
\text{(CT1)}&& \qquad \cos b\,\cos C &= \cot a\,\sin b - \cot A \,\sin C \qquad&&(aCbA)\\[0ex]
\text{(CT2)}&& \cos b\,\cos A &= \cot c\,\sin b - \cot C \,\sin A &&(CbAc)\\[0ex]
\text{(CT3)}&& \cos c\,\cos A &= \cot b\,\sin c - \cot B \,\sin A &&(bAcB)\\[0ex]
\text{(CT4)}&& \cos c\,\cos B &= \cot a\,\sin c - \cot A \,\sin B &&(AcBa)\\[0ex]
\text{(CT5)}&& \cos a\,\cos B &= \cot c\,\sin a - \cot C \,\sin B &&(cBaC)\\[0ex]
\text{(CT6)}&& \cos a\,\cos C &= \cot b\,\sin a - \cot B \,\sin C &&(BaCb)
\end{alignat}
$$
To prove the first formula start from the first cosine rule and on the right-hand side substitute for from the third cosine rule:
$$
\begin{align}
|
https://en.wikipedia.org/wiki/Spherical_trigonometry
|
passage: The simplicity of this definition, which is matched in many other formulas involving the natural logarithm, leads to the term "natural". The definition of the natural logarithm can then be extended to give logarithm values for negative numbers and for all non-zero complex numbers, although this leads to a multi-valued function: see complex logarithm for more.
The natural logarithm function, if considered as a real-valued function of a positive real variable, is the inverse function of the exponential function, leading to the identities:
$$
\begin{align}
e^{\ln x} &= x \qquad \text{ if } x \in \R_{+}\\
\ln e^x &= x \qquad \text{ if } x \in \R
\end{align}
$$
Like all logarithms, the natural logarithm maps multiplication of positive numbers into addition:
$$
\ln( x \cdot y ) = \ln x + \ln y~.
$$
Logarithms can be defined for any positive base other than 1, not only . However, logarithms in other bases differ only by a constant multiplier from the natural logarithm, and can be defined in terms of the latter,
$$
\log_b x = \ln x / \ln b = \ln x \cdot \log_b e
$$
.
Logarithms are useful for solving equations in which the unknown appears as the exponent of some other quantity.
|
https://en.wikipedia.org/wiki/Natural_logarithm
|
passage: The localization sequence can be extended to the left using a generalization of Chow groups, (Borel–Moore) motivic homology groups, also known as higher Chow groups.
For any morphism
$$
f: X\to Y
$$
of smooth schemes over
$$
k
$$
, there is a pullback homomorphism
$$
f^*: CH^i(Y)\to CH^i(X)
$$
, which is in fact a ring homomorphism
$$
CH^*(Y)\to CH^*(X)
$$
.
### Examples of flat pullbacks
Note that non-examples can be constructed using blowups; for example, if we take the blowup of the origin in
$$
\mathbb{A}^2
$$
then the fiber over the origin is isomorphic to
$$
\mathbb{P}^1
$$
.
#### Branched coverings of curves
Consider the branched covering of curves
$$
f: \operatorname{Spec}\left( \frac{\mathbb{C}[x,y]}{(f(x) - g(x,y))} \right) \to \mathbb{A}^1_x
$$
Since the morphism ramifies whenever
$$
f(\alpha) = 0
$$
we get a factorization
$$
g(\alpha,y) = (y - a_1)^{e_1}\cdots(y-a_k)^{e_k}
$$
where one of the
$$
e_i>1
$$
.
|
https://en.wikipedia.org/wiki/Chow_group
|
passage: A few smartphones designed around specific purposes are equipped with uncommon hardware such as a projector (Samsung Beam i8520 and Samsung Galaxy Beam i8530), optical zoom lenses (Samsung Galaxy S4 Zoom and Samsung Galaxy K Zoom), thermal camera, and even PMR446 (walkie-talkie radio) transceiver.
### Central processing unit
Smartphones have central processing units (CPUs), similar to those in computers, but optimised to operate in low power environments. In smartphones, the CPU is typically integrated in a CMOS (complementary metal–oxide–semiconductor) system-on-a-chip (SoC) application processor.
The performance of mobile CPU depends not only on the clock rate (generally given in multiples of hertz) but also on the memory hierarchy. Because of these challenges, the performance of mobile phone CPUs is often more appropriately given by scores derived from various standardized tests to measure the real effective performance in commonly used applications.
### Buttons
Smartphones are typically equipped with a power button and volume buttons. Some pairs of volume buttons are unified. Some are equipped with a dedicated camera shutter button. Units for outdoor use may be equipped with an "SOS" emergency call and "PTT" (push-to-talk button). The presence of physical front-side buttons such as the home and navigation buttons has decreased throughout the 2010s, increasingly becoming replaced by capacitive touch sensors and simulated (on-screen) buttons.
|
https://en.wikipedia.org/wiki/Smartphone
|
passage: The result is a more gradual curve rising to the asymptotes than shown in the present graphs, because the fR is not at a fixed level from the outset.
### Contributions from ancestral genepools
In the section on "Pedigree analysis",
$$
\left( \tfrac {1}{2} \right)^n
$$
was used to represent probabilities of autozygous allele descent over n generations down branches of the pedigree. This formula arose because of the rules imposed by sexual reproduction: (i) two parents contributing virtually equal shares of autosomal genes, and (ii) successive dilution for each generation between the zygote and the "focus" level of parentage. These same rules apply also to any other viewpoint of descent in a two-sex reproductive system. One such is the proportion of any ancestral gene-pool (also known as 'germplasm') which is contained within any zygote's genotype.
Therefore, the proportion of an ancestral genepool in a genotype is:
$$
\gamma_n = \left( \frac{1}{2}\right) ^n
$$
where n = number of sexual generations between the zygote and the focus ancestor.
For example, each parent defines a genepool contributing
$$
\left( \tfrac{1}{2} \right)^1
$$
to its offspring; while each great-grandparent contributes
$$
\left( \tfrac{ 1}{2} \right)^3
$$
to its great-grand-offspring.
|
https://en.wikipedia.org/wiki/Quantitative_genetics
|
passage: Generally, the three keys are generated by taking 24 bytes from a strong random generator, and only keying option 1 should be used (option 2 needs only 16 random bytes, but strong random generators are hard to assert and it is considered best practice to use only option 1).
## Encryption of more than one block
As with all block ciphers, encryption and decryption of multiple blocks of data may be performed using a variety of modes of operation, which can generally be defined independently of the block cipher algorithm. However, ANS X9.52 specifies directly, and NIST SP 800-67 specifies via SP 800-38A, that some modes shall only be used with certain constraints on them that do not necessarily apply to general specifications of those modes. For example, ANS X9.52 specifies that for cipher block chaining, the initialization vector shall be different each time, whereas ISO/IEC 10116 does not. FIPS PUB 46-3 and ISO/IEC 18033-3 define only the single-block algorithm, and do not place any restrictions on the modes of operation for multiple blocks.
## Security
In general, Triple DES with three independent keys (keying option 1) has a key length of 168 bits (three 56-bit DES keys), but due to the meet-in-the-middle attack, the effective security it provides is only 112 bits. Keying option 2 reduces the effective key size to 112 bits (because the third key is the same as the first).
|
https://en.wikipedia.org/wiki/Triple_DES
|
passage: The difference between planetary bodies located inside and outside the frost line can be stark. Earth's mass is 0.000023% water, while Tethys, a moon of Saturn, is almost entirely made of water.
## Reactions
### Acid–base reactions
Water is amphoteric: it has the ability to act as either an acid or a base in chemical reactions. According to the Brønsted-Lowry definition, an acid is a proton () donor and a base is a proton acceptor. When reacting with a stronger acid, water acts as a base; when reacting with a stronger base, it acts as an acid. For instance, water receives an ion from HCl when hydrochloric acid is formed:
+ +
In the reaction with ammonia, , water donates a ion, and is thus acting as an acid:
+ +
Because the oxygen atom in water has two lone pairs, water often acts as a Lewis base, or electron-pair donor, in reactions with Lewis acids, although it can also react with Lewis bases, forming hydrogen bonds between the electron pair donors and the hydrogen atoms of water. HSAB theory describes water as both a weak hard acid and a weak hard base, meaning that it reacts preferentially with other hard species:
+ →
+ →
+ →
When a salt of a weak acid or of a weak base is dissolved in water, water can partially hydrolyze the salt, producing the corresponding base or acid, which gives aqueous solutions of soap and baking soda their basic pH:
+ NaOH +
|
https://en.wikipedia.org/wiki/Properties_of_water
|
passage: ## Distinguishing features
No single feature distinguishes annelids from other invertebrate phyla, but they have a distinctive combination of features. Their bodies are long, with segments that are divided externally by shallow ring-like constrictions called annuli and internally by septa ("partitions") at the same points, although in some species the septa are incomplete and in a few cases missing. Most of the segments contain the same sets of organs, although sharing a common gut, circulatory system and nervous system makes them inter-dependent. Their bodies are covered by a cuticle (outer covering) that does not contain cells but is secreted by cells in the skin underneath, is made of tough but flexible collagen and does not molt – on the other hand arthropods' cuticles are made of the more rigid α-chitin, and molt until the arthropods reach their full size. Most annelids have closed circulatory systems, where the blood makes its entire circuit via blood vessels.
+ Summary of distinguishing features Annelida Recently merged into Annelida Closely related Similar-looking phyla Echiura Sipuncula Nemertea Arthropoda Onychophora External segmentation Yes No Only in a few species Yes, except in mites No Repetition of internal organs Yes No Yes
|
https://en.wikipedia.org/wiki/Annelid
|
passage: | -0.1
|-
|
|(1.6)
|(0.1)
|-
|0.53333
|(1.6)
|
|}
Incrementing x0 by the difference of the two projections gives the first iteration of the difference map, D(x0) = x1 :
{| class="wikitable" style="background-color:white;text-align: center;"
|-
|width="80pt"|-0.96666
|width="80pt"|-1.4
|width="80pt"|
|-
|(-1.93333)
|
| 0.3
|-
|
|(0.9)
|(0.3)
|-
|0.03333
|(0.7)
|
|}
Here is the second iteration, D(x1) = x2 :
{| class="wikitable" style="background-color:white;text-align: center;"
|-
|width="80pt"|-0.3
|width="80pt"|-1.4
|width="80pt"|
|-
|(-2.6)
|
| -0.7
|-
|
|(0.9)
|(-0.7)
|-
|0.7
|(0.7)
|
|}
This is a fixed point: D(x2) = x2. The iterate is unchanged because the two projections agree.
|
https://en.wikipedia.org/wiki/Difference-map_algorithm
|
passage: Let
$$
C
$$
be the compact set of the union of all squares with distance
$$
\leq\delta/4
$$
from
$$
A
$$
. Then
$$
C\cap B=\varnothing
$$
and
$$
\partial C
$$
does not meet
$$
A
$$
or
$$
B
$$
: it consists of finitely many horizontal and vertical segments in
$$
G
$$
forming a finite number of closed rectangular paths
$$
\gamma_j\in G
$$
. Taking
$$
C_i
$$
to be all the squares covering
$$
A
$$
, then
$$
\frac{1}{2\pi}\int_{\partial C}\mathrm{d}\mathrm{arg}(z-a)
$$
equals the sum of the winding numbers of
$$
C_i
$$
over
$$
a
$$
, thus giving
$$
1
$$
. On the other hand the sum of the winding numbers of
$$
\gamma_j
$$
about
$$
a
$$
equals
$$
1
$$
. Hence the winding number of at least one of the
$$
\gamma_j
$$
about
$$
a
$$
is non-zero.
(7) ⇒ (1) This is a purely topological argument. Let
$$
\gamma
$$
be a piecewise smooth closed curve based at
$$
z_0\in G
$$
.
|
https://en.wikipedia.org/wiki/Riemann_mapping_theorem
|
passage: This is one of the reasons for adopting two's complement representation for representing signed integers in computers.
Another example of a non-dictionary use of lexicographical ordering appears in the ISO 8601 standard for dates, which expresses a date as YYYY-MM-DD. This formatting scheme has the advantage that the lexicographical order on sequences of characters that represent dates coincides with the chronological order: an earlier CE date is smaller in the lexicographical order than a later date up to year 9999. This date ordering makes computerized sorting of dates easier by avoiding the need for a separate sorting algorithm.
## Monoid of words
The over an alphabet is the free monoid over . That is, the elements of the monoid are the finite sequences (words) of elements of (including the empty sequence, of length 0), and the operation (multiplication) is the concatenation of words. A word is a prefix (or 'truncation') of another word if there exists a word such that . By this definition, the empty word (
$$
\varepsilon
$$
) is a prefix of every word, and every word is a prefix of itself (with
$$
= \varepsilon
$$
); care must be taken if these cases are to be excluded.
|
https://en.wikipedia.org/wiki/Lexicographic_order
|
passage: Water vapor is the "working medium" of the atmospheric thermodynamic engine which transforms heat energy from sun irradiation into mechanical energy in the form of winds. Transforming thermal energy into mechanical energy requires an upper and a lower temperature level, as well as a working medium which shuttles forth and back between both. The upper temperature level is given by the soil or water surface of the Earth, which absorbs the incoming sun radiation and warms up, evaporating water. The moist and warm air at the ground is lighter than its surroundings and rises up to the upper limit of the troposphere. There the water molecules radiate their thermal energy into outer space, cooling down the surrounding air. The upper atmosphere constitutes the lower temperature level of the atmospheric thermodynamic engine. The water vapor in the now cold air condenses out and falls down to the ground in the form of rain or snow. The now heavier cold and dry air sinks down to ground as well; the atmospheric thermodynamic engine thus establishes a vertical convection, which transports heat from the ground into the upper atmosphere, where the water molecules can radiate it to outer space. Due to the Earth's rotation and the resulting Coriolis forces, this vertical atmospheric convection is also converted into a horizontal convection, in the form of cyclones and anticyclones, which transport the water evaporated over the oceans into the interior of the continents, enabling vegetation to grow.
|
https://en.wikipedia.org/wiki/Water_vapor
|
passage: This expansion follows from the identity for the sums over Dirichlet convolutions given on the divisor sum identities page (a standard trick for these sums).
Dirichlet inverse
### Examples
Given an arithmetic function
$$
f
$$
its Dirichlet inverse
$$
g = f^{-1}
$$
may be calculated recursively: the value of
$$
g(n)
$$
is in terms of
$$
g(m)
$$
for
$$
m<n
$$
.
For
$$
n=1
$$
:
$$
(f * g) (1) = f(1) g(1) = \varepsilon(1) = 1
$$
, so
$$
g(1) = 1/f(1)
$$
. This implies that
$$
f
$$
does not have a Dirichlet inverse if
$$
f(1) = 0
$$
.
|
https://en.wikipedia.org/wiki/Dirichlet_convolution
|
passage: The spermatogenesis is less efficient at lower and higher temperatures than 33 °C. Because the testes are located outside the body, the smooth tissue of the scrotum can move them closer or further away from the body. The temperature of the testes is maintained at 34.4 °C, a little below body temperature, as temperatures above 36.7 °C impede spermatogenesis. There are a number of mechanisms to maintain the testes at the optimum temperature.
The cremasteric muscle covers the testicles and the spermatic cord. When this muscle contracts, the cord shortens and the testicles move closer up toward the body, which provides slightly more warmth to maintain optimal testicular temperature. When cooling is required, the cremasteric muscle relaxes and the testicles lower away from the warm body and are able to cool. Contraction also occurs in response to physical stress, such as blunt trauma; the testicles withdraw and the scrotum shrinks very close to the body in an effort to protect them.
The cremasteric reflex will reflexively raise the testicles. The testicles can also be lifted voluntarily using the pubococcygeus muscle, which partially activates related muscles.
### Gene and protein expression
The human genome includes approximately 20,000 protein coding genes: 80% of these genes are expressed in adult testes. The testes have the highest fraction of tissue type-specific genes compared to other organs and tissues. About 1000 of them are highly specific for the testes, and about 2,200 show an elevated pattern of expression.
|
https://en.wikipedia.org/wiki/Testicle
|
passage: In algebra, a split-complex number (or hyperbolic number, also perplex number, double number) is based on a hyperbolic unit satisfying
$$
j^2=1
$$
, where
$$
j \neq \pm 1
$$
. A split-complex number has two real number components and , and is written
$$
z=x+yj .
$$
The conjugate of is
$$
z^*=x-yj.
$$
Since
$$
j^2=1,
$$
the product of a number with its conjugate is
$$
N(z) := zz^* = x^2 - y^2,
$$
an isotropic quadratic form.
The collection of all split-complex numbers
$$
z=x+yj
$$
for forms an algebra over the field of real numbers. Two split-complex numbers and have a product that satisfies
$$
N(wz)=N(w)N(z).
$$
This composition of over the algebra product makes a composition algebra.
A similar algebra based on and component-wise operations of addition and multiplication, where is the quadratic form on also forms a quadratic space. The ring isomorphism
$$
\begin{align}
BLOCK0 x + yj &\mapsto (x - y, x + y)
\end{align}
$$
is an isometry of quadratic spaces.
Split-complex numbers have many other names; see below. See the article Motor variable for functions of a split-complex number.
|
https://en.wikipedia.org/wiki/Split-complex_number
|
passage: This is the Padovan cuboid spiral. Successive sides of this spiral have lengths that are
the Padovan numbers multiplied by the square root of 2.
## Pascal's triangle
Erv Wilson in his paper The Scales of Mt. Meru observed certain diagonals in Pascal's triangle (see diagram) and drew them on paper in 1993. The Padovan numbers were discovered in 1994. Paul Barry (2004) observed that these diagonals generate the Padovan sequence by summing the diagonal numbers.
## References
- Ian Stewart, A Guide to Computer Dating (Feedback), Scientific American, Vol. 275, No. 5, November 1996, Pg. 118.
## External links
-
- A Padovan sequence calculator
Category:Integer sequences
Category: Recurrence relations
|
https://en.wikipedia.org/wiki/Padovan_sequence
|
passage: ### Ruby
In Ruby, constructors are created by defining a method called `initialize`. This method is executed to initialize each new instance.
```rbcon
irb(main):001:0> class ExampleClass
irb(main):002:1> def initialize
irb(main):003:2> puts "Hello there"
irb(main):004:2> end
irb(main):005:1> end
=> nil
irb(main):006:0> ExampleClass.new
Hello there
=> #<ExampleClass:0x007fb3f4299118>
```
Visual Basic .NET
In Visual Basic .NET, constructors use a method declaration with the name "`New`".
```vbnet
Class Foobar
Private strData As String
' Constructor
Public Sub New(ByVal someParam As String)
strData = someParam
End Sub
End Class
```
```vbnet
' code somewhere else
' instantiating an object with the above constructor
Dim foo As New Foobar(".NET")
```
|
https://en.wikipedia.org/wiki/Constructor_%28object-oriented_programming%29
|
passage: A family of examples: Suppose that
$$
X
$$
is equal to
$$
\Complex
$$
(if considered as a complex vector space) or equal to
$$
\R^2
$$
(if considered as a real vector space). Regardless of whether
$$
X
$$
is a real or complex vector space, every barrel in
$$
X
$$
is necessarily a neighborhood of the origin (so
$$
X
$$
is an example of a barrelled space). Let
$$
R : [0, 2\pi) \to (0, \infty]
$$
be any function and for every angle
$$
\theta \in [0, 2 \pi),
$$
let
$$
S_{\theta}
$$
denote the closed line segment from the origin to the point
$$
R(\theta) e^{i \theta} \in \Complex.
$$
Let
$$
S := \bigcup_{\theta \in [0, 2 \pi)} S_{\theta}.
$$
Then
$$
S
$$
is always an absorbing subset of
$$
\R^2
$$
(a real vector space) but it is an absorbing subset of
$$
\Complex
$$
(a complex vector space) if and only if it is a neighborhood of the origin. Moreover,
$$
S
$$
is a balanced subset of
$$
\R^2
$$
|
https://en.wikipedia.org/wiki/Barrelled_space
|
passage: $$
The product with
$$
V(f)
$$
is thereby reduced to a discrete-frequency function:
$$
\begin{align}
\mathcal{F}\{u_{_N} * v\}(f) &= G_{_N}(f) V(f) \\
&= \frac{1}{N} \sum_{k=-\infty}^{\infty} \left(\scriptstyle{\rm DFT}\displaystyle\{u_{_N}\}[k]\right)\cdot V(f)\cdot \delta\left(f-k/N\right)\\
&= \frac{1}{N} \sum_{k=-\infty}^{\infty} \left(\scriptstyle{\rm DFT}\displaystyle\{u_{_N}\}[k]\right)\cdot V(k/N)\cdot \delta\left(f-k/N\right)\\
&= \frac{1}{N} \sum_{k=-\infty}^{\infty} \left(\scriptstyle{\rm DFT}\displaystyle\{u_{_N}\}[k]\right)\cdot \left(\scriptstyle{\rm DFT}\displaystyle\{v_{_N}\}[k]\right) \cdot \delta\left(f-k/N\right), \quad \scriptstyle \mathsf{(Eq.5b)}
|
https://en.wikipedia.org/wiki/Convolution_theorem
|
passage: This eventually evolved into the American-style roulette game. The American game was developed in the gambling dens across the new territories where makeshift games had been set up, whereas the French game evolved with style and leisure in Monte Carlo.
During the first part of the 20th century, the only casino towns of note were Monte Carlo with the traditional single zero French wheel, and Las Vegas with the American double zero wheel. In the 1970s, casinos began to flourish around the world. In 1996 the first online casino, generally believed to be InterCasino, made it possible to play roulette online. By 2008, there were several hundred casinos worldwide offering roulette games. The double zero wheel is found in the United States, Canada, South America, and the Caribbean, while the single zero wheel is predominant elsewhere.
## Rules of play against a casino
Roulette players have a variety of betting options. "Inside" bets involve selecting either the exact number on which the ball will land, or a small group of numbers adjacent to each other on the layout. "Outside" bets, by contrast, allow players to select a larger group of numbers based on properties such as their color or parity (odd/even). The payout odds for each type of bet are based on its probability.
The roulette table usually imposes minimum and maximum bets, and these rules usually apply separately for all of a player's inside and outside bets for each spin. For inside bets at roulette tables, some casinos may use separate roulette table chips of various colors to distinguish players at the table.
|
https://en.wikipedia.org/wiki/Roulette
|
passage: They start like induction motors, but when slip rate decreases sufficiently, the rotor (a smooth cylinder) becomes temporarily magnetized. Its distributed poles make it act like a permanent magnet synchronous motor. The rotor material, like that of a common nail, stays magnetized, but can be demagnetized with little difficulty. Once running, the rotor poles stay in place; they do not drift.
Low-power synchronous timing motors (such as those for traditional electric clocks) may have multi-pole permanent magnet external cup rotors, and use shading coils to provide starting torque. Telechron clock motors have shaded poles for starting torque, and a two-spoke ring rotor that performs like a discrete two-pole rotor.
### Doubly-fed electric machine
Doubly fed electric motors have two independent multiphase winding sets, which contribute active (i.e., working) power to the energy conversion process, with at least one of the winding sets electronically controlled for variable speed operation. Two independent multiphase winding sets (i.e., dual armature) are the maximum provided in a single package without topology duplication. Doubly-fed electric motors have an effective constant torque speed range that is twice synchronous speed for a given frequency of excitation. This is twice the constant torque speed range as singly-fed electric machines, which have only one active winding set.
A doubly-fed motor allows for a smaller electronic converter but the cost of the rotor winding and slip rings may offset the saving in the power electronics components.
|
https://en.wikipedia.org/wiki/Electric_motor
|
passage: Outside of this time frame, it cannot be determined how these immeasurable factors behave both qualitatively and quantitatively.
Research results by mathematicians, statisticians, econometricians, and economists have been published in response to those questions. For example, detailed notes on the meaning of linear time trends in the regression model are given in Cameron (2005); Granger, Engle, and many other econometricians have written on stationarity, unit root testing, co-integration, and related issues (a summary of some of the works in this area can be found in an information paper by the Royal Swedish Academy of Sciences (2003)); and Ho-Trieu & Tucker (1990) have written on logarithmic time trends with results indicating linear time trends are special cases of cycles.
### Noisy time series
It is harder to see a trend in a noisy time series. For example, if the true series is 0, 1, 2, 3, all plus some independent normally distributed "noise" e of standard deviationE, and a sample series of length 50 is given, then if E=0.1, the trend will be obvious; if E=100, the trend will probably be visible; but if E=10000, the trend will be buried in the noise.
Consider a concrete example, such as the global surface temperature record of the past 140 years as presented by the IPCC. The interannual variation is about 0.2°C, and the trend is about 0.6°C over 140 years, with 95% confidence limits of 0.2°C (by coincidence, about the same value as the interannual variation). Hence, the trend is statistically different from 0.
|
https://en.wikipedia.org/wiki/Linear_trend_estimation
|
passage: 3. Suppose that exactly two of U, V, W are 0. Without loss of generality we assume
(***)
$$
U \neq 0, V = W = 0.\,
$$
It follows that
$$
z = 0,\,
$$
(since
$$
z \neq 0,\,
$$
implies that
$$
x = y = 0,\,
$$
and hence
$$
U = 0,\,
$$
contradicting (***).)
a. In the subcase where
$$
|U| \leq \frac{1}{2},
$$
if we determine x and y by
$$
x^2 = \frac{1 + \sqrt{1 - 4 U^2}}{2}
$$
and
$$
y^2 = \frac{1 - \sqrt{1 - 4 U^2}}{2},
$$
this ensures that (*) holds. It is easy to verify that
$$
x^2 y^2 = U^2,\,
$$
and hence choosing the signs of x and y appropriately will guarantee
$$
x y = U.\,
$$
Since also
$$
y z = 0 = V\text{ and }z x = 0 = W,\,
$$
this shows that this subcase leads to the desired converse.
b. In this remaining subcase of the case 3., we have
$$
|U| > \frac{1}{2}.
$$
Since
$$
x^2 + y^2 = 1,\,
$$
it is easy to check that
$$
xy \leq \frac{1}{2},
$$
and
|
https://en.wikipedia.org/wiki/Roman_surface
|
passage: This fact is the beginning of Grothendieck's Galois theory, a far-reaching extension of Galois theory applicable to algebro-geometric objects.
## Invariants of fields
Basic invariants of a field include the characteristic and the transcendence degree of over its prime field. The latter is defined as the maximal number of elements in that are algebraically independent over the prime field. Two algebraically closed fields and are isomorphic precisely if these two data agree. This implies that any two uncountable algebraically closed fields of the same cardinality and the same characteristic are isomorphic. For example, and are isomorphic (but not isomorphic as topological fields).
### Model theory of fields
In model theory, a branch of mathematical logic, two fields and are called elementarily equivalent if every mathematical statement that is true for is also true for and conversely. The mathematical statements in question are required to be first-order sentences (involving , , the addition and multiplication). A typical example, for , an integer, is
= "any polynomial of degree in has a zero in "
The set of such formulas for all expresses that is algebraically closed.
The Lefschetz principle states that is elementarily equivalent to any algebraically closed field of characteristic zero. Moreover, any fixed statement holds in if and only if it holds in any algebraically closed field of sufficiently high characteristic.
|
https://en.wikipedia.org/wiki/Field_%28mathematics%29
|
passage: The quadratic subfield of the prime cyclotomic field
A classical example of the construction of a quadratic field is to take the unique quadratic field inside the cyclotomic field generated by a primitive
$$
p
$$
th root of unity, with
$$
p
$$
an odd prime number. The uniqueness is a consequence of Galois theory, there being a unique subgroup of index
$$
2
$$
in the Galois group over
$$
\mathbf{Q}
$$
. As explained at Gaussian period, the discriminant of the quadratic field is
$$
p
$$
for
$$
p=4n+1
$$
and
$$
-p
$$
for
$$
p=4n+3
$$
. This can also be predicted from enough ramification theory. In fact,
$$
p
$$
is the only prime that ramifies in the cyclotomic field, so
$$
p
$$
is the only prime that can divide the quadratic field discriminant. That rules out the 'other' discriminants
$$
-4p
$$
and
$$
4p
$$
in the respective cases.
### Other cyclotomic fields
If one takes the other cyclotomic fields, they have Galois groups with extra
$$
2
$$
-torsion, so contain at least three quadratic fields. In general a quadratic field of field discriminant
$$
D
$$
can be obtained as a subfield of a cyclotomic field of
$$
D
$$
-th roots of unity. This expresses the fact that the conductor of a quadratic field is the absolute value of its discriminant, a special case of the conductor-discriminant formula.
|
https://en.wikipedia.org/wiki/Quadratic_field
|
passage: The gravitational instability of mature oceanic lithosphere has the effect that at subduction zones, oceanic lithosphere invariably sinks underneath the overriding lithosphere, which can be oceanic or continental. New oceanic lithosphere is constantly being produced at mid-ocean ridges and is recycled back to the mantle at subduction zones. As a result, oceanic lithosphere is much younger than continental lithosphere: the oldest oceanic lithosphere is about 170 million years old, while parts of the continental lithosphere are billions of years old.
##### Subducted lithosphere
Geophysical studies in the early 21st century posit that large pieces of the lithosphere have been subducted into the mantle as deep as to near the core-mantle boundary, while others "float" in the upper mantle. Yet others stick down into the mantle as far as but remain "attached" to the continental plate above, similar to the extent of the old concept of "tectosphere" revisited by Jordan in 1988. Subducting lithosphere remains rigid (as demonstrated by deep earthquakes along Wadati–Benioff zone) to a depth of about .
Continental lithosphere
Continental lithosphere has a range in thickness from about to perhaps ; the upper approximately of typical continental lithosphere is crust. The crust is distinguished from the upper mantle by the change in chemical composition that takes place at the Moho discontinuity.
|
https://en.wikipedia.org/wiki/Lithosphere
|
passage: A mushroom or toadstool is the fleshy, spore-bearing fruiting body of a fungus, typically produced above ground on soil or another food source. Toadstool generally refers to a poisonous mushroom.
The standard for the name "mushroom" is the cultivated white button mushroom, Agaricus bisporus; hence, the word "mushroom" is most often applied to those fungi (Basidiomycota, Agaricomycetes) that have a stem (stipe), a cap (pileus), and gills (lamellae, sing. lamella) on the underside of the cap. "Mushroom" also describes a variety of other gilled fungi, with or without stems; therefore the term is used to describe the fleshy fruiting bodies of some Ascomycota. The gills produce microscopic spores which help the fungus spread across the ground or its occupant surface.
Forms deviating from the standard morphology usually have more specific names, such as "bolete", "truffle", "puffball", "stinkhorn", and "morel", and gilled mushrooms themselves are often called "agarics" in reference to their similarity to Agaricus or their order Agaricales.
## Etymology
The terms "mushroom" and "toadstool" go back centuries and were never precisely defined, nor was there consensus on application. During the 15th and 16th centuries, the terms mushrom, mushrum, muscheron, mousheroms, mussheron, or musserouns were used.
|
https://en.wikipedia.org/wiki/Mushroom
|
passage: There are several methods to compute this percentage. The two main ways are the path method and the tabular method.
Typical coancestries between relatives are as follows:
- Father/daughter or mother/son → 25% ()
- Brother/sister → 25% ()
- Grandfather/granddaughter or grandmother/grandson → 12.5% ()
- Half-brother/half-sister, Double cousins → 12.5% ()
- Uncle/niece or aunt/nephew → 12.5% ()
- Great-grandfather/great-granddaughter or great-grandmother/great-grandson → 6.25% ()
- Half-uncle/niece or half-aunt/nephew → 6.25% ()
- First cousins → 6.25% ()
Animals
### Wild animals
- Banded mongoose females regularly mate with their fathers and brothers.
- Bed bugs: North Carolina State University found that bedbugs, in contrast to most other insects, tolerate incest and are able to genetically withstand the effects of inbreeding quite well.
- Common fruit fly females prefer to mate with their own brothers over unrelated males.
- Cottony cushion scales: 'It turns out that females in these hermaphrodite insects are not really fertilizing their eggs themselves, but instead are having this done by a parasitic tissue that infects them at birth,' says Laura Ross of Oxford University's Department of Zoology.
|
https://en.wikipedia.org/wiki/Inbreeding
|
passage: They are formed as a glass, and then partially crystallized by heat treatment, producing both amorphous and crystalline phases so that crystalline grains are embedded within a non-crystalline intergranular phase.
Glass-ceramics are used to make cookware (originally known by the brand name CorningWare) and stovetops that have high resistance to thermal shock and extremely low permeability to liquids. The negative coefficient of thermal expansion of the crystalline ceramic phase can be balanced with the positive coefficient of the glassy phase. At a certain point (~70% crystalline) the glass-ceramic has a net coefficient of thermal expansion close to zero. This type of glass-ceramic exhibits excellent mechanical properties and can sustain repeated and quick temperature changes up to 1000 °C.
Glass ceramics may also occur naturally when lightning strikes the crystalline (e.g. quartz) grains found in most beach sand. In this case, the extreme and immediate heat of the lightning (~2500 °C) creates hollow, branching rootlike structures called fulgurite via fusion.
### Organic solids
Organic chemistry studies the structure, properties, composition, reactions, and preparation by synthesis (or other means) of chemical compounds of carbon and hydrogen, which may contain any number of other elements such as nitrogen, oxygen and the halogens: fluorine, chlorine, bromine and iodine. Some organic compounds may also contain the elements phosphorus or sulfur.
|
https://en.wikipedia.org/wiki/Solid
|
passage: and CE are the shortest edges, with length 5, and AD has been arbitrarily chosen, so it is highlighted. CE is now the shortest edge that does not form a cycle, with length 5, so it is highlighted as the second edge. The next edge, DF with length 6, is highlighted using much the same method. The next-shortest edges are AB and BE, both with length 7. AB is chosen arbitrarily, and is highlighted. The edge BD has been highlighted in red, because there already exists a path (in green) between B and D, so it would form a cycle (ABD) if it were chosen. The process continues to highlight the next-smallest edge, BE with length 7. Many more edges are highlighted in red at this stage: BC because it would form the loop BCE, DE because it would form the loop DEBA, and FE because it would form FEBAD.Finally, the process finishes with the edge EG of length 9, and the minimum spanning tree is found.
## Proof of correctness
The proof consists of two parts. First, it is proved that the algorithm produces a spanning tree. Second, it is proved that the constructed spanning tree is of minimal weight.
### Spanning tree
Let
$$
G
$$
be a connected, weighted graph and let
$$
Y
$$
be the subgraph of
$$
G
$$
produced by the algorithm.
$$
Y
$$
cannot have a cycle, as by definition an edge is not added if it results in a cycle.
|
https://en.wikipedia.org/wiki/Kruskal%27s_algorithm
|
passage: Time complexity: (or time to compute the root weight)
A concatenation can be performed simply by creating a new root node with and , which is constant time. The weight of the parent node is set to the length of the left child S1, which would take time, if the tree is balanced.
As most rope operations require balanced trees, the tree may need to be re-balanced after concatenation.
Split
Definition: `Split (i, S)`: split the string S into two new strings S1 and S2, and .
Time complexity:
There are two cases that must be dealt with:
1. The split point is at the end of a string (i.e. after the last character of a leaf node)
1. The split point is in the middle of a string.
The second case reduces to the first by splitting the string at the split point to create two new leaf nodes, then creating a new node that is the parent of the two component strings.
For example, to split the 22-character rope pictured in Figure 2.3 into two equal component ropes of length 11, query the 12th character to locate the node K at the bottom level. Remove the link between K and G. Go to the parent of G and subtract the weight of K from the weight of D. Travel up the tree and remove any right links to subtrees covering characters past position 11, subtracting the weight of K from their parent nodes (only node D and A, in this case).
|
https://en.wikipedia.org/wiki/Rope_%28data_structure%29
|
passage: In mathematics a topological space is called countably compact if every countable open cover has a finite subcover.
## Equivalent definitions
A topological space X is called countably compact if it satisfies any of the following equivalent conditions:
(1) Every countable open cover of X has a finite subcover.
(2) Every infinite set A in X has an ω-accumulation point in X.
(3) Every sequence in X has an accumulation point in X.
(4) Every countable family of closed subsets of X with an empty intersection has a finite subfamily with an empty intersection.
(1) (2): Suppose (1) holds and A is an infinite subset of X without
$$
\omega
$$
-accumulation point. By taking a subset of A if necessary, we can assume that A is countable.
Every
$$
x\in X
$$
has an open neighbourhood
$$
O_x
$$
such that
$$
O_x\cap A
$$
is finite (possibly empty), since x is not an ω-accumulation point. For every finite subset F of A define
$$
O_F = \cup\{O_x: O_x\cap A=F\}
$$
. Every
$$
O_x
$$
is a subset of one of the
$$
O_F
$$
, so the
$$
O_F
$$
cover X. Since there are countably many of them, the
$$
O_F
$$
form a countable open cover of X. But every
$$
O_F
$$
intersect A in a finite subset (namely F), so finitely many of them cannot cover A, let alone X.
|
https://en.wikipedia.org/wiki/Countably_compact_space
|
passage: Iterative methodologies, such as Rational Unified Process and dynamic systems development method, focus on stabilizing project scope and iteratively expanding or improving products. Sequential or big-design-up-front (BDUF) models, such as waterfall, focus on complete and correct planning to guide larger projects and limit risks to successful and predictable results. Anamorphic development is guided by project scope and adaptive iterations.
In project management a project can include both a project life cycle (PLC) and an SDLC, during which somewhat different activities occur. According to Taylor (2004), "the project life cycle encompasses all the activities of the project, while the systems development life cycle focuses on realizing the product requirements".
SDLC is not a methodology per se, but rather a description of the phases that a methodology should address. The list of phases is not definitive, but typically includes planning, analysis, design, build, test, implement, and maintenance/support. In the Scrum framework, for example, one could say a single user story goes through all the phases of the SDLC within a two-week sprint. By contrast the waterfall methodology, where every business requirement is translated into feature/functional descriptions which are then all implemented typically over a period of months or longer.
## History
According to Elliott (2004), SDLC "originated in the 1960s, to develop large scale functional business systems in an age of large scale business conglomerates. Information systems activities revolved around heavy data processing and number crunching routines".
|
https://en.wikipedia.org/wiki/Systems_development_life_cycle
|
passage: $$
its derivative
$$
d\omega \cong (D \wedge A)^{\dagger} \cdot d^{k+1}X = (D \wedge A) \cdot \left(d^{k+1}X \right)^{\dagger},
$$
and its Hodge dual
$$
\star\omega \cong (I^{-1} A)^{\dagger} \cdot d^kX,
$$
embed the theory of differential forms within geometric calculus.
|
https://en.wikipedia.org/wiki/Geometric_calculus
|
passage: For genus 1 the dimension is the Hodge number h1,0 which is therefore 1. It is known that all curves of genus one have equations of form y2 = x3 + ax + b. These obviously depend on two parameters, a and b, whereas the isomorphism classes of such curves have only one parameter. Hence there must be an equation relating those a and b which describe isomorphic elliptic curves. It turns out that curves for which b2a−3 has the same value, describe isomorphic curves. I.e. varying a and b is one way to deform the structure of the curve y2 = x3 + ax + b, but not all variations of a,b actually change the isomorphism class of the curve.
One can go further with the case of genus g > 1, using Serre duality to relate the H1 to
$$
H^0(\Omega^{[2]})
$$
where Ω is the holomorphic cotangent bundle and the notation Ω[2] means the tensor square (not the second exterior power). In other words, deformations are regulated by holomorphic quadratic differentials on a Riemann surface, again something known classically. The dimension of the moduli space, called Teichmüller space in this case, is computed as 3g − 3, by the Riemann–Roch theorem.
These examples are the beginning of a theory applying to holomorphic families of complex manifolds, of any dimension.
|
https://en.wikipedia.org/wiki/Deformation_%28mathematics%29
|
passage: ### Inequalities
The number serves appears in similar eigenvalue problems in higher-dimensional analysis. As mentioned above, it can be characterized via its role as the best constant in the isoperimetric inequality: the area enclosed by a plane Jordan curve of perimeter satisfies the inequality
$$
4\pi A\le P^2,
$$
and equality is clearly achieved for the circle, since in that case and .
Ultimately, as a consequence of the isoperimetric inequality, appears in the optimal constant for the critical Sobolev inequality in n dimensions, which thus characterizes the role of in many physical phenomena as well, for example those of classical potential theory. In two dimensions, the critical Sobolev inequality is
$$
2\pi\|f\|_2 \le \|\nabla f\|_1
$$
for f a smooth function with compact support in ,
$$
\nabla f
$$
is the gradient of f, and
$$
\|f\|_2
$$
and
$$
\|\nabla f\|_1
$$
refer respectively to the and -norm. The Sobolev inequality is equivalent to the isoperimetric inequality (in any dimension), with the same best constants.
Wirtinger's inequality also generalizes to higher-dimensional Poincaré inequalities that provide best constants for the Dirichlet energy of an n-dimensional membrane.
|
https://en.wikipedia.org/wiki/Pi
|
passage: If a random variable X has this distribution, we write .
The exponential distribution exhibits infinite divisibility.
### Cumulative distribution function
The cumulative distribution function is given by
$$
F(x;\lambda) = \begin{cases}
1-e^{-\lambda x} & x \ge 0, \\
0 & x < 0.
\end{cases}
$$
### Alternative parametrization
The exponential distribution is sometimes parametrized in terms of the scale parameter , which is also the mean:
$$
f(x;\beta) = \begin{cases}
\frac{1}{\beta} e^{-x/\beta} & x \ge 0, \\
BLOCK0 \end{cases} \qquad\qquad
F(x;\beta) = \begin{cases}
1- e^{-x/\beta} & x \ge 0, \\
BLOCK1 \end{cases}
$$
## Properties
### Mean, variance, moments, and median
The mean or expected value of an exponentially distributed random variable X with rate parameter λ is given by
$$
\operatorname{E}[X] = \frac{1}{\lambda}.
$$
In light of the examples given below, this makes sense; a person who receives an average of two telephone calls per hour can expect that the time between consecutive calls will be 0.5 hour, or 30 minutes.
|
https://en.wikipedia.org/wiki/Exponential_distribution
|
passage: The algorithm evaluates each leaf node using a heuristic evaluation function, obtaining the values shown. The moves where the maximizing player wins are assigned with positive infinity, while the moves that lead to a win of the minimizing player are assigned with negative infinity. At level 3, the algorithm will choose, for each node, the smallest of the child node values, and assign it to that same node (e.g. the node on the left will choose the minimum between "10" and "+∞", therefore assigning the value "10" to itself). The next step, in level 2, consists of choosing for each node the largest of the child node values. Once again, the values are assigned to each parent node. The algorithm continues evaluating the maximum and minimum values of the child nodes alternately until it reaches the root node, where it chooses the move with the largest value (represented in the figure with a blue arrow). This is the move that the player should make in order to minimize the maximum possible loss.
## Minimax for individual decisions
### Minimax in the face of uncertainty
Minimax theory has been extended to decisions where there is no other player, but where the consequences of decisions depend on unknown facts. For example, deciding to prospect for minerals entails a cost, which will be wasted if the minerals are not present, but will bring major rewards if they are.
|
https://en.wikipedia.org/wiki/Minimax
|
passage: It is often referred to as the second fundamental form.
An immediate corollary is the Gauss equation for the curvature tensor. For
$$
X, Y, Z, W \in TM
$$
,
$$
\langle R'(X, Y)Z, W\rangle = \langle R(X, Y)Z, W\rangle + \langle \alpha(X, Z), \alpha(Y, W)\rangle - \langle \alpha(Y, Z), \alpha(X, W)\rangle
$$
where
$$
R'
$$
is the Riemann curvature tensor of P and R is that of M.
The Weingarten equation is an analog of the Gauss formula for a connection in the normal bundle. Let
$$
X \in TM
$$
and
$$
\xi
$$
a normal vector field. Then decompose the ambient covariant derivative of
$$
\xi
$$
along X into tangential and normal components:
$$
\nabla'_X\xi = \top \left(\nabla'_X\xi\right) + \bot\left(\nabla'_X\xi\right) = -A_\xi(X) + D_X(\xi).
$$
Then
1. Weingarten's equation: BLOCK171. DX is a metric connection in the normal bundle.
There are thus a pair of connections: ∇, defined on the tangent bundle of M; and D, defined on the normal bundle of M.
|
https://en.wikipedia.org/wiki/Gauss%E2%80%93Codazzi_equations
|
passage: Instead, it is often sufficient (as well as faster, and more economical for storage) to compute a reduced version of the SVD. The following can be distinguished for an matrix of rank :
### Thin SVD
The thin, or economy-sized, SVD of a matrix is given by
$$
\mathbf{M} = \mathbf{U}_k \mathbf \Sigma_k \mathbf{V}^*_k,
$$
where
$$
k = \min(m, n),
$$
the matrices and contain only the first columns of and and contains only the first singular values from The matrix is thus is diagonal, and is
The thin SVD uses significantly less space and computation time if The first stage in its calculation will usually be a QR decomposition of which can make for a significantly quicker calculation in this case.
### Compact SVD
The compact SVD of a matrix is given by
$$
\mathbf{M} = \mathbf U_r \mathbf \Sigma_r \mathbf V_r^*.
$$
Only the column vectors of and row vectors of corresponding to the non-zero singular values are calculated. The remaining vectors of and are not calculated. This is quicker and more economical than the thin SVD if The matrix is thus is diagonal, and is
### Truncated SVD
In many applications the number of the non-zero singular values is large making even the Compact SVD impractical to compute.
|
https://en.wikipedia.org/wiki/Singular_value_decomposition
|
passage: The component on the right hand side is the Gauss's law component, and this is the component that is relevant to the conservation of charge argument above. The second term on the right-hand side is the one relevant to the electromagnetic wave equation, because it is the term that contributes to the curl of . Because of the vector identity that says the curl of a gradient is zero, does not contribute to .
## History and interpretation
Maxwell's displacement current was postulated in part III of his 1861 paper 'On Physical Lines of Force'. Few topics in modern physics have caused as much confusion and misunderstanding as that of displacement current. This is in part due to the fact that Maxwell used a sea of molecular vortices in his derivation, while modern textbooks operate on the basis that displacement current can exist in free space. Maxwell's derivation is unrelated to the modern day derivation for displacement current in the vacuum, which is based on consistency between Ampère's circuital law for the magnetic field and the continuity equation for electric charge.
Maxwell's purpose is stated by him at (Part I, p. 161):
He is careful to point out the treatment is one of analogy:
In part III, in relation to displacement current, he says
Clearly Maxwell was driving at magnetization even though the same introduction clearly talks about dielectric polarization.
Maxwell compared the speed of electricity measured by Wilhelm Eduard Weber and Rudolf Kohlrausch (193,088 miles/second) and the speed of light determined by the Fizeau experiment (195,647 miles/second).
|
https://en.wikipedia.org/wiki/Displacement_current
|
passage: In probability theory, a pairwise independent collection of random variables is a set of random variables any two of which are independent. Any collection of mutually independent random variables is pairwise independent, but some pairwise independent collections are not mutually independent. Pairwise independent random variables with finite variance are uncorrelated.
A pair of random variables X and Y are independent if and only if the random vector (X, Y) with joint cumulative distribution function (CDF)
$$
F_{X,Y}(x,y)
$$
satisfies
$$
F_{X,Y}(x,y) = F_X(x) F_Y(y),
$$
or equivalently, their joint density
$$
f_{X,Y}(x,y)
$$
satisfies
$$
f_{X,Y}(x,y) = f_X(x) f_Y(y).
$$
That is, the joint distribution is equal to the product of the marginal distributions.
Unless it is not clear in context, in practice the modifier "mutual" is usually dropped so that independence means mutual independence. A statement such as " X, Y, Z are independent random variables" means that X, Y, Z are mutually independent.
## Example
Pairwise independence does not imply mutual independence, as shown by the following example attributed to S. Bernstein.
Suppose X and Y are two independent tosses of a fair coin, where we designate 1 for heads and 0 for tails.
|
https://en.wikipedia.org/wiki/Pairwise_independence
|
passage: It follows that the smallest key in the entire heap is one of the roots.
The second property implies that a binomial heap with
$$
n
$$
nodes consists of at most
$$
1+\log_2 n
$$
binomial trees, where
$$
\log_2
$$
is the binary logarithm. The number and orders of these trees are uniquely determined by the number of nodes
$$
n
$$
: there is one binomial tree for each nonzero bit in the binary representation of the number
$$
n
$$
. For example, the decimal number 13 is 1101 in binary,
$$
2^3 + 2^2 + 2^0
$$
, and thus a binomial heap with 13 nodes will consist of three binomial trees of orders 3, 2, and 0 (see figure below).
The number of different ways that
$$
n
$$
items with distinct keys can be arranged into a binomial heap equals the largest odd divisor of
$$
n!
$$
. For
$$
n=1,2,3,\dots
$$
these numbers are
1, 1, 3, 3, 15, 45, 315, 315, 2835, 14175, ...
If the
$$
n
$$
items are inserted into a binomial heap in a uniformly random order, each of these arrangements is equally likely.
## Implementation
Because no operation requires random access to the root nodes of the binomial trees, the roots of the binomial trees can be stored in a linked list, ordered by increasing order of the tree.
|
https://en.wikipedia.org/wiki/Binomial_heap
|
passage: To aid in the diagnosis of infectious diseases, immunoassays can detect or measure antigens from either infectious agents or proteins generated by an infected organism in response to a foreign agent. For example, immunoassay A may detect the presence of a surface protein from a virus particle. Immunoassay B on the other hand may detect or measure antibodies produced by an organism's immune system that are made to neutralize and allow the destruction of the virus.
Instrumentation can be used to read extremely small signals created by secondary reactions linked to the antibody – antigen binding. Instrumentation can control sampling, reagent use, reaction times, signal detection, calculation of results, and data management to yield a cost-effective automated process for diagnosis of infectious disease.
### PCR-based diagnostics
Technologies based upon the polymerase chain reaction (PCR) method will become nearly ubiquitous gold standards of diagnostics of the near future, for several reasons. First, the catalog of infectious agents has grown to the point that virtually all of the significant infectious agents of the human population have been identified. Second, an infectious agent must grow within the human body to cause disease; essentially it must amplify its own nucleic acids to cause a disease. This amplification of nucleic acid in infected tissue offers an opportunity to detect the infectious agent by using PCR. Third, the essential tools for directing PCR, primers, are derived from the genomes of infectious agents, and with time those genomes will be known if they are not already.
|
https://en.wikipedia.org/wiki/Infection
|
passage: It states that the following conditions are equivalent:
- CM([a,b]) is quasi-analytic.
-
$$
\sum 1/L_j = \infty
$$
where
$$
L_j= \inf_{k\ge j}(k\cdot M_k^{1/k})
$$
.
-
$$
\sum_j \frac{1}{j}(M_j^*)^{-1/j} = \infty
$$
, where Mj* is the largest log convex sequence bounded above by Mj.
-
$$
\sum_j\frac{M_{j-1}^*}{(j+1)M_j^*} = \infty.
$$
The proof that the last two conditions are equivalent to the second uses Carleman's inequality.
Example: pointed out that if Mn is given by one of the sequences
$$
1,\, {(\ln n)}^n,\, {(\ln n)}^n\,{(\ln \ln n)}^n,\, {(\ln n)}^n\,{(\ln \ln n)}^n\,{(\ln \ln \ln n)}^n, \dots,
$$
then the corresponding class is quasi-analytic. The first sequence gives analytic functions.
|
https://en.wikipedia.org/wiki/Quasi-analytic_function
|
passage: Normal rhythm produces four entities – a P wave, a QRS complex, a T wave, and a U wave – that each have a fairly unique pattern.
- The P wave represents atrial depolarization.
- The QRS complex represents ventricular depolarization.
- The T wave represents ventricular repolarization.
- The U wave represents papillary muscle repolarization.
Changes in the structure of the heart and its surroundings (including blood composition) change the patterns of these four entities.
The U wave is not typically seen and its absence is generally ignored. Atrial repolarization is typically hidden in the much more prominent QRS complex and normally cannot be seen without additional, specialized electrodes.
### Background grid
ECGs are normally printed on a grid.
The horizontal axis represents time and the vertical axis represents voltage.
The standard values on this grid are shown in the adjacent image at 25mm/sec (or 40ms per mm):
- A small box is 1 mm × 1 mm and represents 0.1 mV × 0.04 seconds.
- A large box is 5 mm × 5 mm and represents 0.5 mV × 0.20 seconds.
The "large" box is represented by a heavier line weight than the small boxes.
The standard printing speed in the United States is 25 mm per sec (5 big boxes per second), but in other countries it can be 50 mm per sec.
Faster speeds such as 100 and 200 mm per sec are used during electrophysiology studies.
|
https://en.wikipedia.org/wiki/Electrocardiography
|
passage: Thus much of classification theory was concerned with analyzing stable theories and various subsets of stable theories given by further dividing lines, such as superstable theories.
One of the key features of stable theories developed by Shelah is that they admit a general notion of independence called non-forking independence, generalizing linear independence from vector spaces and algebraic independence from field theory. Although non-forking independence makes sense in arbitrary theories, and remains a key tool beyond stable theories, it has particularly good geometric and combinatorial properties in stable theories. As with linear independence, this allows the definition of independent sets and of local dimensions as the cardinalities of maximal instances of these independent sets, which are well-defined under additional hypotheses. These local dimensions then give rise to the cardinal-invariants classifying models up to isomorphism.
## Definition and alternate characterizations
Let T be a complete first-order theory.
For a given infinite cardinal
$$
\kappa
$$
, T is -stable if for every set A of cardinality
$$
\kappa
$$
in a model of T, the set S(A) of complete types over A also has cardinality
$$
\kappa
$$
. This is the smallest the cardinality of S(A) can be, while it can be as large as
$$
2^\kappa
$$
. For the case
$$
\kappa = \aleph_0
$$
, it is common to say T is -stable rather than
$$
\aleph_0
$$
-stable.
|
https://en.wikipedia.org/wiki/Stable_theory
|
passage: In June 2019, Russia conceded that it was "possible" its electrical grid is under cyber-attack by the United States. The New York Times reported that American hackers from the United States Cyber Command planted malware potentially capable of disrupting the Russian electrical grid.
## Records
- Highest capacity system: 12 GW Zhundong–Wannan (准东-皖南)±1100 kV HVDC.
- Highest transmission voltage (AC):
- planned: 1.20 MV (Ultra-High Voltage) on Wardha-Aurangabad line (India), planned to initially operate at 400 kV.
- worldwide: 1.15 MV (Ultra-High Voltage) on Ekibastuz-Kokshetau line (Kazakhstan) (operating at 500kv)
- Largest double-circuit transmission, Kita-Iwaki Powerline (Japan).
- Highest towers: Yangtze River Crossing (China) (height: )
- Longest power line: Inga-Shaba (Democratic Republic of Congo) (length: )
- Longest span of power line: at Ameralik Span (Greenland, Denmark)
- Longest submarine cables:
- North Sea Link, (Norway/United Kingdom) – (length of submarine cable: )
- NorNed, North Sea (Norway/Netherlands) – (length of submarine cable: )
- Basslink, Bass Strait, (Australia) – (length of submarine cable: , total length: )
- Baltic Cable, Baltic Sea (Germany/Sweden) – (length of submarine cable: , HVDC length: , total length: )
- Longest underground cables:
- Murraylink, Riverland/Sunraysia (Australia) – (length of underground cable: )
|
https://en.wikipedia.org/wiki/Electric_power_transmission
|
passage: If M is an oriented spin manifold with Dirac operator ð, then one may form the spin Laplacian Δ = ð2 on the spin bundle. On the other hand, the Levi-Civita connection extends to the spin bundle to yield a differential operator
$$
\nabla:SM\rightarrow T^*M\otimes SM.
$$
As in the case of Riemannian manifolds, let
$$
\Delta'=\nabla^*\nabla
$$
. This is another self-adjoint operator and, moreover, has the same leading symbol as the spin Laplacian. The Weitzenböck formula yields:
$$
\Delta' - \Delta = -\frac{1}{4}Sc
$$
where Sc is the scalar curvature. This result is also known as the Lichnerowicz formula.
## Complex differential geometry
If M is a compact Kähler manifold, there is a Weitzenböck formula relating the
$$
\bar{\partial}
$$
-Laplacian (see Dolbeault complex) and the Euclidean Laplacian on (p,q)-forms. Specifically, let
$$
\Delta = \bar{\partial}^*\bar{\partial}+\bar{\partial}\bar{\partial}^*,
$$
and
$$
\Delta' = -\sum_k\nabla_k\nabla_{\bar{k}}
$$
in a unitary frame at each point.
|
https://en.wikipedia.org/wiki/Weitzenb%C3%B6ck_identity
|
passage: Poisson processes is another Poisson process. If a point
$$
x
$$
is sampled from a countable
$$
n
$$
union of Poisson processes, then the probability that the point
$$
\textstyle x
$$
belongs to the
$$
j
$$
th Poisson process
$$
N_j
$$
is given by:
$$
\Pr \{x\in N_j\}=\frac{\Lambda_j}{\sum_{i=1}^n\Lambda_i}.
$$
For two homogeneous Poisson processes with intensities
$$
\lambda_1,\lambda_2\dots
$$
, the two previous expressions reduce to
$$
\lambda=\sum_{i=1}^\infty \lambda_i,
$$
and
$$
\Pr \{x\in N_j\}=\frac{\lambda_j}{\sum_{i=1}^n \lambda_i}.
$$
### Clustering
The operation clustering is performed when each point
$$
\textstyle x
$$
of some point process
$$
\textstyle {N}
$$
is replaced by another (possibly different) point process. If the original process
$$
\textstyle {N}
$$
is a Poisson point process, then the resulting process
$$
\textstyle {N}_c
$$
is called a Poisson cluster point process.
|
https://en.wikipedia.org/wiki/Poisson_point_process
|
passage: The divergence theorem states that any such continuity equation can be written in a differential form (in terms of a divergence) and an integral form (in terms of a flux).
### Inverse-square laws
Any inverse-square law can instead be written in a Gauss's law-type form (with a differential and integral form, as described above). Two examples are Gauss's law (in electrostatics), which follows from the inverse-square Coulomb's law, and Gauss's law for gravity, which follows from the inverse-square Newton's law of universal gravitation. The derivation of the Gauss's law-type equation from the inverse-square formulation or vice versa is exactly the same in both cases; see either of those articles for details.
## History
Joseph-Louis Lagrange introduced the notion of surface integrals in 1760 and again in more general terms in 1811, in the second edition of his Mécanique Analytique. Lagrange employed surface integrals in his work on fluid mechanics. He discovered the divergence theorem in 1762.
Carl Friedrich Gauss was also using surface integrals while working on the gravitational attraction of an elliptical spheroid in 1813, when he proved special cases of the divergence theorem. He proved additional special cases in 1833 and 1839. But it was Mikhail Ostrogradsky, who gave the first proof of the general theorem, in 1826, as part of his investigation of heat flow.
|
https://en.wikipedia.org/wiki/Divergence_theorem
|
passage: A geodesic is a shortest possible path between any two of its points.
A geodesic metric space is a metric space which admits a geodesic between any two of its points. The spaces
$$
(\R^2,d_1)
$$
and
$$
(\R^2,d_2)
$$
are both geodesic metric spaces. In
$$
(\R^2,d_2)
$$
, geodesics are unique, but in
$$
(\R^2,d_1)
$$
, there are often infinitely many geodesics between two points, as shown in the figure at the top of the article.
The space is a length space (or the metric is intrinsic) if the distance between any two points and is the infimum of lengths of paths between them. Unlike in a geodesic metric space, the infimum does not have to be attained. An example of a length space which is not geodesic is the Euclidean plane minus the origin: the points and can be joined by paths of length arbitrarily close to 2, but not by a path of length 2. An example of a metric space which is not a length space is given by the straight-line metric on the sphere: the straight line between two points through the center of the Earth is shorter than any path along the surface.
Given any metric space , one can define a new, intrinsic distance function on by setting the distance between points and to be the infimum of the -lengths of paths between them. For instance, if is the straight-line distance on the sphere, then is the great-circle distance.
|
https://en.wikipedia.org/wiki/Metric_space
|
passage: The full collineation group is of order 168 and is isomorphic to the group PSL(2,7) ≈ PSL(3,2), which in this special case is also isomorphic to the general linear group .
### Order of planes
A finite plane of order n is one such that each line has n points (for an affine plane), or such that each line has n + 1 points (for a projective plane). One major open question in finite geometry is:
Is the order of a finite plane always a prime power?
This is conjectured to be true.
Affine and projective planes of order n exist whenever n is a prime power (a prime number raised to a positive integer exponent), by using affine and projective planes over the finite field with elements. Planes not derived from finite fields also exist (e.g. for
$$
n=9
$$
), but all known examples have order a prime power.
The best general result to date is the Bruck–Ryser theorem of 1949, which states:
If n is a positive integer of the form or and n is not equal to the sum of two integer squares, then n does not occur as the order of a finite plane.
The smallest integer that is not a prime power and not covered by the Bruck–Ryser theorem is 10; 10 is of the form , but it is equal to the sum of squares . The non-existence of a finite plane of order 10 was proven in a computer-assisted proof that finished in 1989 – see for details.
|
https://en.wikipedia.org/wiki/Finite_geometry
|
passage: - Some observers with sub-light relative motion will disagree about which occurs first of any two events that are separated by a space-like interval. In other words, any travel that is faster-than-light will be seen as traveling backwards in time in some other, equally valid, frames of reference, or need to assume the speculative hypothesis of possible Lorentz violations at a presently unobserved scale (for instance the Planck scale). Therefore, any theory which permits "true" FTL also has to cope with time travel and all its associated paradoxes, or else to assume the Lorentz invariance to be a symmetry of thermodynamical statistical nature (hence a symmetry broken at some presently unobserved scale).
- In special relativity the coordinate speed of light is only guaranteed to be c in an inertial frame; in a non-inertial frame the coordinate speed may be different from c. In general relativity no coordinate system on a large region of curved spacetime is "inertial", so it is permissible to use a global coordinate system where objects travel faster than c, but in the local neighborhood of any point in curved spacetime we can define a "local inertial frame" and the local speed of light will be c in this frame, with massive objects moving through this local neighborhood always having a speed less than c in the local inertial frame.
## Justifications
### Casimir vacuum and quantum tunnelling
Special relativity postulates that the speed of light in vacuum is invariant in inertial frames. That is, it will be the same from any frame of reference moving at a constant speed.
|
https://en.wikipedia.org/wiki/Faster-than-light
|
passage: ## Naming convention
The theorem is also known variously as the Hermite–Lindemann theorem and the Hermite–Lindemann–Weierstrass theorem. Charles Hermite first proved the simpler theorem where the exponents are required to be rational integers and linear independence is only assured over the rational integers, a result sometimes referred to as Hermite's theorem. Although that appears to be a special case of the above theorem, the general result can be reduced to this simpler case. Lindemann was the first to allow algebraic numbers into Hermite's work in 1882. Shortly afterwards Weierstrass obtained the full result, and further simplifications have been made by several mathematicians, most notably by David Hilbert and Paul Gordan.
## Transcendence of and
The transcendence of and are direct corollaries of this theorem.
Suppose is a non-zero algebraic number; then is a linearly independent set over the rationals, and therefore by the first formulation of the theorem is an algebraically independent set; or in other words is transcendental. In particular, is transcendental. (A more elementary proof that is transcendental is outlined in the article on transcendental numbers.)
Alternatively, by the second formulation of the theorem, if is a non-zero algebraic number, then is a set of distinct algebraic numbers, and so the set is linearly independent over the algebraic numbers and in particular cannot be algebraic and so it is transcendental.
To prove that is transcendental, we prove that it is not algebraic.
|
https://en.wikipedia.org/wiki/Lindemann%E2%80%93Weierstrass_theorem
|
passage: In functional analysis, a closed linear span of a set of vectors is the minimal closed set which contains the linear span of that set.
Suppose that is a normed vector space and let be any non-empty subset of . The closed linear span of , denoted by
$$
\overline{\operatorname{Sp}}(E)
$$
or
$$
\overline{\operatorname{Span}}(E)
$$
, is the intersection of all the closed linear subspaces of which contain .
One mathematical formulation of this is
$$
\overline{\operatorname{Sp}}(E) = \{u\in X | \forall\varepsilon > 0\,\exists x\in\operatorname{Sp}(E) : \|x - u\|<\varepsilon\}.
$$
The closed linear span of the set of functions xn on the interval [0, 1], where n is a non-negative integer, depends on the norm used. If the L2 norm is used, then the closed linear span is the Hilbert space of square-integrable functions on the interval. But if the maximum norm is used, the closed linear span will be the space of continuous functions on the interval. In either case, the closed linear span contains functions that are not polynomials, and so are not in the linear span itself. However, the cardinality of the set of functions in the closed linear span is the cardinality of the continuum, which is the same cardinality as for the set of polynomials.
|
https://en.wikipedia.org/wiki/Linear_span
|
passage: That is to say,
$$
\int_{S^{n-1}} f\bar{g} \, \mathrm{d}\Omega = 0
$$
for and for .
- Conversely, the spaces are precisely the eigenspaces of . In particular, an application of the spectral theorem to the Riesz potential
$$
\Delta_{S^{n-1}}^{-1}
$$
gives another proof that the spaces are pairwise orthogonal and complete in .
- Every homogeneous polynomial can be uniquely written in the form
$$
p(x) = p_\ell(x) + |x|^2p_{\ell-2} + \cdots + \begin{cases}
|x|^\ell p_0 & \ell \rm{\ even}\\
|x|^{\ell-1} p_1(x) & \ell\rm{\ odd}
\end{cases}
$$
where .
|
https://en.wikipedia.org/wiki/Spherical_harmonics
|
passage: ## Matrix form
A quadratic form can be written in terms of matrices as
$$
x^\mathsf{T} A \, x
$$
where is any ×1 Cartesian vector
$$
\; [x_1, \cdots , x_n]^\mathsf{T} \;
$$
in which at least one element is not 0; is an symmetric matrix; and superscript denotes a matrix transpose. If is diagonal this is equivalent to a non-matrix form containing solely terms involving squared variables; but if has any non-zero off-diagonal elements, the non-matrix form will also contain some terms involving products of two different variables.
Positive or negative-definiteness or semi-definiteness, or indefiniteness, of this quadratic form is equivalent to the same property of, which can be checked by considering all eigenvalues of or by checking the signs of all of its principal minors.
## Optimization
Definite quadratic forms lend themselves readily to optimization problems. Suppose the matrix quadratic form is augmented with linear terms, as
$$
x^\mathsf{T} A \, x + b^\mathsf{T} x \;,
$$
where is an ×1 vector of constants.
|
https://en.wikipedia.org/wiki/Definite_quadratic_form
|
passage: Canada Tracy 176 m Tallest electricity pylon in CanadaLínea de Transmisión Carapongo – Carabayllo 2011 Peru Lima 170.5 m Crossing of Rimac River in a 1055 metres long spanDoel Schelde Powerline Crossing 1 1974 Belgium Antwerp 170 m Group of 2 towers with 1 pylon situated in the middle of Schelde RiverSunshine Mississippi Powerline Crossing 1967United States St. Gabriel, Louisiana 164.6 m Tallest electricity pylons in the United StatesLekkerkerk Crossing 1 1970Netherlands Lekkerkerk163 m Tallest crossing in the NetherlandsBosporus overhead line crossing III 1999 Turkey Istanbul 160 m Balakovo 500 kV Wolga Crossing, Tower West 1983–1984 Russia Balakovo 159 m Pylons of Cadiz 1957–1960 Spain Cadiz 158 m Maracaibo Bay Powerline Crossing ? Venezuela Maracaibo150 m Towers on caissonsMeredosia-Ipava Illinois River Crossing 2017 United States Beardstown 149.35 m Aust Severn Powerline Crossing 1959 UK Aust 148.75 m 132 kV Thames Crossing 1932 UK West Thurrock148.4 m Demolished in 1987Karmsundet Powerline Crossing ? Norway Karmsundet 143.5 m Limfjorden Overhead powerline crossing 2 ? Denmark Raerup 141.7 m Saint Lawrence River HVDC Quebec-New England Overhead Powerline Crossing 1989 Canada Deschambault-Grondines 140 m Dismantled in 1992Pylons of Voerde 1926 Germany Voerde 138 m Köhlbrand Powerline Crossing ? Germany Hamburg 138 m Bremen-Farge Weser Powerline Crossing ?
|
https://en.wikipedia.org/wiki/Transmission_tower
|
passage: The Galois group of is , by a basic result of Emil Artin. acts on by restriction of action of . If the fixed field of this action is , then, by the fundamental theorem of Galois theory, the Galois group of is .
On the other hand, it is an open problem whether every finite group is the Galois group of a field extension of the field of the rational numbers. Igor Shafarevich proved that every solvable finite group is the Galois group of some extension of . Various people have solved the inverse Galois problem for selected non-Abelian simple groups. Existence of solutions has been shown for all but possibly one (Mathieu group ) of the 26 sporadic simple groups. There is even a polynomial with integral coefficients whose Galois group is the Monster group.
## Inseparable extensions
In the form mentioned above, including in particular the fundamental theorem of Galois theory, the theory only considers Galois extensions, which are in particular separable. General field extensions can be split into a separable, followed by a purely inseparable field extension. For a purely inseparable extension F / K, there is a Galois theory where the Galois group is replaced by the vector space of derivations,
$$
Der_K(F, F)
$$
, i.e., K-linear endomorphisms of F satisfying the Leibniz rule. In this correspondence, an intermediate field E is assigned
$$
Der_E(F, F) \subset Der_K(F, F)
$$
.
|
https://en.wikipedia.org/wiki/Galois_theory
|
passage: For low-dose exposure, for example among nuclear workers, who receive an average yearly radiation dose of 19 mSv, the risk of dying from cancer (excluding leukemia) increases by 2 percent. For a dose of 100 mSv, the risk increase is 10 percent. By comparison, risk of dying from cancer was increased by 32 percent for the survivors of the atomic bombing of Hiroshima and Nagasaki.
## Units of measurement and exposure
The following table shows radiation quantities in SI and non-SI units:
The measure of the ionizing effect of gamma and X-rays in dry air is called the exposure, for which a legacy unit, the röntgen, was used from 1928. This has been replaced by kerma, now mainly used for instrument calibration purposes but not for received dose effect. The effect of gamma and other ionizing radiation on living tissue is more closely related to the amount of energy deposited in tissue rather than the ionisation of air, and replacement radiometric units and quantities for radiation protection have been defined and developed from 1953 onwards. These are:
- The gray (Gy), is the SI unit of absorbed dose, which is the amount of radiation energy deposited in the irradiated material. For gamma radiation this is numerically equivalent to equivalent dose measured by the sievert, which indicates the stochastic biological effect of low levels of radiation on human tissue. The radiation weighting conversion factor from absorbed dose to equivalent dose is 1 for gamma, whereas alpha particles have a factor of 20, reflecting their greater ionising effect on tissue.
-
|
https://en.wikipedia.org/wiki/Gamma_ray%23Radioactive_decay_%28gamma_decay%29
|
passage: ### Proof by mathematical induction
Despite its name, mathematical induction is a method of deduction, not a form of inductive reasoning. In proof by mathematical induction, a single "base case" is proved, and an "induction rule" is proved that establishes that any arbitrary case implies the next case. Since in principle the induction rule can be applied repeatedly (starting from the proved base case), it follows that all (usually infinitely many) cases are provable. This avoids having to prove each case individually. A variant of mathematical induction is proof by infinite descent, which can be used, for example, to prove the irrationality of the square root of two.
A common application of proof by mathematical induction is to prove that a property known to hold for one number holds for all natural numbers:
Let } be the set of natural numbers, and let be a mathematical statement involving the natural number belonging to such that
- (i) is true, i.e., is true for .
- (ii) is true whenever is true, i.e., is true implies that is true.
- Then is true for all natural numbers .
For example, we can prove by induction that all positive integers of the form are odd. Let represent " is odd":
(i) For , , and is odd, since it leaves a remainder of when divided by . Thus is true.
(ii) For any , if is odd (), then must also be odd, because adding to an odd number results in an odd number. But , so is odd ().
|
https://en.wikipedia.org/wiki/Mathematical_proof
|
passage: For example, on a ringed space (X,OX), it follows that the Picard group of invertible sheaves on X is isomorphic to the sheaf cohomology group H1(X,OX*), where OX* is the sheaf of units in OX.
## Relative cohomology
For a subset Y of a topological space X and a sheaf E of abelian groups on X, one can define relative cohomology groups:
$$
H^j_Y(X,E)=H^j(X,X-Y;E)
$$
for integers j. Other names are the cohomology of X with support in Y, or (when Y is closed in X) local cohomology. A long exact sequence relates relative cohomology to sheaf cohomology in the usual sense:
$$
\cdots \to H^j_Y(X,E)\to H^j(X,E)\to H^j(X-Y,E)\to H^{j+1}_Y(X,E)\to\cdots.
$$
When Y is closed in X, cohomology with support in Y can be defined as the derived functors of the functor
$$
H^0_Y(X,E):=\{s\in E(X): s|_{X-Y}=0\},
$$
the group of sections of E that are supported on Y.
There are several isomorphisms known as excision.
|
https://en.wikipedia.org/wiki/Sheaf_cohomology
|
passage: $$
Thus the wave function is:
$$
\psi_k(x)=\sum_{K}\tilde{u}_k(K)\,e^{i(k+K)x}.
$$
Putting this into the Schrödinger equation, we get:
$$
\left[\frac{\hbar^2(k+K)^2}{2m}-E_k\right] \tilde{u}_k(K)+\sum_{K'}\tilde{V}(K-K')\,\tilde{u}_k(K') = 0
$$
or rather:
$$
\left[\frac{\hbar^2(k+K)^2}{2m}-E_k\right] \tilde{u}_k(K)+\frac{A}{a}\sum_{K'}\tilde{u}_k(K')=0
$$
Now we recognize that:
$$
u_k(0)=\sum_{K'}\tilde{u}_k(K')
$$
Plug this into the Schrödinger equation:
$$
\left[\frac{\hbar^2(k+K)^2}{2m}-E_k\right] \tilde{u}_k(K)+\frac{A}{a}u_k(0)=0
$$
Solving this for
$$
\tilde{u}_k(K)
$$
we get:
$$
|
https://en.wikipedia.org/wiki/Particle_in_a_one-dimensional_lattice
|
passage: ## In computational complexity
In computational complexity theory, the strict definition of in-place algorithms includes all algorithms with space complexity, the class DSPACE(1). This class is very limited; it equals the regular languages. In fact, it does not even include any of the examples listed above.
Algorithms are usually considered in L, the class of problems requiring additional space, to be in-place. This class is more in line with the practical definition, as it allows numbers of size as pointers or indices. This expanded definition still excludes quicksort, however, because of its recursive calls.
Identifying the in-place algorithms with L has some interesting implications; for example, it means that there is a (rather complex) in-place algorithm to determine whether a path exists between two nodes in an undirected graph, a problem that requires extra space using typical algorithms such as depth-first search (a visited bit for each node). This in turn yields in-place algorithms for problems such as determining if a graph is bipartite or testing whether two graphs have the same number of connected components.
## Role of randomness
In many cases, the space requirements of an algorithm can be drastically cut by using a randomized algorithm. For example, if one wishes to know if two vertices in a graph of vertices are in the same connected component of the graph, there is no known simple, deterministic, in-place algorithm to determine this.
|
https://en.wikipedia.org/wiki/In-place_algorithm
|
passage: The degree of freedom of a kinematic chain is computed from the number of links and the number and type of joints using the mobility formula. This formula can also be used to enumerate the topologies of kinematic chains that have a given degree of freedom, which is known as type synthesis in machine design.
#### Examples
The planar one degree-of-freedom linkages assembled from N links and j hinges or sliding joints are:
- N = 2, j = 1 : a two-bar linkage that is the lever;
- N = 4, j = 4 : the four-bar linkage;
- N = 6, j = 7 : a six-bar linkage. This must have two links ("ternary links") that support three joints. There are two distinct topologies that depend on how the two ternary linkages are connected. In the Watt topology, the two ternary links have a common joint; in the Stephenson topology, the two ternary links do not have a common joint and are connected by binary links.
- N = 8, j = 10 : eight-bar linkage with 16 different topologies;
- N = 10, j = 13 : ten-bar linkage with 230 different topologies;
- N = 12, j = 16 : twelve-bar linkage with 6,856 topologies.
For larger chains and their linkage topologies, see R. P. Sunkari and L. C. Schmidt, "Structural synthesis of planar kinematic chains by adapting a Mckay-type algorithm", Mechanism and Machine Theory #41, pp. 1021–1030 (2006).
|
https://en.wikipedia.org/wiki/Kinematics
|
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