text stringlengths 82 2.62k | source stringlengths 31 108 |
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passage: Sets are uniquely characterized by their elements; this means that two sets that have precisely the same elements are equal (they are the same set). In a formalized set theory, this is usually defined by an axiom called the Axiom of extensionality.
For example, using set builder notation, the following states... | https://en.wikipedia.org/wiki/Equality_%28mathematics%29 |
passage: A proper example of a smooth Bump function would be:
$$
u(x)=\begin{cases} 1,\text{if } x=0, \\ 0, \text{if } |x|\geq 1, \\ \frac{1}{1+e^{\frac{1-2|x|}{x^2-|x|}}}, \text{otherwise}, \end{cases}
$$
A proper example of a smooth transition function will be:
$$
w(x)=\begin{cases}\frac{1}{1+e^{\frac{2x-1}{x^2-x}}}&... | https://en.wikipedia.org/wiki/Bump_function |
passage: Rank 3GroupCartanGroupCartanG22<5,3,2>2 G23[5,3]G24 [1 1 14]4 G253[3]3[3]3G263[3]3[4]2G27 [1 1 15]4
+ Rank 4GroupCartanGroupCartanG28[3,4,3]G29[1 1 2]4G30[5,3,3]G323[3]3[3]3
+ Rank 5GroupCartanGroupCartanG31O4 G33[1 2 2]3 | https://en.wikipedia.org/wiki/Complex_reflection_group |
passage: #### Example augmented matrix
Suppose you have three points that define a non-degenerate triangle in a plane, or four points that define a non-degenerate tetrahedron in 3-dimensional space, or generally points , ..., that define a non-degenerate simplex in -dimensional space. Suppose you have corresponding... | https://en.wikipedia.org/wiki/Affine_transformation |
passage: This inequation is satisfied for any x, so the largest such x is 1.
Furthermore, the rule of modus ponens allows us to derive the formula Q from the formulas P and P→Q. But in any Heyting algebra, if P has the value 1, and P→Q has the value 1, then it means that
$$
P \land 1 \le Q
$$
, and so
$$
1 \land 1 \... | https://en.wikipedia.org/wiki/Heyting_algebra |
passage: Until 2005, those wishing to become a general practitioner of medicine had to do a minimum of the following postgraduate training:
- One year as a pre-registration house officer (PRHO) (formerly called a house officer), in which the trainee would usually spend six months on a general surgical ward and six mont... | https://en.wikipedia.org/wiki/General_practitioner |
passage: Bits are deleted according to a puncturing matrix. The following puncturing matrices are the most frequently used:
Code rate Puncturing matrix Free distance (for NASA standard K=7 convolutional code) 1/2(No perf.) 1 1 10 2/3 1 0 1 1 6 3/4 1 0 1 1 1 0 5 5/6 1 0 1 0 1 1 1 0 1 0 4 7/8 1 0 0 0 1 0 1 1 1 1 1 0 1 0... | https://en.wikipedia.org/wiki/Convolutional_code |
passage: In mathematics, the theory of fiber bundles with a structure group
$$
G
$$
(a topological group) allows an operation of creating an associated bundle, in which the typical fiber of a bundle changes from
$$
F_1
$$
to
$$
F_2
$$
, which are both topological spaces with a group action of
$$
G
$$
. For a fi... | https://en.wikipedia.org/wiki/Associated_bundle |
passage: For example, if a language allows new types to be declared, a CFG cannot predict the names of such types nor the way in which they should be used. Even if a language has a predefined set of types, enforcing proper usage usually requires some context. Another example is duck typing, where the type of an element... | https://en.wikipedia.org/wiki/Abstract_syntax_tree |
passage: HealthLeaders Media, May 27, 2009.
Funding for primary care varies a great deal between different countries: general taxation, national insurance systems, private insurance and direct payment by patients are all used, sometimes in combination. The payment system for primary care physicians also varies. Some a... | https://en.wikipedia.org/wiki/Primary_care |
passage: In particular, one can ask whether the Hell–Nešetřil theorem can be extended to directed graphs. By the above theorem, this is equivalent to the Feder–Vardi conjecture (aka CSP conjecture, dichotomy conjecture) on CSP dichotomy, which states that for every constraint language Γ, CSP(Γ) is NP-complete or in P. ... | https://en.wikipedia.org/wiki/Graph_homomorphism |
passage: ## Modifications
- At the extreme values of the coefficient BM25 turns into ranking functions known as BM11 (for
$$
b=1
$$
) and BM15 (for
$$
b=0
$$
).
- BM25F (or the BM25 model with Extension to Multiple Weighted Fields) is a modification of BM25 in which the document is considered to be composed from sev... | https://en.wikipedia.org/wiki/Okapi_BM25 |
passage: Depending on the operating system, utility and remote file system, a file transfer might silently strip data streams. A safe way of copying or moving files is to use the BackupRead and BackupWrite system calls, which allow programs to enumerate streams, to verify whether each stream should be written to the de... | https://en.wikipedia.org/wiki/NTFS |
passage: Using Euler's formula and taking only the real part of the solution it is the same cosine solution for the 1 DOF system. The exponential solution is only used because it is easier to manipulate mathematically.
The equation then becomes:
$$
\begin{bmatrix}-\omega^2 \begin{bmatrix} M \end{bmatrix} + \begin{bm... | https://en.wikipedia.org/wiki/Vibration |
passage: Aureomycin was the best known of the second generation.
Lithium was discovered in the 19th century for nervous disorders and its possible mood-stabilizing or prophylactic effect; it was cheap and easily produced. As lithium fell out of favor in France, valpromide came into play. This antibiotic was the origin... | https://en.wikipedia.org/wiki/Medication |
passage: This is in fact the first printed version of Green's theorem in the form appearing in modern textbooks. George Green, An Essay on the Application of Mathematical Analysis to the Theories of Electricity and Magnetism (Nottingham, England: T. Wheelhouse, 1828). Green did not actually derive the form of "Green's ... | https://en.wikipedia.org/wiki/Green%27s_theorem |
passage: Another compounding factor was that fossils of apparently marine animals were found in parts of the world that were well above sea-level. Some suggested that these fossils had accumulated in horizontal layers under the sea and that subsequent tectonic activity had displaced them from their original positions. ... | https://en.wikipedia.org/wiki/Paleontology |
passage: But since
$$
f \in L^1(\mathbb R^n)
$$
, fact 5 says that
$$
\lim_{\varepsilon\to 0}(\varphi_{\varepsilon} * f) (x) = f(x).
$$
Putting together the above we have shown that
$$
\int_{\mathbb{R}^n} e^{2\pi i x\cdot\xi}(\mathcal{F}f)(\xi)\,d\xi = f(x). \qquad\square
$$
Fourier integral theorem
The theorem can be... | https://en.wikipedia.org/wiki/Fourier_inversion_theorem |
passage: Examples of such formulas encountered in practice can be very large, for example with 100,000 variables and 1,000,000 conjuncts.
A formula in CNF can be converted into an equisatisfiable formula in "kCNF" (for k≥3) by replacing each conjunct with more than k variables
$$
X_1 \vee \ldots \vee X_k \vee \ldots ... | https://en.wikipedia.org/wiki/Conjunctive_normal_form |
passage: With the emergence of biochemistry, classifications of organisms are now often based on DNA sequence data or a combination of DNA and morphology. Many systematists contend that only monophyletic taxa should be recognized as named groups. The degree to which classification depends on inferred evolutionary hi... | https://en.wikipedia.org/wiki/Phylogenetics |
passage: The Dirac
$$
\delta
$$
-function stands for the conservation of energy. In addition, the term
$$
\langle k|H'|k' \rangle
$$
, generally referred to as the matrix element, mathematically represents an inner product of the initial and final wave functions of the carrier:
$$
\langle k|H'|k' \rangle = \frac{1}{V... | https://en.wikipedia.org/wiki/Monte_Carlo_methods_for_electron_transport |
passage: Contours
Contours are the class of curves on which we define contour integration. A contour is a directed curve which is made up of a finite sequence of directed smooth curves whose endpoints are matched to give a single direction. This requires that the sequence of curves
$$
\gamma_1,\dots,\gamma_n
$$
be su... | https://en.wikipedia.org/wiki/Contour_integration |
passage: In quantum mechanics, it often occurs that little or no information about the inner product of two arbitrary (state) kets is present, while it is still possible to say something about the expansion coefficients and of those vectors with respect to a specific (orthonormalized) basis. In this case, it is part... | https://en.wikipedia.org/wiki/Bra%E2%80%93ket_notation |
passage: This can be corrected by dividing by the square root of the right hand side of the equation above, when
$$
n=m
$$
.
Although it does not yield an orthonormal basis, an alternative normalization is sometimes preferred due to its simplicity:
$$
P_n^{(\alpha, \beta)} (1) = {n+\alpha\choose n}.
$$
### Symmetry ... | https://en.wikipedia.org/wiki/Jacobi_polynomials |
passage: Let be a curve of class and let and denote the circular points at infinity. Draw the tangents to through each of and . There are two sets of lines which will have points of intersection, with exceptions in some cases due to singularities, etc. These points of intersection are the defined to be the foc... | https://en.wikipedia.org/wiki/Focus_%28geometry%29 |
passage: However, trial division is still used, with a smaller limit than the square root on the divisor size, to quickly discover composite numbers with small factors, before using more complicated methods on the numbers that pass this filter.
### Sieves
Before computers, mathematical tables listing all of the primes... | https://en.wikipedia.org/wiki/Prime_number |
passage:
$$
\text{s} = \int_{a}^{b} \sqrt{\mathrm{d}x^2+\mathrm{d}y^2} = \int_{a}^{b} \sqrt{1+y'^2}\,\mathrm{d}x,
$$
the integrand function being
$$
L(x,y, y') = \sqrt{1+y'^2}
$$
.
The partial derivatives of L are:
$$
\frac{\partial L(x, y, y')}{\partial y'} = \frac{y'}{\sqrt{1 + y'^2}} \quad \text{and} \quad
\frac{... | https://en.wikipedia.org/wiki/Euler%E2%80%93Lagrange_equation |
passage: ## History
The idea of an isoperimetric function for a finitely presented group goes back to the work of Max Dehn in 1910s. Dehn proved that the word problem for the standard presentation of the fundamental group of a closed oriented surface of genus at least two is solvable by what is now called Dehn's algori... | https://en.wikipedia.org/wiki/Dehn_function |
passage: They are probabilistically complete, meaning the probability that they will produce a solution approaches 1 as more time is spent. However, they cannot determine if no solution exists.
Given basic visibility conditions on Cfree, it has been proven that as the number of configurations N grows higher, the proba... | https://en.wikipedia.org/wiki/Motion_planning |
passage: Note that all irreducible representations belonging to the same isotype appear with a multiplicity equal to
$$
\dim (\text{Hom}_G(V_\eta,V_I))=\langle V_\eta,V_I \rangle_G.
$$
Let
$$
(\rho, V_\rho)
$$
be a representation of
$$
G,
$$
then there exists a canonical isomorphism
$$
T: \text{Hom}_G(V_\rho, I^G_... | https://en.wikipedia.org/wiki/Representation_theory_of_finite_groups |
passage: Viable inbred offspring are also likely to be inflicted with physical deformities and genetically inherited diseases. Studies have confirmed an increase in several genetic disorders due to inbreeding such as blindness, hearing loss, neonatal diabetes, limb malformations, disorders of sex development, schizophr... | https://en.wikipedia.org/wiki/Inbreeding |
passage: Actual impedances and admittances must be normalised before using them on a Smith chart. Once the result is obtained it may be de-normalised to obtain the actual result.
### The normalised impedance Smith chart
Using transmission-line theory, if a transmission line is terminated in an impedance (
$$
Z_\text{T... | https://en.wikipedia.org/wiki/Smith_chart |
passage: ### Calculations of moments
The moment-generating function is so called because if it exists on an open interval around , then it is the exponential generating function of the moments of the probability distribution:
$$
m_n = \operatorname{E}\left[ X^n \right] = M_X^{(n)}(0) = \left. \frac{d^n M_X}{dt^n}\righ... | https://en.wikipedia.org/wiki/Moment-generating_function |
passage: ### France
Léandre Pourcelot, a researcher and teacher at INSA (Institut National des Sciences Appliquées), Lyon, co-published a report in 1965 at the Académie des sciences, "Effet Doppler et mesure du débit sanguin" ("Doppler effect and measure of the blood flow"), the basis of his design of a Doppler flow me... | https://en.wikipedia.org/wiki/Medical_ultrasound |
passage: When a fluid is flowing in the x-direction parallel to a solid surface, the fluid has x-directed momentum, and its concentration is υxρ. By random diffusion of molecules there is an exchange of molecules in the z-direction. Hence the x-directed momentum has been transferred in the z-direction from the faster- ... | https://en.wikipedia.org/wiki/Transport_phenomena |
passage: In a pie chart, the arc length of each slice (and consequently its central angle and area), is proportional to the quantity it represents.
- For example, as shown in the graph to the right, the proportion of English native speakers worldwide Line chart- x position
- y position
- symbol/glyph
- color
- size- ... | https://en.wikipedia.org/wiki/Data_and_information_visualization |
passage: This led to Darwin adopting some Lamarckian ideas in later editions of On the Origin of Species and his later biological works. Darwin's primary approach to heredity was to outline how it appeared to work (noticing that traits that were not expressed explicitly in the parent at the time of reproduction could b... | https://en.wikipedia.org/wiki/Heredity |
passage: Cooperation can occur willingly between individuals when both benefit directly as well. Cooperative breeding, where one individual cares for the offspring of another, occurs in several species, including wedge-capped capuchin monkeys.
Cooperative behavior may also be enforced, where their failure to coopera... | https://en.wikipedia.org/wiki/Behavioral_ecology |
passage: Such a tensor
$$
A \in (K^{n})^{\otimes d}
$$
defines polynomial maps
$$
K^n \to K^n
$$
and
$$
\mathbb{P}^{n-1} \to \mathbb{P}^{n-1}
$$
with coordinates:
$$
\psi_i(x_1, \ldots, x_n) = \sum_{j_2=1}^n \sum_{j_3=1}^n \cdots \sum_{j_d = 1}^n a_{i j_2 j_3 \cdots j_d} x_{j_2} x_{j_3}\cdots x_{j_d} \;\; \mbox{... | https://en.wikipedia.org/wiki/Tensor_product |
passage: #### Isolated vesicles
Producing membrane vesicles is one of the methods to investigate various membranes of the cell. After the living tissue is crushed into suspension, various membranes form tiny closed bubbles. Big fragments of the crushed cells can be discarded by low-speed centrifugation and later the fr... | https://en.wikipedia.org/wiki/Vesicle_%28biology_and_chemistry%29 |
passage: In January 2017, the spiral was tweeted by Bernie Sanders and the U.S. National Park Service, both conveying how almost all recorded warmest years have been recent years.
A 2022 study emphasized the importance of user-centered design in climate data visualizations, highlighting tools like the climate spiral ... | https://en.wikipedia.org/wiki/Climate_spiral |
passage: For any term
$$
t
$$
,
$$
(\mathbf{\lambda} x . x x x) t \rightarrow t t t
$$
But consider what happens when we apply
$$
\lambda x . x x x
$$
to itself:
$$
\begin{align}
(\mathbf{\lambda} x . x x x) (\lambda x . x x x)
& \rightarrow (\mathbf{\lambda} x . x x x) (\lambda x . x x x) (\lambda x . x x x)
\\
& \... | https://en.wikipedia.org/wiki/Normal_form_%28abstract_rewriting%29 |
passage: gives several more precise versions of this result, called zero density estimates, which bound the number of zeros in regions with imaginary part at most T and real part at least .
### Hardy–Littlewood conjectures
In 1914 Godfrey Harold Hardy proved that
$$
\zeta\left(\tfrac{1}{2}+it\right)
$$
has infinitel... | https://en.wikipedia.org/wiki/Riemann_hypothesis |
passage: #### Natural language
Natural languages such as English have words for several Boolean operations, in particular conjunction (and), disjunction (or), negation (not), and implication (implies). But not is synonymous with and not. When used to combine situational assertions such as "the block is on the table" an... | https://en.wikipedia.org/wiki/Boolean_algebra |
passage: Another example is through the use of utility applications. Some AR applications, such as Augment, enable users to apply digital objects into real environments, allowing businesses to use augmented reality devices as a way to preview their products in the real world. Similarly, it can also be used to demo what... | https://en.wikipedia.org/wiki/Augmented_reality |
passage: Developers may request additional documentation such as a real-time video of the bug's manifestation.
- Analysis. The developer responsible for the bug, such as an artist, programmer or game designer checks the malfunction. This is outside the scope of game tester duties, although inconsistencies in the report... | https://en.wikipedia.org/wiki/Game_testing |
passage: If all circuit components were linear or the circuit was linearized beforehand, the equation system at this point is a system of linear equations and is solved with numerical linear algebra methods. Otherwise, it is a nonlinear algebraic equation system and is solved with nonlinear numerical methods such as Ro... | https://en.wikipedia.org/wiki/Network_analysis_%28electrical_circuits%29 |
passage: Much of corporate finance theory, by contrast, considers investment under "certainty" (Fisher separation theorem, "theory of investment value", and Modigliani–Miller theorem). Here, theory and methods are developed for the decisioning about funding, dividends, and capital structure discussed above. A recent de... | https://en.wikipedia.org/wiki/Finance |
passage: ### Factorial
For any positive integer n, the product of the integers less than or equal to n is a unary operation called factorial. In the context of complex numbers, the gamma function is a unary operation extension of factorial.
### Trigonometry
In trigonometry, the trigonometric functions, such as
$$
\si... | https://en.wikipedia.org/wiki/Unary_operation |
passage: The property of categoriality (categoricity) ensures the completeness of a system, however the converse is not true: Completeness does not ensure the categoriality (categoricity) of a system, since two models can differ in properties that cannot be expressed by the semantics of the system.
### Example
As an ... | https://en.wikipedia.org/wiki/Axiomatic_system |
passage: Rather, an external laser injects counter-propagating beams into an optical fiber ring, where rotation causes a relative phase shift between those beams when interfered after their pass through the fiber ring. The phase shift is proportional to the rate of rotation. This is less sensitive in a single traverse ... | https://en.wikipedia.org/wiki/Ring_laser_gyroscope |
passage: In reality, a diagnostic procedure may involve components of multiple methods.
### Differential diagnosis
The method of differential diagnosis is based on finding as many candidate diseases or conditions as possible that can possibly cause the signs or symptoms, followed by a process of elimination or at leas... | https://en.wikipedia.org/wiki/Medical_diagnosis |
passage: It is convenient to imagine this gravitational force concentrated at the center of mass of the object.
If an object with weight is displaced upwards or downwards a vertical distance , the work done on the object is:
$$
W = F_g (y_2 - y_1) = F_g\Delta y = mg\Delta y
$$
where Fg is weight (pounds in imperial ... | https://en.wikipedia.org/wiki/Work_%28physics%29 |
passage: - It may be more cost-efficient to obtain the desired level of performance by using a cluster of several low-end computers, in comparison with a single high-end computer.
Examples
Examples of distributed systems and applications of distributed computing include the following:
- telecommunications networks:
... | https://en.wikipedia.org/wiki/Distributed_computing |
passage: Finally, we substitute into our
$$
AX + UY = I
$$
, and we have
$$
AX + U\left(C^{-1} + VA^{-1}U\right)^{-1}VA^{-1} = I
$$
. Thus,
$$
(A + UCV)^{-1} = X = A^{-1} - A^{-1}U\left(C^{-1} + VA^{-1}U\right)^{-1}VA^{-1}.
$$
We have derived the Woodbury matrix identity.
We start by the matrix
$$
\begin{bmatrix} A ... | https://en.wikipedia.org/wiki/Woodbury_matrix_identity |
passage: People with schizophrenia perform worse on these behavioral tasks, which relate to perception and continuous recognition memory. The neurobiological basis of gamma dysfunction in schizophrenia is thought to lie with GABAergic interneurons involved in known brain wave rhythm-generating networks. Antipsychotic t... | https://en.wikipedia.org/wiki/Gamma_wave |
passage: || || || ||
|-
! 4
| 1 || 3 || 6 || 11 || 3 || || || || || || || || || || ||
|-
! 5
| 1 || 4 || 12 || 35 || 48 || 20 || || || || || || || || || ||
|-
! 6
| 1 || 5 || 20 || 79 || 199 || 281 || 133 || 2 || || || || || || || ||
|-
! 7
| 1 || 6 || 30 || 149 || 543 || 1357 || 1903 || 1016 || 35 || || || || || || ||... | https://en.wikipedia.org/wiki/Pancake_sorting |
passage: In constrained optimization, a field of mathematics, a barrier function is a continuous function whose value increases to infinity as its argument approaches the boundary of the feasible region of an optimization problem. Such functions are used to replace inequality constraints by a penalizing term in the ob... | https://en.wikipedia.org/wiki/Barrier_function |
passage: In particular, the above is equivalent to
$$
\Delta f = \frac{\partial^2 f}{\partial r^2} + \frac{2}{r}\frac{\partial f}{\partial r} + \frac{1}{r^2}\Delta_{S^2} f ,
$$
where
$$
\Delta_{S^2}f
$$
is the Laplace-Beltrami operator on the unit sphere.
In general curvilinear coordinates ():
$$
\Delta = \nabla \x... | https://en.wikipedia.org/wiki/Laplace_operator |
passage: ### Data mining
is the process of extracting and finding patterns in massive data sets involving methods at the intersection of machine learning, statistics, and database systems. Data mining is an interdisciplinary subfield of computer science and statistics with an overall goal of extracting information (wi... | https://en.wikipedia.org/wiki/Data_mining |
passage: According to proponents of climate justice, the costs of climate adaptation should be paid by those most responsible for climate change, while the beneficiaries of payments should be those suffering impacts. One way this can be addressed in practice is to have wealthy nations pay poorer countries to adapt.
Ox... | https://en.wikipedia.org/wiki/Climate_change |
passage: This is precisely the Fréchet derivative, and the same construction can be made to work for a function between any Banach spaces.
Another fruitful point of view is to define the differential directly as a kind of directional derivative:
$$
df(\mathbf{x},\mathbf{h}) = \lim_{t\to 0}\frac{f(\mathbf{x}+t\mathbf{h... | https://en.wikipedia.org/wiki/Differential_of_a_function |
passage: ### Marketing documentation
For many applications it is necessary to have some promotional materials to encourage casual observers to spend more time learning about the product. This form of documentation has three purposes:
1. To excite the potential user about the product and instill in them a desire to be... | https://en.wikipedia.org/wiki/Software_documentation |
passage: The singular points of a degenerate quadric are the points whose projective coordinates belong to the null space of the matrix .
A quadric is reducible if and only if the rank of is one (case of a double hyperplane) or two (case of two hyperplanes).
## Normal form of projective quadrics
In real projective ... | https://en.wikipedia.org/wiki/Quadric |
passage: So the electrons in the circuit flow in the opposite direction to the direction of conventional current.
From the standpoint of electric power, components in an electric circuit can be divided into two categories:
### Active devices (power sources)
If conventional electric current (positive charge) is forc... | https://en.wikipedia.org/wiki/Electric_power |
passage: $$
If the auxiliary worldsheet metric tensor
$$
\sqrt{-h}
$$
is calculated from the equations of motion:
$$
\sqrt{-h} = \frac{2 \sqrt{-G}}{h^{cd} G_{cd}}
$$
and substituted back to the action, it becomes the Nambu–Goto action:
$$
S = {T \over 2}\int \mathrm{d}^2 \sigma \sqrt{-h} h^{ab} G_{ab} = {T \over 2}\... | https://en.wikipedia.org/wiki/Polyakov_action |
passage: In mathematics, Hurwitz's theorem is a theorem of Adolf Hurwitz (1859–1919), published posthumously in 1923, solving the Hurwitz problem for finite-dimensional unital real non-associative algebras endowed with a nondegenerate positive-definite quadratic form. The theorem states that if the quadratic form defin... | https://en.wikipedia.org/wiki/Hurwitz%27s_theorem_%28composition_algebras%29 |
passage: Expanding both numerators on the right hand side of this formula into sums of divisors of
$$
n_i
$$
results in the desired Egyptian fraction representation. use a similar technique involving a different sequence of practical numbers to show that every rational number
$$
x/y
$$
has an Egyptian fraction re... | https://en.wikipedia.org/wiki/Practical_number |
passage: Smooth normals are specified per vertex. .ply Polygon File Format Stanford University Various Binary and ASCII .pmd Polygon Movie Maker data Yu Higuchi MikuMikuDance Proprietary binary file format for storing humanoid model geometry with rigging, material, and physics information. .stl Stereolithog... | https://en.wikipedia.org/wiki/Polygon_mesh |
passage: Some but not all polynomial equations with rational coefficients have a solution that is an algebraic expression that can be found using a finite number of operations that involve only those same types of coefficients (that is, can be solved algebraically). This can be done for all such equations of degree one... | https://en.wikipedia.org/wiki/Algebraic_equation |
passage: The first test of Newton's law of gravitation between masses in the laboratory was the Cavendish experiment conducted by the British scientist Henry Cavendish in 1798. It took place 111 years after the publication of Newton's Principia and approximately 71 years after his death.
Newton's law of gravitation re... | https://en.wikipedia.org/wiki/Newton%27s_law_of_universal_gravitation |
passage: &= 963^3 + 804^3 \\
&= 1134^3 - 357^3 \\
&= 1155^3 - 504^3 \\
&= 1246^3 - 805^3 \\
&= 2115^3 - 2004^3 \\
&= 4746^3 - 4725^3 \\[6pt]
\mathrm{Cabtaxi}(7) =& \ 11302198488 \\
&= 1926^3 + 1608^3 \\
&= 1939^3 + 1589^3 \\
&= 2268^3 - 714^3 \\
&= 2310^3 - 1008^3 \\
&= 2492^3 - 1610^3 \\
&= 423... | https://en.wikipedia.org/wiki/Cabtaxi_number |
passage: The field operators transform under Lorentz transformations according to the spin of the particle that they create, by definition.
Additionally, the assumption (known as microcausality) that spacelike-separated fields either commute or anticommute can be made only for relativistic theories with a time directi... | https://en.wikipedia.org/wiki/Spin%E2%80%93statistics_theorem |
passage: {|
|
$$
p_{0,0}(x) = y_0 \,
$$
|-
| ||
$$
p_{0,1}(x) \,
$$
|-
|
$$
p_{1,1}(x) = y_1 \,
$$
|| ||
$$
p_{0,2}(x) \,
$$
|-
| ||
$$
p_{1,2}(x) \,
$$
|| ||
$$
p_{0,3}(x) \,
$$
|-
|
$$
p_{2,2}(x) = y_2 \,
$$
|| ||
$$
p_{1,3}(x) \,
$$
|| || style="border: 1px solid;" |
$$
p_{0,4}(x) \,
$$
|-
| ||
$$
p_{... | https://en.wikipedia.org/wiki/Neville%27s_algorithm |
passage: ## Proof that the Moore plane is not normal
The fact that this space
$$
\Gamma
$$
is not normal can be established by the following counting argument (which is very similar to the argument that the Sorgenfrey plane is not normal):
1. On the one hand, the countable set
$$
S:=\{(p,q) \in \mathbb Q\times \mat... | https://en.wikipedia.org/wiki/Moore_plane |
passage: Similarly, one can derive an equivalent formula for identical charged particles of charge in a uniform electric field of magnitude , where is replaced with the electrostatic force . Equating these two expressions yields the Einstein relation for the diffusivity, independent of or or other such forces:
$$
\... | https://en.wikipedia.org/wiki/Brownian_motion |
passage: These are often called Calabi–Yau manifolds. However, the term is often used in slightly different ways by various authors — for example, some uses may refer to the complex manifold while others might refer to a complex manifold together with a particular Ricci-flat Kähler metric.
This special case can equiva... | https://en.wikipedia.org/wiki/Calabi_conjecture |
passage: ReRAM involves generating defects in a thin oxide layer, known as oxygen vacancies (oxide bond locations where the oxygen has been removed), which can subsequently charge and drift under an electric field. The motion of oxygen ions and vacancies in the oxide would be analogous to the motion of electrons and ho... | https://en.wikipedia.org/wiki/Non-volatile_memory |
passage: The proof sketched in the previous paragraph that the consistency of ZFC implies the consistency of ZFC + "there is not an inaccessible cardinal" can be formalized in ZFC. However, assuming that ZFC is consistent, no proof that the consistency of ZFC implies the consistency of ZFC + "there is an inaccessible c... | https://en.wikipedia.org/wiki/Inaccessible_cardinal |
passage: Then the family of waves in question consists of all functions
$$
F
$$
that satisfy those constraints – that is, all solutions of the equation.
This approach is extremely important in physics, because the constraints usually are a consequence of the physical processes that cause the wave to evolve. For exam... | https://en.wikipedia.org/wiki/Wave |
passage: Given a topological space ... a atlas is a collection of charts such that covers , and such that for all and in , the transition map is a map a smooth or atlas a smooth map an analytic or atlas a real-analytic map a holomorphic atlas a holomorphic map
Since every real-analytic map is smooth, and ... | https://en.wikipedia.org/wiki/Differentiable_manifold |
passage: In a ring R, the set R itself forms a two-sided ideal of R called the unit ideal. It is often also denoted by
$$
(1)
$$
since it is precisely the two-sided ideal generated (see below) by the unity . Also, the set
$$
\{ 0_R \}
$$
consisting of only the additive identity 0R forms a two-sided ideal called the... | https://en.wikipedia.org/wiki/Ideal_%28ring_theory%29 |
passage: Optimize the whole
Modern software systems are not simply the sum of their parts, but also the product of their interactions. Defects in software tend to accumulate during the development process – by decomposing the big tasks into smaller tasks, and by standardizing different stages of development, the root c... | https://en.wikipedia.org/wiki/Lean_software_development |
passage: Dirichlet's theorem on primes in arithmetic progressions then tells us
that
$$
\pi(x;q,a) \sim \frac{\pi(x)}{\varphi(q)}\ \ (x\rightarrow\infty)
$$
where
$$
\varphi
$$
is Euler's totient function. If we then define the error function
$$
E(x;q) = \max_{\text{gcd}(a,q) = 1} \left|\pi(x;q,a) - \frac{\pi(x)}{... | https://en.wikipedia.org/wiki/Elliott%E2%80%93Halberstam_conjecture |
passage: Another possible way of verifying computer-aided proofs is to generate their reasoning steps in a machine readable form, and then use a proof checker program to demonstrate their correctness. Since validating a given proof is much easier than finding a proof, the checker program is simpler than the original as... | https://en.wikipedia.org/wiki/Computer-assisted_proof |
passage: For any single operation, the bottom-up technique is advantageous if the number of downward movements is at least of the height of the tree (when the number of comparisons is times the height for both techniques), and it turns out that this is more than true on average, even for worst-case inputs.
A naïve ... | https://en.wikipedia.org/wiki/Heapsort |
passage: The two writes could have been done by the same processor or by different processors.
As in sequential consistency, reads do not need to reflect changes instantaneously, however, they need to reflect all changes to a variable sequentially.
Sequence 1 2 3 4 1 W(x)1 R(x)1 R(x)1 R(x)1 2 W(x)2 3 W(x)3 R(x)3 R(... | https://en.wikipedia.org/wiki/Consistency_model |
passage: $$
- (See Integral of the secant function. This result was a well-known conjecture in the 17th century.)
-
$$
\int \csc{x} \, dx = -\ln{\left| \csc{x} + \cot{x}\right|} + C = \ln{\left| \csc{x} - \cot{x}\right|} + C = \ln{\left| \tan {\frac{x}{2}} \right|} + C
$$
-
$$
\int \sec^2 x \, dx = \tan x + C
$$
-... | https://en.wikipedia.org/wiki/Lists_of_integrals |
passage: Set a tag array equal to the original array size and initialize to a false value.
1. [Main Sort] Determines whether all buckets of the original array have been sorted. If the sorting is not completed, the [Divide function] is executed.
1. [Divide function] Find the maximum and minimum values in the bucket. I... | https://en.wikipedia.org/wiki/Interpolation_sort |
passage: Designers do not work this way – extensive empirical evidence has demonstrated that designers do not act as the rational model suggests.
1. Unrealistic assumptions – goals are often unknown when a design project begins, and the requirements and constraints continue to change.
### Action-centric model
The act... | https://en.wikipedia.org/wiki/Design |
passage: In particular, the estimated effects may be biased if CCTV is introduced in response to crime trends.
In 2012, cities such as Manchester in the UK are using DVR-based technology to improve accessibility for crime prevention. In 2013, City of Philadelphia Auditor found that the $15 million system was operation... | https://en.wikipedia.org/wiki/Closed-circuit_television |
passage: This is a general issue with area graphs, and area is hard to judge – see "Cleveland's hierarchy". (summarized)
For example, the alternating data 9, 1, 9, 1, 9, 1 yields a spiking radar chart (which goes in and out), while reordering the data as 9, 9, 9, 1, 1, 1 instead yields two distinct wedges (sectors).
... | https://en.wikipedia.org/wiki/Radar_chart |
passage: Using the same logic, Alice also steps forward on turn two.
Assume that there is a third child, Charlie. If only Alice is muddy (
$$
X=1
$$
), she will see no muddy faces and will step forward on turn one. If both Alice and Bob are muddy (
$$
X=2
$$
), neither can step forward on turn one but each will know b... | https://en.wikipedia.org/wiki/Induction_puzzles |
passage: In the Jordan normal form, we have written
$$
V = \bigoplus_{i = 1}^r V_i
$$
where
$$
r
$$
is the number of Jordan blocks and
$$
x |_{V_i}
$$
is one Jordan block. Now let
$$
f(t) = \operatorname{det}(t I - x)
$$
be the characteristic polynomial of
$$
x
$$
. Because
$$
f
$$
splits, it can be written ... | https://en.wikipedia.org/wiki/Jordan%E2%80%93Chevalley_decomposition |
passage: Here is a table showing the conditional probabilities of being hit, depending on the state of the lights. (Note that the columns in this table must add up to 1 because the probability of being hit or not hit is 1 regardless of the state of the light.)
+ Conditional distribution: RedYellowGreen Not Hit 0.99 0.... | https://en.wikipedia.org/wiki/Marginal_distribution |
passage: The Kaplansky density theorem can be used to formulate some approximations with respect to the strong operator topology.
1) If h is a positive operator in (A−)1, then h is in the strong-operator closure of the set of self-adjoint operators in (A+)1, where A+ denotes the set of positive operators in A.
2) If A... | https://en.wikipedia.org/wiki/Kaplansky_density_theorem |
passage: If there is only one page table, different applications running at the same time use different parts of a single range of virtual addresses. If there are multiple page or segment tables, there are multiple virtual address spaces and concurrent applications with separate page tables redirect to different real a... | https://en.wikipedia.org/wiki/Virtual_memory |
passage: // E (1, 2)4.// E (3)5.// I (2, 4) (discharging 2)6. // I (1, 5) (discharging 1)
- Import-export:
$$
P \to (Q \to R) \equiv (P \land Q) \to R
$$
- Negated conditionals:
$$
\neg(P \to Q) \equiv P \land \neg Q
$$
- Or-and-if:
$$
P \to Q \equiv \neg P \lor Q
$$
- Commutativity of antecedents:
$$
\big(P \to (Q... | https://en.wikipedia.org/wiki/Material_conditional |
passage: ATP is synthesized by the ATP synthase enzyme when the chemiosmotic gradient is used to drive the phosphorylation of ADP. The electrons are finally transferred to exogenous oxygen and, with the addition of two protons, water is formed.
## Efficiency of ATP production
The table below describes the reactions in... | https://en.wikipedia.org/wiki/Cellular_respiration |
passage: Let A0, A1, ... An be vector fields on Rd. They are said to satisfy Hörmander's condition if, for every point x ∈ Rd, the vectors
$$
\begin{align}
&A_{j_0} (x)~,\\
&[A_{j_{0}} (x), A_{j_{1}} (x)]~,\\
&[[A_{j_{0}} (x), A_{j_{1}} (x)], A_{j_{2}} (x)]~,\\
&\quad\vdots\quad
\end{align}
\qquad 0 \leq j_{0}, j_{1}... | https://en.wikipedia.org/wiki/H%C3%B6rmander%27s_condition |
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