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passage: Then, linearization gives
$$
\omega(k) \approx \omega_0 + \left(k - k_0\right)\omega'_0
$$
where
$$
\omega_0 = \omega(k_0)
$$
and
$$
\omega'_0 = \left.\frac{\partial \omega(k)}{\partial k}\right|_{k=k_0}
$$
(see next section for discussion of this step). Then, after some algebra,
$$
\alpha(x,t) = e^{i\left(k_0 x - \omega_0 t\right)}\int_{-\infty}^\infty dk \, A(k) e^{i(k - k_0)\left(x - \omega'_0 t\right)}.
$$
There are two factors in this expression. The first factor,
$$
e^{i\left(k_0 x - \omega_0 t\right)}
$$
, describes a perfect monochromatic wave with wavevector , with peaks and troughs moving at the phase velocity
$$
\omega_0/k_0
$$
within the envelope of the wavepacket.
The other factor,
$$
\int_{-\infty}^\infty dk \, A(k) e^{i(k - k_0)\left(x - \omega'_0 t\right)}
$$
,
gives the envelope of the wavepacket.
|
https://en.wikipedia.org/wiki/Group_velocity
|
passage: If, instead of a field, the entries are supposed to belong to a ring, then one must add the condition that belongs to the center of the ring.
One special case where commutativity does occur is when and are two (square) diagonal matrices (of the same size); then . Again, if the matrices are over a general ring rather than a field, the corresponding entries in each must also commute with each other for this to hold.
### Distributivity
The matrix product is distributive with respect to matrix addition. That is, if are matrices of respective sizes , , , and , one has (left distributivity)
$$
\mathbf{A}(\mathbf{B} + \mathbf{C}) = \mathbf{AB} + \mathbf{AC},
$$
and (right distributivity)
$$
(\mathbf{B} + \mathbf{C} )\mathbf{D} = \mathbf{BD} + \mathbf{CD}.
$$
This results from the distributivity for coefficients by
$$
\sum_k a_{ik}(b_{kj} + c_{kj}) = \sum_k a_{ik}b_{kj} + \sum_k a_{ik}c_{kj}
$$
$$
\sum_k (b_{ik} + c_{ik}) d_{kj} = \sum_k b_{ik}d_{kj} + \sum_k c_{ik}d_{kj}.
$$
|
https://en.wikipedia.org/wiki/Matrix_multiplication
|
passage: ### Topics related to first-order theories
- Compactness theorem
- Consistent set
- Deduction theorem
- Lindenbaum's lemma
- Löwenheim–Skolem theorem
## Examples
One way to specify a theory is to define a set of axioms in a particular language. The theory can be taken to include just those axioms, or their logical or provable consequences, as desired. Theories obtained this way include ZFC and Peano arithmetic.
A second way to specify a theory is to begin with a structure, and let the theory be the set of sentences that are satisfied by the structure. This is a method for producing complete theories through the semantic route, with examples including the set of true sentences under the structure (N, +, ×, 0, 1, =), where N is the set of natural numbers, and the set of true sentences under the structure (R, +, ×, 0, 1, =), where R is the set of real numbers. The first of these, called the theory of true arithmetic, cannot be written as the set of logical consequences of any enumerable set of axioms.
The theory of (R, +, ×, 0, 1, =) was shown by Tarski to be decidable; it is the theory of real closed fields (see Decidability of first-order theories of the real numbers for more).
|
https://en.wikipedia.org/wiki/Theory_%28mathematical_logic%29
|
passage: Then, ISGD is equivalent to:
$$
w^\text{new} = w^\text{old} + \xi^\ast x_i,~\text{where}~\xi^\ast = f(\xi^\ast).
$$
The scaling factor
$$
\xi^\ast\in\mathbb{R}
$$
can be found through the bisection method since in most regular models, such as the aforementioned generalized linear models, function
$$
q()
$$
is decreasing, and thus the search bounds for
$$
\xi^\ast
$$
are
$$
[\min(0, f(0)), \max(0, f(0))]
$$
.
### Momentum
Further proposals include the momentum method or the heavy ball method, which in ML context appeared in Rumelhart, Hinton and Williams' paper on backpropagation learning and borrowed the idea from Soviet mathematician Boris Polyak's 1964 article on solving functional equations. Stochastic gradient descent with momentum remembers the update at each iteration, and determines the next update as a linear combination of the gradient and the previous update:
$$
\Delta w := \alpha \Delta w - \eta\, \nabla Q_i(w)
$$
$$
w := w + \Delta w
$$
that leads to:
$$
w := w - \eta\, \nabla Q_i(w) + \alpha \Delta w
$$
where the parameter
$$
w
$$
which minimizes
$$
Q(w)
$$
is to be estimated,
$$
\eta
$$
is a step size (sometimes called the learning rate in machine learning) and
$$
\alpha
$$
is an exponential decay factor between 0 and 1 that determines the relative contribution of the current gradient and earlier gradients to the weight change.
|
https://en.wikipedia.org/wiki/Stochastic_gradient_descent
|
passage: Hot cathode
A hot-cathode ionization gauge is composed mainly of three electrodes acting together as a triode, wherein the cathode is the filament. The three electrodes are a collector or plate, a filament, and a grid. The collector current is measured in picoamperes by an electrometer. The filament voltage to ground is usually at a potential of 30 volts, while the grid voltage at 180–210 volts DC, unless there is an optional electron bombardment feature, by heating the grid, which may have a high potential of approximately 565 volts.
The most common ion gauge is the hot-cathode Bayard–Alpert gauge, with a small ion collector inside the grid. A glass envelope with an opening to the vacuum can surround the electrodes, but usually the nude gauge is inserted in the vacuum chamber directly, the pins being fed through a ceramic plate in the wall of the chamber. Hot-cathode gauges can be damaged or lose their calibration if they are exposed to atmospheric pressure or even low vacuum while hot. The measurements of a hot-cathode ionization gauge are always logarithmic.
Electrons emitted from the filament move several times in back-and-forth movements around the grid before finally entering the grid. During these movements, some electrons collide with a gaseous molecule to form a pair of an ion and an electron (electron ionization).
|
https://en.wikipedia.org/wiki/Pressure_measurement
|
passage: ### Bearing uncertainty due to noise
Many of the causes of bearing error, such as mechanical imperfections in the antenna structure, poor gain matching of receiver gains, or non-ideal antenna gain patterns may be compensated by calibration procedures and corrective look-up tables, but thermal noise will always be a degrading factor. As all systems generate thermal noiseSchwartz M., "Information Transmission, Modulation and Noise", McGraw-Hill, N.Y.,4th Ed., 1990, p.525 then, when the level of the incoming signal is low, the signal-to-noise ratios in the receiver channels will be poor, and the accuracy of the bearing prediction will suffer.
In general, a guide to bearing uncertainty is given by Al-Sharabi K.I.A. and Muhammad D.F., "Design of Wideband Radio Direction Finder Based on Amplitude Comparison", Al-Rafidain Engineering, Vol. 19, Oct 2011, pp.77-86 (Find at: www.iasj.net/iasj?func=fulltext&aid=26752 )>
$$
\Delta \phi_{RMS} = 0.724 \frac{2. \Psi_0}{ \sqrt{SNR_0}}
$$
degrees
for a signal at crossover, but where SNR0 is the signal-to-noise ratio that would apply at boresight.
To obtain more precise predictions at a given bearing, the actual S:N ratios of the signals of interest are used.
|
https://en.wikipedia.org/wiki/Direction_finding
|
passage: The three coordinates form the 3d position vector, written as a column vector
$$
\vec{x}(t) = \begin{bmatrix} x^1(t) \\[0.7ex] x^2(t) \\[0.7ex] x^3(t) \end{bmatrix} \, .
$$
The components of the velocity
$$
\vec{u}
$$
(tangent to the curve) at any point on the world line are
$$
\vec{u} = \begin{bmatrix} u^1 \\ u^2 \\ u^3 \end{bmatrix} = \frac{d \vec{x}}{dt} =
\begin{bmatrix} \tfrac{dx^1}{dt} \\ \tfrac{dx^2}{dt} \\ \tfrac{dx^3}{dt} \end{bmatrix}.
$$
Each component is simply written
$$
u^i = \frac{dx^i}{dt}
$$
## Theory of relativity
In Einstein's theory of relativity, the path of an object moving relative to a particular frame of reference is defined by four coordinate functions , where is a spacetime index which takes the value 0 for the timelike component, and 1, 2, 3 for the spacelike coordinates. The zeroth component is defined as the time coordinate multiplied by ,
$$
x^0 = ct\,,
$$
Each function depends on one parameter τ called its proper time.
|
https://en.wikipedia.org/wiki/Four-velocity
|
passage: ## Classes of Monte Carlo and Las Vegas algorithms
Randomized algorithms are primarily divided by its two main types, Monte Carlo and Las Vegas, however, these represent only a top of the hierarchy and can be further categorized.
- Las Vegas
- Sherwood—"performant and effective special case of Las Vegas"
- Numerical—"numerical Las Vegas"
- Monte Carlo
- Atlantic City—"bounded error special case of Monte Carlo"
- Numerical—"numerical approximation Monte Carlo"
"Both Las Vegas and Monte Carlo are dealing with decisions, i.e., problems in their decision version." "This however should not give a wrong impression and confine these algorithms to such problems—both types of randomized algorithms can be used on numerical problems as well, problems where the output is not simple ‘yes’/‘no’, but where one needs to receive a result that is numerical in nature."
+Comparison of Las Vegas and Monte Carlo algorithmsEfficiencyOptimumFailure (LV) / Error (MC)Las Vegas (LV)ProbabilisticCertainSherwoodCertain, or Sherwood probabilistic
(stronger bound than regular LV)Certain0NumericalProbabilistic, certain, or
Sherwood probabilisticCertainor 0Monte Carlo (MC)CertainProbabilistic(probability which through repeated runs grows sub-exponentially
will inhibit usefulness of the algorithm; typical case is )Atlantic CityCertainProbabilisticNumericalCertainProbabilistic(algorithm type dependent)
Previous table represents a general framework for Monte Carlo and Las Vegas randomized algorithms.
|
https://en.wikipedia.org/wiki/Monte_Carlo_algorithm
|
passage: In addition to causing power loss, in resonant circuits this can reduce the Q factor of the circuit, broadening the bandwidth. In RF inductors specialized construction techniques are used to minimize these losses. The losses are due to these effects:
- Skin effect: The resistance of a wire to high frequency current is higher than its resistance to direct current because of skin effect. Due to induced eddy currents, radio frequency alternating current does not penetrate far into the body of a conductor but travels along its surface. For example, at 6 MHz the skin depth of copper wire is about 0.001 inches (25 μm); most of the current is within this depth of the surface. Therefore, in a solid wire, the interior portion of the wire may carry little current, effectively increasing its resistance.
- Proximity effect: Another similar effect that also increases the resistance of the wire at high frequencies is proximity effect, which occurs in parallel wires that lie close to each other. The individual magnetic field of adjacent turns induces eddy currents in the wire of the coil, which causes the current density in the conductor to be displaced away from the adjacent surfaces. Like skin effect, this reduces the effective cross-sectional area of the wire conducting current, increasing its resistance.
- Dielectric losses: The high frequency electric field near the conductors in a tank coil can cause the motion of polar molecules in nearby insulating materials, dissipating energy as heat.
|
https://en.wikipedia.org/wiki/Inductor
|
passage: ## Transcription factor databases
There are numerous databases cataloging information about transcription factors, but their scope and utility vary dramatically. Some may contain only information about the actual proteins, some about their binding sites, or about their target genes. Examples include the following:
- footprintDB - a metadatabase of multiple databases, including JASPAR and others
- JASPAR: database of transcription factor binding sites for eukaryotes
- PlantTFD: Plant transcription factor database
- TcoF-DB: Database of transcription co-factors and transcription factor interactions
- TFcheckpoint: database of human, mouse and rat TF candidates
- transcriptionfactor.org (now commercial, selling reagents)
- MethMotif.org: An integrative cell-specific database of transcription factor binding motifs coupled with DNA methylation profiles.
|
https://en.wikipedia.org/wiki/Transcription_factor
|
passage: This is the case of:
- The square root is defined only for nonnegative values of the variable, and not differentiable at 0 (it is differentiable for all positive values of the variable).
## General definition
A real-valued function of a real variable is a function that takes as input a real number, commonly represented by the variable x, for producing another real number, the value of the function, commonly denoted f(x). For simplicity, in this article a real-valued function of a real variable will be simply called a function. To avoid any ambiguity, the other types of functions that may occur will be explicitly specified.
Some functions are defined for all real values of the variables (one says that they are everywhere defined), but some other functions are defined only if the value of the variable is taken in a subset X of
$$
\mathbb{R}
$$
, the domain of the function, which is always supposed to contain an interval of positive length. In other words, a real-valued function of a real variable is a function
$$
f: X \to \R
$$
such that its domain X is a subset of
$$
\mathbb{R}
$$
that contains an interval of positive length.
A simple example of a function in one variable could be:
$$
f : X \to \R
$$
$$
X = \{ x \in \R \,:\, x \geq 0\}
$$
$$
f(x) = \sqrt{x}
$$
which is the square root of x.
|
https://en.wikipedia.org/wiki/Function_of_a_real_variable
|
passage: ### Capacitance instability
The capacitance of certain capacitors decreases as the component ages. In ceramic capacitors, this is caused by degradation of the dielectric. The type of dielectric, ambient operating and storage temperatures are the most significant aging factors, while the operating voltage usually has a smaller effect, i.e., usual capacitor design is to minimize voltage coefficient. The aging process may be reversed by heating the component above the Curie point. Aging is fastest near the beginning of life of the component, and the device stabilizes over time. Electrolytic capacitors age as the electrolyte evaporates. In contrast with ceramic capacitors, this occurs towards the end of life of the component.
Temperature dependence of capacitance is usually expressed in parts per million (ppm) per °C. It can usually be taken as a broadly linear function but can be noticeably non-linear at the temperature extremes. The temperature coefficient may be positive or negative, depending mostly on the dielectric material. Some, designated C0G/NP0, but called NPO, have a somewhat negative coefficient at one temperature, positive at another, and zero in between. Such components may be specified for temperature-critical circuits.
Capacitors, especially ceramic capacitors, and older designs such as paper capacitors, can absorb sound waves resulting in a microphonic effect. Vibration moves the plates, causing the capacitance to vary, in turn inducing AC current. Some dielectrics also generate piezoelectricity.
|
https://en.wikipedia.org/wiki/Capacitor
|
passage: Reducing or eliminating these extra steps means the product will be completed sooner and with less wasted material in the process. During the design and prototyping process, potential issues in the design can be corrected earlier in the product development stages to further reduce the production time frame.
The benefits of concurrent design and manufacturing can be sorted in to short term and long term.
### Short term benefits
- Competitive advantage with implementing part into market quickly
- Large amounts of same part produced in a shorter amount of time
- Allows for early correction of part
- Less material wasted
- Less time spent on multiple iterations of essentially the same part
### Long term benefits
- More cost efficient over several parts produced and several years
- Large amounts of different parts produced in a shorter total amount of time
- Better communication between disciplines in company
- Ability to leverage teamwork and make informed decisions
## Using C.E.
Currently, several companies, agencies and universities use CE. Among them can be mentioned:
- Concurrent Design Facility - European Space Agency
- NASA Team X - Jet Propulsion Laboratory
- NASA Integrated Design Center (IDC), Mission Design Lab (MDL), and Instrument Design Lab (IDL) - Goddard Space Flight Center
- NASA Compass Team - Glenn Research Center
- CNES – French Space Agency
- ASI – Italian Space Agency
- Boeing
- EADS Astrium – Satellite Design Office
- Thales Alenia Space
- The Aerospace Corporation Concept Design Center
- STV Incorporated
- German Aerospace Center Concurrent Engineering Facility
- EPFL Space Center
- Schlumberger
- Harley Davidson
- ASML
- Fiberthree
- Toyota
|
https://en.wikipedia.org/wiki/Concurrent_engineering
|
passage: = 0x428a2f98d728ae22, 0x7137449123ef65cd, 0xb5c0fbcfec4d3b2f, 0xe9b5dba58189dbbc, 0x3956c25bf348b538,
0x59f111f1b605d019, 0x923f82a4af194f9b, 0xab1c5ed5da6d8118, 0xd807aa98a3030242, 0x12835b0145706fbe,
0x243185be4ee4b28c, 0x550c7dc3d5ffb4e2, 0x72be5d74f27b896f, 0x80deb1fe3b1696b1, 0x9bdc06a725c71235,
0xc19bf174cf692694, 0xe49b69c19ef14ad2, 0xefbe4786384f25e3, 0x0fc19dc68b8cd5b5, 0x240ca1cc77ac9c65,
0x2de92c6f592b0275, 0x4a7484aa6ea6e483, 0x5cb0a9dcbd41fbd4, 0x76f988da831153b5, 0x983e5152ee66dfab,
0xa831c66d2db43210, 0xb00327c898fb213f, 0xbf597fc7beef0ee4, 0xc6e00bf33da88fc2, 0xd5a79147930aa725,
|
https://en.wikipedia.org/wiki/SHA-2
|
passage: ## Name
The name is derived from cereal germ in a continuation of the sheaf metaphor, as a germ is (locally) the "heart" of a function, as it is for a grain.
## Formal definition
### Basic definition
Given a point x of a topological space X, and two maps
$$
f, g: X \to Y
$$
(where Y is any set), then
$$
f
$$
and
$$
g
$$
define the same germ at x if there is a neighbourhood U of x such that restricted to U, f and g are equal; meaning that
$$
f(u)=g(u)
$$
for all u in U.
Similarly, if S and T are any two subsets of X, then they define the same germ at x if there is again a neighbourhood U of x such that
$$
S \cap U = T \cap U.
$$
It is straightforward to see that defining the same germ at x is an equivalence relation (be it on maps or sets), and the equivalence classes are called germs (map-germs, or set-germs accordingly). The equivalence relation is usually written
$$
f \sim_x g \quad \text{or} \quad S \sim_x T.
$$
Given a map f on X, then its germ at x is usually denoted [f]x. Similarly, the germ at x of a set S is written [S]x.
|
https://en.wikipedia.org/wiki/Germ_%28mathematics%29
|
passage: In contrast, the density matrix of for the pure product state
$$
|\psi\rangle_A \otimes |\phi\rangle_B
$$
discussed above is
$$
\rho_A = |\psi\rangle_A \langle\psi|_A,
$$
the projection operator onto
$$
|\psi\rangle_A
$$
.
In general, a bipartite pure state ρ is entangled if and only if its reduced states are mixed rather than pure.
### Entanglement as a resource
In quantum information theory, entangled states are considered a 'resource', i.e., something costly to produce and that allows implementing valuable transformations. The setting in which this perspective is most evident is that of "distant labs", i.e., two quantum systems labelled "A" and "B" on each of which arbitrary quantum operations can be performed, but which do not interact with each other quantum mechanically. The only interaction allowed is the exchange of classical information, which combined with the most general local quantum operations gives rise to the class of operations called LOCC (local operations and classical communication). These operations do not allow the production of entangled states between systems A and B. But if A and B are provided with a supply of entangled states, then these, together with LOCC operations can enable a larger class of transformations.
If Alice and Bob share an entangled state, Alice can tell Bob over a telephone call how to reproduce a quantum state
$$
|\Psi\rangle
$$
she has in her lab.
|
https://en.wikipedia.org/wiki/Quantum_entanglement
|
passage: The second convergence condition was formulated by Johnson et al. in 2006, when the spectral radius of the matrix
$$
\rho (I - |D^{-1/2}AD^{-1/2}|) < 1 \,
$$
where D = diag(A). Later, Su and Wu established the necessary and sufficient convergence conditions for synchronous GaBP and damped GaBP, as well as another sufficient convergence condition for asynchronous GaBP. For each case, the convergence condition involves verifying 1) a set (determined by A) being non-empty, 2) the spectral radius of a certain matrix being smaller than one, and 3) the singularity issue (when converting BP message into belief) does not occur.
The GaBP algorithm was linked to the linear algebra domain, and it was shown that the GaBP algorithm can be viewed as an iterative algorithm for solving the linear system of equations Ax = b where A is the information matrix and b is the shift vector. Empirically, the GaBP algorithm is shown to converge faster than classical iterative methods like the Jacobi method, the Gauss–Seidel method, successive over-relaxation, and others. Additionally, the GaBP algorithm is shown to be immune to numerical problems of the preconditioned conjugate gradient method
## Syndrome-based BP decoding
The previous description of BP algorithm is called the codeword-based decoding, which calculates the approximate marginal probability
$$
P(x|X)
$$
, given received codeword
$$
X
$$
.
|
https://en.wikipedia.org/wiki/Belief_propagation
|
passage: Determine the inverse Z-transform of the following by series expansion method,
Eliminating negative powers if
$$
z
$$
and dividing by
$$
z
$$
,
$$
\frac{X(z)}{z} = \frac{z^2}{z(z^2 - 1.5z + 0.5)} = \frac{z}{z^2 - 1.5z + 0.5}
$$
By Partial Fraction Expansion,
$$
\begin{align}
\frac{X(z)}{z} &= \frac{z}{(z-1)(z-0.5)} = \frac{A_1}{z-0.5} + \frac{A_2}{z-1} \\[4pt]
& A_1 = \left. \frac{(z-0.5) X(z)}{z} \right\vert_{z=0.5} = \frac{0.5}{(0.5-1)} = -1 \\[4pt]
& A_2 = \left. \frac{(z-1) X(z)}{z} \right\vert_{z=1} = \frac{1}{1-0.5} = {2} \\[4pt]
\frac{X(z)}{z} &= \frac{2}{z-1} - \frac{1}{z-0.5}
\end{align}
$$
Case 1:
ROC:
$$
\left\vert Z \right\vert > 1
$$
Both the terms are causal, hence
$$
x(n)
$$
is causal.
$$
\begin{align}
x(n) &= 2{(1)^n}u(n) - 1{(0.5)^n}u(n) \\
&= (2-0.5^n) u(n) \\
\end{align}
$$
Case 2:
ROC:
$$
\left\vert Z \right\vert < 0.5
$$
Both the terms are anticausal, hence
$$
x(n)
$$
is anticausal.
|
https://en.wikipedia.org/wiki/Z-transform
|
passage: When
$$
C
$$
is the one-object preadditive category corresponding to the ring
$$
R
$$
, this reduces to the ordinary category of (left) -modules. Again, virtually all concepts from the theory of modules can be generalised to this setting.
## -linear categories
More generally, one can consider a category enriched over the monoidal category of modules over a commutative ring , called an -linear category. In other words, each hom-set
$$
\text{Hom}(A,B)
$$
in has the structure of an -module, and composition of morphisms is -bilinear.
When considering functors between two -linear categories, one often restricts to those that are -linear, so those that induce -linear maps on each hom-set.
## Biproducts
Any finite product in a preadditive category must also be a coproduct, and conversely. In fact, finite products and coproducts in preadditive categories can be characterised by the following biproduct condition:
The object B is a biproduct of the objects A1, ..., An if and only if there are projection morphisms pj: B → Aj and injection morphisms ij: Aj → B, such that (i1∘p1) + ··· + (in∘pn) is the identity morphism of B, pj∘ij is the identity morphism of Aj, and pj∘ik is the zero morphism from Ak to Aj whenever j and k are distinct.
|
https://en.wikipedia.org/wiki/Preadditive_category
|
passage: From this interpretation, there is a bilinear map, sometimes called the Yoneda product:
$$
\operatorname{Ext}^i_{\mathbf C}(A,B) \times \operatorname{Ext}^j_{\mathbf C}(B,C) \to \operatorname{Ext}^{i+j}_{\mathbf C}(A,C),
$$
which is simply the composition of morphisms in the derived category.
The Yoneda product can also be described in more elementary terms. For i = j = 0, the product is the composition of maps in the category C. In general, the product can be defined by splicing together two Yoneda extensions.
Alternatively, the Yoneda product can be defined in terms of resolutions. (This is close to the definition of the derived category.) For example, let R be a ring, with R-modules A, B, C, and let P, Q, and T be projective resolutions of A, B, C. Then Ext(A,B) can be identified with the group of chain homotopy classes of chain maps P → Q[i]. The Yoneda product is given by composing chain maps:
$$
P\to Q[i]\to T[i+j].
$$
By any of these interpretations, the Yoneda product is associative.
|
https://en.wikipedia.org/wiki/Ext_functor
|
passage: The bilinear transform maps the
$$
j \omega
$$
axis of the s-plane (which is the domain of
$$
H_a(s)
$$
) to the unit circle of the z-plane,
$$
|z| = 1
$$
(which is the domain of
$$
H_d(z)
$$
), but it is not the same mapping
$$
z = e^{sT}
$$
which also maps the
$$
j \omega
$$
axis to the unit circle. When the actual frequency of
$$
\omega_d
$$
is input to the discrete-time filter designed by use of the bilinear transform, then it is desired to know at what frequency,
$$
\omega_a
$$
, for the continuous-time filter that this
$$
\omega_d
$$
is mapped to.
$$
H_d(z) = H_a \left( \frac{2}{T} \frac{z-1}{z+1}\right)
$$
{|
|-
|
$$
H_d(e^{ j \omega_d T})
$$
|
$$
= H_a \left( \frac{2}{T} \frac{e^{ j \omega_d T} - 1}{e^{ j \omega_d T} + 1}\right)
$$
|-
|
|
$$
|
https://en.wikipedia.org/wiki/Bilinear_transform
|
passage: A topological soliton, also called a topological defect, is any solution of a set of partial differential equations that is stable against decay to the "trivial solution". Soliton stability is due to topological constraints, rather than integrability of the field equations. The constraints arise almost always because the differential equations must obey a set of boundary conditions, and the boundary has a nontrivial homotopy group, preserved by the differential equations. Thus, the differential equation solutions can be classified into homotopy classes.
No continuous transformation maps a soliton in one homotopy class to another. The solitons are truly distinct, and maintain their integrity, even in the face of extremely powerful forces. Examples of topological solitons include the screw dislocation in a crystalline lattice, the Dirac string and the magnetic monopole in electromagnetism, the Skyrmion and the Wess–Zumino–Witten model in quantum field theory, the magnetic skyrmion in condensed matter physics, and cosmic strings and domain walls in cosmology.
## History
In 1834, John Scott Russell described his wave of translation:This passage has been repeated in many papers and books on soliton theory.
Scott Russell spent some time making practical and theoretical investigations of these waves.
|
https://en.wikipedia.org/wiki/Soliton
|
passage: In the case of the oceans, in 2017 Costello et al. analyzed the distribution of 65,000 species of marine animals and plants as then documented in OBIS, and used the results to distinguish 30 distinct marine realms, split between continental-shelf and offshore deep-sea areas.
Since it is self evident that compilations of species occurrence records cannot cover with any completeness, areas that have received either limited or no sampling, a number of methods have been developed to produce arguably more complete "predictive" or "modelled" distributions for species based on their associated environmental or other preferences (such as availability of food or other habitat requirements); this approach is known as either Environmental niche modelling (ENM) or Species distribution modelling (SDM). Depending on the reliability of the source data and the nature of the models employed (including the scales for which data are available), maps generated from such models may then provide better representations of the "real" biogeographic distributions of either individual species, groups of species, or biodiversity as a whole, however it should also be borne in mind that historic or recent human activities (such as hunting of great whales, or other human-induced exterminations) may have altered present-day species distributions from their potential "full" ecological footprint.
|
https://en.wikipedia.org/wiki/Biogeography
|
passage: Wav2vec 2.0 discretizes the audio waveform into timesteps via temporal convolutions, and then trains a transformer on masked prediction of random timesteps using a contrastive loss. This is similar to the BERT language model, except as in many SSL approaches to video, the model chooses among a set of options rather than over the entire word vocabulary.
### Multimodal
Self-supervised learning has also been used to develop joint representations of multiple data types. Approaches usually rely on some natural or human-derived association between the modalities as an implicit label, for instance video clips of animals or objects with characteristic sounds, or captions written to describe images. CLIP produces a joint image-text representation space by training to align image and text encodings from a large dataset of image-caption pairs using a contrastive loss. MERLOT Reserve trains a transformer-based encoder to jointly represent audio, subtitles and video frames from a large dataset of videos through 3 joint pretraining tasks: contrastive masked prediction of either audio or text segments given the video frames and surrounding audio and text context, along with contrastive alignment of video frames with their corresponding captions.
Multimodal representation models are typically unable to assume direct correspondence of representations in the different modalities, since the precise alignment can often be noisy or ambiguous. For example, the text "dog" could be paired with many different pictures of dogs, and correspondingly a picture of a dog could be captioned with varying degrees of specificity.
|
https://en.wikipedia.org/wiki/Feature_learning
|
passage: Biochemists often refer to four distinct aspects of a protein's structure:
- Primary structure: the amino acid sequence. A protein is a polyamide.
- Secondary structure: regularly repeating local structures stabilized by hydrogen bonds. The most common examples are the α-helix, β-sheet and turns. Because secondary structures are local, many regions of distinct secondary structure can be present in the same protein molecule.
- Tertiary structure: the overall shape of a single protein molecule; the spatial relationship of the secondary structures to one another. Tertiary structure is generally stabilized by nonlocal interactions, most commonly the formation of a hydrophobic core, but also through salt bridges, hydrogen bonds, disulfide bonds, and even post-translational modifications. The term "tertiary structure" is often used as synonymous with the term fold. The tertiary structure is what controls the basic function of the protein.
- Quaternary structure: the structure formed by several protein molecules (polypeptide chains), usually called protein subunits in this context, which function as a single protein complex.
- Quinary structure: the signatures of protein surface that organize the crowded cellular interior. Quinary structure is dependent on transient, yet essential, macromolecular interactions that occur inside living cells.
Proteins are not entirely rigid molecules. In addition to these levels of structure, proteins may shift between several related structures while they perform their functions.
|
https://en.wikipedia.org/wiki/Protein
|
passage: ```cpp
1. include <iostream>
1. include <array>
constexpr int TABLE_SIZE = 10;
/**
- Variadic template for a recursive helper struct.
- /
template<int INDEX = 0, int ...D>
struct Helper : Helper<INDEX + 1, D..., INDEX * INDEX> { };
/**
- Specialization of the template to end the recursion when the table size reaches TABLE_SIZE.
- /
template<int ...D>
struct Helper<TABLE_SIZE, D...> {
static constexpr std::array<int, TABLE_SIZE> table = { D... };
};
constexpr std::array<int, TABLE_SIZE> table = Helper<>::table;
enum {
FOUR = table[2] // compile time use
};
int main() {
for (int i=0; i < TABLE_SIZE; i++) {
std::cout << table[i] << std::endl; // run time use
}
std::cout << "FOUR: " << FOUR << std::endl;
}
|
https://en.wikipedia.org/wiki/Template_metaprogramming
|
passage: |
$$
c_3
$$
||
$$
a_{32}
$$
|-
| style="border-right:1px solid;" |
$$
\vdots
$$
||
$$
\vdots
$$
|| ||
$$
\ddots
$$
|-
| style="border-right:1px solid; border-bottom:1px solid;" |
$$
c_s
$$
| style="border-bottom:1px solid;" |
$$
a_{s2}
$$
| style="border-bottom:1px solid;" |
$$
a_{s3}
$$
| style="border-bottom:1px solid;" |
$$
\cdots
$$
| style="border-bottom:1px solid;" |
$$
a_{s,s-1}
$$
|| style="border-bottom:1px solid;" |
|-
| style="border-right:1px solid;" | ||
$$
b_2
$$
||
$$
b_3
$$
||
$$
\cdots
$$
||
$$
b_{s-1}
$$
||
$$
b_s
$$
|-
| style="border-right:1px solid;" | ||
$$
\bar{b}_2
$$
||
$$
\bar{b}_3
$$
||
$$
\cdots
$$
||
|
https://en.wikipedia.org/wiki/Exponential_integrator
|
passage: Another definition of Euclidean spaces by means of vector spaces and linear algebra has been shown to be equivalent to the axiomatic definition. It is this definition that is more commonly used in modern mathematics, and detailed in this article. In all definitions, Euclidean spaces consist of points, which are defined only by the properties that they must have for forming a Euclidean space.
There is essentially only one Euclidean space of each dimension; that is, all Euclidean spaces of a given dimension are isomorphic. Therefore, it is usually possible to work with a specific Euclidean space, denoted
$$
\mathbf{E}^n
$$
or
$$
\mathbb{E}^n
$$
, which can be represented using
### Cartesian coordinates
as the real -space
$$
\R^n
$$
equipped with the standard dot product.
## Definition
### History of the definition
Euclidean space was introduced by ancient Greeks as an abstraction of our physical space. Their great innovation, appearing in Euclid's Elements was to build and prove all geometry by starting from a few very basic properties, which are abstracted from the physical world, and cannot be mathematically proved because of the lack of more basic tools. These properties are called postulates, or axioms in modern language. This way of defining Euclidean space is still in use under the name of synthetic geometry.
In 1637, René Descartes introduced Cartesian coordinates, and showed that these allow reducing geometric problems to algebraic computations with numbers.
|
https://en.wikipedia.org/wiki/Euclidean_space
|
passage: then
$$
f(x)
$$
is close to
$$
f(A).
$$
Instead of specifying topological spaces by their open subsets, any topology on
$$
X
$$
can alternatively be determined by a closure operator or by an interior operator.
Specifically, the map that sends a subset
$$
A
$$
of a topological space
$$
X
$$
to its topological closure
$$
\operatorname{cl}_X A
$$
satisfies the Kuratowski closure axioms. Conversely, for any closure operator
$$
A \mapsto \operatorname{cl} A
$$
there exists a unique topology
$$
\tau
$$
on
$$
X
$$
(specifically,
$$
\tau := \{ X \setminus \operatorname{cl} A : A \subseteq X \}
$$
) such that for every subset
$$
A \subseteq X,
$$
$$
\operatorname{cl} A
$$
is equal to the topological closure
$$
\operatorname{cl}_{(X, \tau)} A
$$
of
$$
A
$$
in
$$
(X, \tau).
$$
|
https://en.wikipedia.org/wiki/Continuous_function
|
passage: ## Definitions and statement
The theorem equates two quantities: the maximum flow through a network, and the minimum capacity of a cut of the network. To state the theorem, each of these notions must first be defined.
### Network
A network consists of
- a finite directed graph , where V denotes the finite set of vertices and is the set of directed edges;
- a source and a sink ;
- a capacity function, which is a mapping
$$
c:E\to\R^+
$$
denoted by or for . It represents the maximum amount of flow that can pass through an edge.
### Flows
A flow through a network is a mapping
$$
f:E\to\R^+
$$
denoted by
$$
f_{uv}
$$
or
$$
f(u, v)
$$
, subject to the following two constraints:
1. Capacity Constraint: For every edge
$$
(u, v) \in E
$$
, BLOCK41. Conservation of Flows: For each vertex
$$
v
$$
apart from
$$
s
$$
and
$$
t
$$
(i.e. the source and sink, respectively), the following equality holds:
$$
\sum\nolimits_{\{ u : (u,v)\in E\}} f_{uv} = \sum\nolimits_{\{w : (v,w)\in E\}} f_{vw}.
$$
A flow can be visualized as a physical flow of a fluid through the network, following the direction of each edge.
|
https://en.wikipedia.org/wiki/Max-flow_min-cut_theorem
|
passage: The human body, as a biological machine, has many functions that can be modeled using engineering methods.
The heart for example functions much like a pump, the skeleton is like a linked structure with levers, the brain produces electrical signals etc. These similarities as well as the increasing importance and application of engineering principles in medicine, led to the development of the field of biomedical engineering that uses concepts developed in both disciplines.
Newly emerging branches of science, such as systems biology, are adapting analytical tools traditionally used for engineering, such as systems modeling and computational analysis, to the description of biological systems.
### Art
There are connections between engineering and art, for example, architecture, landscape architecture and industrial design (even to the extent that these disciplines may sometimes be included in a university's Faculty of Engineering).MIT World:The Art of Engineering: Inventor James Dyson on the Art of Engineering: quote: A member of the British Design Council, James Dyson has been designing products since graduating from the Royal College of Art in 1970.
The Art Institute of Chicago, for instance, held an exhibition about the art of NASA's aerospace design. Robert Maillart's bridge design is perceived by some to have been deliberately artistic. At the University of South Florida, an engineering professor, through a grant with the National Science Foundation, has developed a course that connects art and engineering.quote:..the tools of artists and the perspective of engineers..
Among famous historical figures, Leonardo da Vinci is a well-known Renaissance artist and engineer, and a prime example of the nexus between art and engineering.
|
https://en.wikipedia.org/wiki/Engineering
|
passage: ## Multiplication theorem
The Bessel functions obey a multiplication theorem
$$
\lambda^{-\nu} J_\nu(\lambda z) = \sum_{n=0}^\infty \frac{1}{n!} \left(\frac{\left(1 - \lambda^2\right)z}{2}\right)^n J_{\nu+n}(z),
$$
where and may be taken as arbitrary complex numbers. For , the above expression also holds if is replaced by . The analogous identities for modified Bessel functions and are
$$
\lambda^{-\nu} I_\nu(\lambda z) = \sum_{n=0}^\infty \frac{1}{n!} \left(\frac{\left(\lambda^2 - 1\right)z}{2}\right)^n I_{\nu+n}(z)
$$
and
$$
\lambda^{-\nu} K_\nu(\lambda z) = \sum_{n=0}^\infty \frac{(-1)^n}{n!} \left(\frac{\left(\lambda^2 - 1\right)z}{2}\right)^n K_{\nu+n}(z).
$$
## Zeros of the Bessel function
### Bourget's hypothesis
Bessel himself originally proved that for nonnegative integers , the equation has an infinite number of solutions in .
|
https://en.wikipedia.org/wiki/Bessel_function
|
passage: Moreover, cluster analyses revealed that L-arginine and its main metabolites L-citrulline, L-ornithine and agmatine formed distinct groups, which were altered in the schizophrenia group. Despite this, the biological basis of schizophrenia is still poorly understood, a number of factors, such as dopamine hyperfunction, glutamatergic hypofunction, GABAergic deficits, cholinergic system dysfunction, stress vulnerability and neurodevelopmental disruption, have been linked to the aetiology and/or pathophysiology of the disease.
### Raynaud's phenomenon
Oral L-arginine has been shown to reverse digital necrosis in Raynaud syndrome.
## Safety and potential drug interactions
L-arginine is recognized as safe (GRAS-status) at intakes of up to 20 grams per day. L-arginine is found in many foods, such as fish, poultry, and dairy products, and is used as a dietary supplement. It may interact with various prescription drugs and herbal supplements.
|
https://en.wikipedia.org/wiki/Arginine
|
passage: The cross-section of the elements are similar to the previously described types: one-dimensional for thin plates and shells, and two-dimensional for solids, thick plates and shells.
- Three-dimensional elements for modeling 3-D solids such as machine components, dams, embankments or soil masses. Common element shapes include tetrahedrals and hexahedrals. Nodes are placed at the vertexes and possibly on element faces or within the element.
### Element interconnection and displacement
The elements are interconnected only at the exterior nodes, and altogether they should cover the entire domain as accurately as possible. Nodes will have nodal (vector) displacements or degrees of freedom which may include translations, rotations, and for special applications, higher order derivatives of displacements. When the nodes displace, they will drag the elements along in a certain manner dictated by the element formulation. In other words, displacements of any points in the element will be interpolated from the nodal displacements, and this is the main reason for the approximate nature of the solution.
## Practical considerations
From the application point of view, it is important to model the system such that:
- Symmetry or anti-symmetry conditions are exploited in order to reduce the size of the model.
- Displacement compatibility, including any required discontinuity, is ensured at the nodes, and preferably, along the element edges as well, particularly when adjacent elements are of different types, material or thickness.
|
https://en.wikipedia.org/wiki/Finite_element_method_in_structural_mechanics
|
passage: This collection of 286 nuclides are known as primordial nuclides. Finally, an additional 53 short-lived nuclides are known to occur naturally, as daughter products of primordial nuclide decay (such as radium from uranium), or as products of natural energetic processes on Earth, such as cosmic ray bombardment (for example, carbon-14).For more recent updates see Brookhaven National Laboratory's Interactive Chart of Nuclides ] .
For 80 of the chemical elements, at least one stable isotope exists. As a rule, there is only a handful of stable isotopes for each of these elements, the average being 3.1 stable isotopes per element. Twenty-six "monoisotopic elements" have only a single stable isotope, while the largest number of stable isotopes observed for any element is ten, for the element tin. Elements 43, 61, and all elements numbered 83 or higher have no stable isotopes.
Stability of isotopes is affected by the ratio of protons to neutrons, and also by the presence of certain "magic numbers" of neutrons or protons that represent closed and filled quantum shells. These quantum shells correspond to a set of energy levels within the shell model of the nucleus; filled shells, such as the filled shell of 50 protons for tin, confers unusual stability on the nuclide. Of the 251 known stable nuclides, only four have both an odd number of protons and odd number of neutrons: hydrogen-2 (deuterium), lithium-6, boron-10, and nitrogen-14.
|
https://en.wikipedia.org/wiki/Atom
|
passage: Loss of pain sensation predisposes to trauma that can lead to diabetic foot problems (such as ulceration), the most common cause of non-traumatic lower-limb amputation.
Hearing loss is another long-term complication associated with diabetes.
Based on extensive data and numerous cases of gallstone disease, it appears that a causal link might exist between type 2 diabetes and gallstones. People with diabetes are at a higher risk of developing gallstones compared to those without diabetes.
There is a link between cognitive deficit and diabetes; studies have shown that diabetic individuals are at a greater risk of cognitive decline, and have a greater rate of decline compared to those without the disease. Diabetes increases the risk of dementia, and the earlier that one is diagnosed with diabetes, the higher the risk becomes. The condition also predisposes to falls in the elderly, especially those treated with insulin.
## Types
+Comparison of type 1 and 2 diabetes Feature Type 1 diabetes Type 2 diabetes Onset SuddenGradual, Insidious Age at onset Any age; average age at diagnosis being 24. Mostly in adults Body size Thin or normal Often obese Ketoacidosis Common Rare Autoantibodies Usually present Absent Endogenous insulin Low or absent Normal, decreased or increased Heritability 0.69 to 0.88 0.47 to 0.77 Prevalence
(age standardized)
|
https://en.wikipedia.org/wiki/Diabetes
|
passage: The geodesic flow is a generalization of the idea of moving in a "straight line" on a curved surface: such straight lines are geodesics. One of the earliest cases studied is Hadamard's billiards, which describes geodesics on the Bolza surface, topologically equivalent to a donut with two holes. Ergodicity can be demonstrated informally, if one has a sharpie and some reasonable example of a two-holed donut: starting anywhere, in any direction, one attempts to draw a straight line; rulers are useful for this. It doesn't take all that long to discover that one is not coming back to the starting point. (Of course, crooked drawing can also account for this; that's why we have proofs.)
These results extend to higher dimensions. The geodesic flow for negatively curved compact Riemannian manifolds is ergodic. A classic example for this is the Anosov flow, which is the horocycle flow on a hyperbolic manifold. This can be seen to be a kind of Hopf fibration. Such flows commonly occur in classical mechanics, which is the study in physics of finite-dimensional moving machinery, e.g. the double pendulum and so-forth. Classical mechanics is constructed on symplectic manifolds. The flows on such systems can be deconstructed into stable and unstable manifolds; as a general rule, when this is possible, chaotic motion results.
|
https://en.wikipedia.org/wiki/Ergodicity
|
passage: #### Independent (unpaired) samples
The independent samples t-test is used when two separate sets of independent and identically distributed samples are obtained, and one variable from each of the two populations is compared. For example, suppose we are evaluating the effect of a medical treatment, and we enroll 100 subjects into our study, then randomly assign 50 subjects to the treatment group and 50 subjects to the control group. In this case, we have two independent samples and would use the unpaired form of the t-test.
#### Paired samples
Paired samples t-tests typically consist of a sample of matched pairs of similar units, or one group of units that has been tested twice (a "repeated measures" t-test).
A typical example of the repeated measures t-test would be where subjects are tested prior to a treatment, say for high blood pressure, and the same subjects are tested again after treatment with a blood-pressure-lowering medication. By comparing the same patient's numbers before and after treatment, we are effectively using each patient as their own control. That way the correct rejection of the null hypothesis (here: of no difference made by the treatment) can become much more likely, with statistical power increasing simply because the random interpatient variation has now been eliminated. However, an increase of statistical power comes at a price: more tests are required, each subject having to be tested twice. Because half of the sample now depends on the other half, the paired version of Student's t-test has only degrees of freedom (with being the total number of observations).
|
https://en.wikipedia.org/wiki/Student%27s_t-test
|
passage: (Although
$$
\mathcal T
$$
is often called the "time-ordering operator", strictly speaking it is neither an operator on states nor a superoperator on operators.)
For two operators A(x) and B(y) that depend on spacetime locations x and y we define:
$$
\mathcal T \left\{A(x) B(y)\right\} := \begin{cases} A(x) B(y) & \text{if } \tau_x > \tau_y, \\ \pm B(y)A(x) & \text{if } \tau_x < \tau_y. \end{cases}
$$
Here
$$
\tau_x
$$
and
$$
\tau_y
$$
denote the invariant scalar time-coordinates of the points x and y.
Explicitly we have
$$
\mathcal T \left\{A(x) B(y)\right\} := \theta (\tau_x - \tau_y) A(x) B(y) \pm \theta (\tau_y - \tau_x) B(y) A(x),
$$
where
$$
\theta
$$
denotes the Heaviside step function and the
$$
\pm
$$
depends on if the operators are bosonic or fermionic in nature. If bosonic, then the + sign is always chosen, if fermionic then the sign will depend on the number of operator interchanges necessary to achieve the proper time ordering.
|
https://en.wikipedia.org/wiki/Path-ordering
|
passage: ### Beam waist
The shape of a Gaussian beam of a given wavelength is governed solely by one parameter, the beam waist . This is a measure of the beam size at the point of its focus ( in the above equations) where the beam width (as defined above) is the smallest (and likewise where the intensity on-axis () is the largest). From this parameter the other parameters describing the beam geometry are determined. This includes the Rayleigh range and asymptotic beam divergence , as detailed below.
### Rayleigh range and confocal parameter
The Rayleigh distance or Rayleigh range is determined given a Gaussian beam's waist size:
$$
z_\mathrm{R} = \frac{\pi w_0^2 n}{\lambda}.
$$
Here is the wavelength of the light, is the index of refraction. At a distance from the waist equal to the Rayleigh range , the width of the beam is larger than it is at the focus where , the beam waist. That also implies that the on-axis () intensity there is one half of the peak intensity (at ). That point along the beam also happens to be where the wavefront curvature () is greatest.
The distance between the two points is called the confocal parameter or depth of focus of the beam.
### Beam divergence
|
https://en.wikipedia.org/wiki/Gaussian_beam
|
passage: The symbol
$$
X_a^b
$$
, with
$$
-\infty \le a \le b \le \infty
$$
denotes a sub-σ-algebra of the σ-algebra; it is the set of cylinder sets that are specified between times a and b, i.e. the σ-algebra generated by .
The process
$$
(X_t)_{-\infty < t < \infty}
$$
is said to be strongly mixing if
$$
\alpha(s)\to 0
$$
as . That is to say, a strongly mixing process is such that, in a way that is uniform over all times
$$
t
$$
and all events, the events before time
$$
t
$$
and the events after time
$$
t+s
$$
tend towards being independent as
$$
s \to \infty
$$
; more colloquially, the process, in a strong sense, forgets its history.
### Mixing in Markov processes
Suppose
$$
(X_t)
$$
were a stationary Markov process with stationary distribution
$$
\mathbb{Q}
$$
and let
$$
L^2(\mathbb{Q})
$$
denote the space of Borel-measurable functions that are square-integrable with respect to the measure
$$
\mathbb{Q}
$$
.
|
https://en.wikipedia.org/wiki/Mixing_%28mathematics%29
|
passage: Surface tension indeed leads to surface minimization and, thus, to film disappearance. The hole aperture is not instantaneous and is slowed by the liquid inertia. The balance between the forces of inertia and surface tension leads to the opening velocity:
$$
V=\sqrt{\frac{2\gamma}{\rho h}}
$$
where
$$
\gamma
$$
is the liquid surface tension,
$$
\rho
$$
is the liquid density and
$$
h
$$
is the film thickness.
## References
### General sources
-
Category:Minimal surfaces
Category:Bubbles (physics)
he:קרום סבון
|
https://en.wikipedia.org/wiki/Soap_film
|
passage: It contains six exons, giving rise to 22 different mRNAs, which produce four complete isoforms whose form of expression is probably dependent on the type of tissue they are found in. It also has two different DNA promoters. It has been noted that the sequences translated from this locus and from that of β-actin are very similar to the predicted ones, suggesting a common ancestral sequence that suffered duplication and genetic conversion.
In terms of pathology, it has been associated with processes such as amyloidosis, retinitis pigmentosa, infection mechanisms, kidney diseases, and various types of congenital hearing loss.
Six autosomal-dominant point mutations in the sequence have been found to cause various types of hearing loss, particularly sensorineural hearing loss linked to the DFNA 20/26 locus. It seems that they affect the stereocilia of the ciliated cells present in the inner ear's Organ of Corti. β-actin is the most abundant protein found in human tissue, but it is not very abundant in ciliated cells, which explains the location of the pathology. On the other hand, it appears that the majority of these mutations affect the areas involved in linking with other proteins, particularly actomyosin. Some experiments have suggested that the pathological mechanism for this type of hearing loss relates to the F-actin in the mutations being more sensitive to cofilin than normal.
|
https://en.wikipedia.org/wiki/Actin
|
passage: The predominant cells of dermal tissue are epidermal cells.
Ground tissue usually consists mainly of parenchyma, collenchyma and sclerenchyma cells, and they surround vascular tissue. Ground tissue is important in aiding metabolic activities (eg. respiration, photosynthesis, transport, storage) as well as acting as structural support and forming new meristems. Most or all ground tissue may be lost in woody stems.
Vascular tissue, consisting of xylem, phloem and cambium; provides long distance transport of water, minerals and metabolites (sugars, amino acids); whilst aiding structural support and growth. The arrangement of the vascular tissues varies widely among plant species.
### Dicot stems
Dicot stems with primary growth have pith in the center, with vascular bundles forming a distinct ring visible when the stem is viewed in cross section. The outside of the stem is covered with an epidermis, which is covered by a waterproof cuticle. The epidermis also may contain stomata for gas exchange and multicellular stem hairs called trichomes. A cortex consisting of hypodermis (collenchyma cells) and endodermis (starch containing cells) is present above the pericycle and vascular bundles.
Woody dicots and many nonwoody dicots have secondary growth originating from their lateral or secondary meristems: the vascular cambium and the cork cambium or phellogen.
|
https://en.wikipedia.org/wiki/Plant_stem
|
passage: Let denote the first exit time of
$$
X
$$
from
$$
D
$$
.
In this notation, the candidate solution for (P1) is:
$$
u(x) = \mathbb{E}^{x} \left[ g \big( X_{\tau_{D}} \big) \cdot \chi_{\{ \tau_{D} < + \infty \}} \right] + \mathbb{E}^{x} \left[ \int_{0}^{\tau_{D}} f(X_{t}) \, \mathrm{d} t \right]
$$
provided that
$$
g
$$
is a bounded function and that:
$$
\mathbb{E}^{x} \left[ \int_{0}^{\tau_{D}} \big| f(X_{t}) \big| \, \mathrm{d} t \right] < + \infty
$$
It turns out that one further condition is required:
$$
\mathbb{P}^{x} \big( \tau_{D} < \infty \big) = 1, \quad \forall x \in D
$$
For all
$$
x
$$
, the process
$$
X
$$
starting at
$$
x
$$
almost surely leaves
$$
D
$$
in finite time.
|
https://en.wikipedia.org/wiki/Stochastic_processes_and_boundary_value_problems
|
passage: George noted that humans are distinct from other species, because unlike most species humans can use their minds to leverage the reproductive forces of nature to their advantage. He wrote, "Both the hawk and the man eat chickens; but the more hawks, the fewer chickens, while the more men, the more chickens."
D. E. C. Eversley observed that Malthus appeared unaware of the extent of industrialisation, and either ignored or discredited the possibility that it could improve living conditions of the poorer classes.
Barry Commoner believed in The Closing Circle (1971) that technological progress will eventually reduce the demographic growth and environmental damage created by civilisation. He also opposed coercive measures postulated by neo-malthusian movements of his time arguing that their cost will fall disproportionately on the low-income population who are struggling already.
Ester Boserup suggested that expanding population leads to agricultural intensification and development of more productive and less labor-intensive methods of farming. Thus, human population levels determines agricultural methods, rather than agricultural methods determining population.
Environmentalist founder of Ecomodernism, Stewart Brand, summarised how the Malthusian predictions of The Population Bomb and The Limits to Growth failed to materialise due to radical changes in fertility that peaked at a growth of 2 percent per year in 1963 globally and has since rapidly declined.
Short-term trends, even on the scale of decades or centuries, cannot prove or disprove the existence of mechanisms promoting a Malthusian catastrophe over longer periods.
|
https://en.wikipedia.org/wiki/Malthusianism
|
passage: The overhand knot is the simplest single-strand stopper knot.
T
### Turn
A turn is one round of rope on a pin or cleat, or one round of a coil.
W
### Whipping
A whipping is a binding knot tied around the end of a rope to prevent the rope from unraveling.
### Working end
The working end (or working part) of a rope is the part active in knot tying. The opposite end is the standing end.
|
https://en.wikipedia.org/wiki/List_of_knot_terminology
|
passage: The intersection of two lines may be found in the same way, using duality, as the cross product of the vectors representing the lines, .
## Embedding into 4-dimensional space
The projective plane embeds into 4-dimensional Euclidean space. The real projective plane P2(R) is the quotient of the two-sphere
S2 = {(x, y, z) ∈ R3 : x2 + y2 + z2 = 1}
by the antipodal relation . Consider the function given by . This map restricts to a map whose domain is S2 and, since each component is a homogeneous polynomial of even degree, it takes the same values in R4 on each of any two antipodal points on S2. This yields a map . Moreover, this map is an embedding. Notice that this embedding admits a projection into R3 which is the Roman surface.
## Higher non-orientable surfaces
By gluing together projective planes successively we get non-orientable surfaces of higher demigenus. The gluing process consists of cutting out a little disk from each surface and identifying (gluing) their boundary circles. Gluing two projective planes creates the Klein bottle.
The article on the fundamental polygon describes the higher non-orientable surfaces.
|
https://en.wikipedia.org/wiki/Real_projective_plane
|
passage: The goal is that rather than organizing the web using keywords as most applications (e.g. Google) do today the web can be organized by concepts organized in an ontology.
The name of the OWL language itself provides a good example of the value of a Semantic Web. If one were to search for "OWL" using the Internet today most of the pages retrieved would be on the bird Owl rather than the standard OWL. With a Semantic Web it would be possible to specify the concept "Web Ontology Language" and the user would not need to worry about the various possible acronyms or synonyms as part of the search. Likewise, the user would not need to worry about homonyms crowding the search results with irrelevant data such as information about birds of prey as in this simple example.
In addition to OWL, various standards and technologies that are relevant to the Semantic Web and were influenced by Frame languages include OIL and DAML. The Protege Open Source software tool from Stanford University provides an ontology editing capability that is built on OWL and has the full capabilities of a classifier. However it ceased to explicitly support frames as of version 3.5 (which is maintained for those preferring frame orientation), the version current in 2017 being 5. The justification for moving from explicit frames being that OWL DL is more expressive and "industry standard".
### Comparison of frames and objects
Frame languages have a significant overlap with object-oriented languages.
|
https://en.wikipedia.org/wiki/Frame_%28artificial_intelligence%29
|
passage: Intuitively, the velocity increases linearly, so the average velocity multiplied by time is the distance traveled while increasing the velocity from to , as can be illustrated graphically by plotting velocity against time as a straight line graph. Algebraically, it follows from solving [1] for
$$
\mathbf{a} = \frac{(\mathbf{v} - \mathbf{v}_0)}{t}
$$
and substituting into [2]
$$
\mathbf{r} = \mathbf{r}_0 + \mathbf{v}_0 t + \frac{t}{2}(\mathbf{v} - \mathbf{v}_0) \,,
$$
then simplifying to get
$$
\mathbf{r} = \mathbf{r}_0 + \frac{t}{2}(\mathbf{v} + \mathbf{v}_0)
$$
or in magnitudes
$$
r = r_0 + \left( \frac{v+v_0}{2} \right )t \quad [3]
$$
From [3],
$$
t = \left( r - r_0 \right)\left( \frac{2}{v+v_0} \right )
$$
substituting for in [1]:
$$
\begin{align}
v & = a\left( r - r_0 \right)\left( \frac{2}{v+v_0} \right )+v_0 \\
|
https://en.wikipedia.org/wiki/Equations_of_motion
|
passage: Achieving constant orbital radius while supplying the proper accelerating electric field requires that the magnetic flux linking the orbit be somewhat independent of the magnetic field on the orbit, bending the particles into a constant radius curve. These machines have in practice been limited by the large radiative losses suffered by the electrons moving at nearly the speed of light in a relatively small radius orbit.
### Linear accelerators
In a linear particle accelerator (linac), particles are accelerated in a straight line with a target of interest at one end. They are often used to provide an initial low-energy kick to particles before they are injected into circular accelerators. The longest linac in the world is the Stanford Linear Accelerator, SLAC, which is long. SLAC was originally an electron–positron collider but is now a X-ray Free-electron laser.
Linear high-energy accelerators use a linear array of plates (or drift tubes) to which an alternating high-energy field is applied. As the particles approach a plate they are accelerated towards it by an opposite polarity charge applied to the plate. As they pass through a hole in the plate, the polarity is switched so that the plate now repels them and they are now accelerated by it towards the next plate. Normally a stream of "bunches" of particles are accelerated, so a carefully controlled AC voltage is applied to each plate to continuously repeat this process for each bunch.
|
https://en.wikipedia.org/wiki/Particle_accelerator
|
passage: & = \sum_{\sigma \in S_n} \sgn(\sigma)
\left( \left(b_{\sigma(j)}\prod_{i = 1, i \neq j}^n a_{\sigma(i)}^i\right) + \left(a_{\sigma(j)}^j\prod_{i = 1, i \neq j}^n a_{\sigma(i)}^i\right)\right)\\
& = \left(\sum_{\sigma \in S_n} \sgn(\sigma) b_{\sigma(j)}\prod_{i = 1, i \neq j}^n a_{\sigma(i)}^i\right)
+ \left(\sum_{\sigma \in S_n} \sgn(\sigma) \prod_{i = 1}^n a_{\sigma(i)}^i\right)\\
&= F(A^1, \dots, b, \dots) + F(A^1, \dots, A^j, \dots)\\
\\
\end{align}
$$
Alternating:
|
https://en.wikipedia.org/wiki/Leibniz_formula_for_determinants
|
passage: In mathematical logic, a conservative extension is a supertheory of a theory which is often convenient for proving theorems, but proves no new theorems about the language of the original theory. Similarly, a non-conservative extension is a supertheory which is not conservative, and can prove more theorems than the original.
More formally stated, a theory
$$
T_2
$$
is a (proof theoretic) conservative extension of a theory
$$
T_1
$$
if every theorem of
$$
T_1
$$
is a theorem of
$$
T_2
$$
, and any theorem of
$$
T_2
$$
in the language of
$$
T_1
$$
is already a theorem of
$$
T_1
$$
.
More generally, if
$$
\Gamma
$$
is a set of formulas in the common language of
$$
T_1
$$
and
$$
T_2
$$
, then
$$
T_2
$$
is
$$
\Gamma
$$
-conservative over
$$
T_1
$$
if every formula from
$$
\Gamma
$$
provable in
$$
T_2
$$
is also provable in
$$
T_1
$$
.
Note that a conservative extension of a consistent theory is consistent. If it were not, then by the principle of explosion, every formula in the language of
$$
T_2
$$
would be a theorem of
$$
T_2
$$
, so every formula in the language of
$$
T_1
$$
would be a theorem of
$$
T_1
$$
, so
$$
T_1
$$
would not be consistent.
|
https://en.wikipedia.org/wiki/Conservative_extension
|
passage: $$
The inverse relation is usually written in the reduced form
$$
\begin{bmatrix}\varepsilon_{11} \\ \varepsilon_{22} \\ 2\varepsilon_{12} \end{bmatrix} \,=\, \frac{1}{E}
\begin{bmatrix} 1 & -\nu & 0 \\
-\nu & 1 & 0 \\
0 & 0 & 2+2\nu \end{bmatrix}
\begin{bmatrix}\sigma_{11} \\ \sigma_{22} \\ \sigma_{12} \end{bmatrix}
$$
|
https://en.wikipedia.org/wiki/Hooke%27s_law
|
passage: An application to open shop scheduling," Technical report TR/IRIDIA/2003-17, 2003.
- Permutation flow shop problem (PFSP)
- Single machine total tardiness problem (SMTTP)
- Single machine total weighted tardiness problem (SMTWTP)M, den Bseten, T. Stützle and M. Dorigo, "Ant colony optimization for the total weighted tardiness problem," Proceedings of PPSN-VI, Sixth International Conference on Parallel Problem Solving from Nature, vol. 1917 of Lecture Notes in Computer Science, pp.611-620, 2000.
- Resource-constrained project scheduling problem (RCPSP)
- Group-shop scheduling problem (GSP)
- Single-machine total tardiness problem with sequence dependent setup times (SMTTPDST)
- Multistage flowshop scheduling problem (MFSP) with sequence dependent setup/changeover times
- Assembly sequence planning (ASP) problems
### Vehicle routing problem
- Capacitated vehicle routing problem (CVRP)J. M. Belenguer, and E. Benavent, "A cutting plane algorithm for capacitated arc routing problem," Computers & Operations Research, vol.30, no.5, pp.705-728, 2003.
- Multi-depot vehicle routing problem (MDVRP)
- Period vehicle routing problem (PVRP)
- Split delivery vehicle routing problem (SDVRP)
- Stochastic vehicle routing problem (SVRP)
- Vehicle routing problem with pick-up and delivery (VRPPD)R. Bent and P.V. Hentenryck, "A two-stage hybrid algorithm for pickup and delivery vehicle routing problems with time windows," Computers & Operations Research, vol.33, no.4, pp.875-893, 2003.
- Vehicle routing problem with time windows (VRPTW)
- Time dependent vehicle routing problem with time windows (TDVRPTW)
- Vehicle routing problem with time windows and multiple service workers (VRPTWMS)
|
https://en.wikipedia.org/wiki/Ant_colony_optimization_algorithms
|
passage: In algebraic geometry, a branch of mathematics, a Hilbert scheme is a scheme that is the parameter space for the closed subschemes of some projective space (or a more general projective scheme), refining the Chow variety. The Hilbert scheme is a disjoint union of projective subschemes corresponding to Hilbert polynomials. The basic theory of Hilbert schemes was developed by . Hironaka's example shows that non-projective varieties need not have Hilbert schemes.
## Hilbert scheme of projective space
The Hilbert scheme
$$
\mathbf{Hilb}(n)
$$
of
$$
\mathbb{P}^n
$$
classifies closed subschemes of projective space in the following sense: For any locally Noetherian scheme , the set of -valued points
$$
\operatorname{Hom}(S, \mathbf{Hilb}(n))
$$
of the Hilbert scheme is naturally isomorphic to the set of closed subschemes of
$$
\mathbb{P}^n \times S
$$
that are flat over . The closed subschemes of
$$
\mathbb{P}^n \times S
$$
that are flat over can informally be thought of as the families of subschemes of projective space parameterized by . The Hilbert scheme
$$
\mathbf{Hilb}(n)
$$
breaks up as a disjoint union of pieces
$$
\mathbf{Hilb}(n, P)
$$
corresponding to the Hilbert scheme of the subschemes of projective space with Hilbert polynomial .
|
https://en.wikipedia.org/wiki/Hilbert_scheme
|
passage: then the map
$$
f
$$
will be a quotient map if and only if
$$
X
$$
has the final topology induced by the maps
$$
f_i.
$$
### Effects of changing the family of maps
Throughout, let
$$
\mathcal{F} := \left\{ f_i : i \in I \right\}
$$
be a family of
$$
X
$$
-valued maps with each map being of the form
$$
f_i : \left(Y_i, \upsilon_i\right) \to X
$$
and let
$$
\tau_{\mathcal{F}}
$$
denote the final topology on
$$
X
$$
induced by
$$
\mathcal{F}.
$$
The definition of the final topology guarantees that for every index
$$
i,
$$
the map
$$
f_i : \left(Y_i, \upsilon_i\right) \to \left(X, \tau_{\mathcal{F}}\right)
$$
is continuous.
For any subset
$$
\mathcal{S} \subseteq \mathcal{F},
$$
the final topology
$$
\tau_{\mathcal{S}}
$$
on
$$
X
$$
will be than (and possibly equal to) the topology
$$
\tau_{\mathcal{F}}
$$
; that is,
$$
\mathcal{S} \subseteq \mathcal{F}
$$
implies
$$
|
https://en.wikipedia.org/wiki/Final_topology
|
passage: Moreover, if a manifold is separable and paracompact then it is also second-countable.
Every compact manifold is second-countable and paracompact.
### Dimensionality
By invariance of domain, a non-empty n-manifold cannot be an m-manifold for n ≠ m. The dimension of a non-empty n-manifold is n. Being an n-manifold is a topological property, meaning that any topological space homeomorphic to an n-manifold is also an n-manifold.
## Coordinate charts
By definition, every point of a locally Euclidean space has a neighborhood homeomorphic to an open subset of
$$
\mathbb R^n
$$
. Such neighborhoods are called Euclidean neighborhoods. It follows from invariance of domain that Euclidean neighborhoods are always open sets. One can always find Euclidean neighborhoods that are homeomorphic to "nice" open sets in
$$
\mathbb R^n
$$
. Indeed, a space M is locally Euclidean if and only if either of the following equivalent conditions holds:
- every point of M has a neighborhood homeomorphic to an open ball in
$$
\mathbb R^n
$$
.
- every point of M has a neighborhood homeomorphic to
$$
\mathbb R^n
$$
itself.
A Euclidean neighborhood homeomorphic to an open ball in
$$
\mathbb R^n
$$
is called a Euclidean ball. Euclidean balls form a basis for the topology of a locally Euclidean space.
|
https://en.wikipedia.org/wiki/Topological_manifold
|
passage: #### Source moving towards (blueshift)
To apply this to a situation where the source is moving straight towards the observer (), this becomes:
$$
\frac{\omega_{\mathrm{obs}}}{\omega_s} = \frac{1}{\gamma (1 - \beta)} = \frac{\sqrt{1-\beta^2}}{1-\beta} = \frac{\sqrt{(1+\beta)(1-\beta)}}{1-\beta} = \frac{\sqrt{1+\beta}}{\sqrt{1-\beta}}
$$
#### Source moving tangentially (transverse Doppler effect)
To apply this to a situation where the source is moving transversely with respect to the observer (), this becomes:
$$
\frac{\omega_{\mathrm{obs}}}{\omega_s} = \frac{1}{\gamma (1 - 0)} = \frac{1}{\gamma}
$$
|
https://en.wikipedia.org/wiki/Wave_vector
|
passage: Another example, relevant to the two side bands of amplitude modulation of AM radio, is:
$$
\begin{align}
\cos((\omega + \alpha)t) + \cos\left((\omega - \alpha)t\right)
BLOCK0\end{align}
$$
### In physics
#### Electromagnetism and electrical engineering
In electrical engineering, the Fourier transform is used to analyze varying electric currents and voltages. The treatment of resistors, capacitors, and inductors can then be unified by introducing imaginary, frequency-dependent resistances for the latter two and combining all three in a single complex number called the impedance. This approach is called phasor calculus.
In electrical engineering, the imaginary unit is denoted by , to avoid confusion with , which is generally in use to denote electric current, or, more particularly, , which is generally in use to denote instantaneous electric current.
|
https://en.wikipedia.org/wiki/Complex_number
|
passage: The nucleoplasm is also a route for many molecules to travel through. Smaller molecules are able to pass freely through the nuclear pore to get into and out of the nucleoplasm, while larger proteins need the help of receptors on the surface of the nuclear envelope.
The nuclear matrix is also believed to be contained in the nucleoplasm where it functions to maintain the size and shape of the nucleus, in a role similar to that of the cytoskeleton found in the cytoplasm. However, the existence and the exact function of the nuclear matrix remain unclear and heavily debated.
## Composition
The nucleoplasm is a highly viscous liquid that is enveloped by the nuclear membrane and consists mainly of water, proteins, dissolved ions, and a variety of other substances including nucleic acids and minerals.
Proteins
There are around 20,000 protein-coding genes in humans, and nearly a third of these have been found to localize to the nucleoplasm via targeting by a nuclear localization sequence (NLS). Cytosolic proteins, known as importins, act as receptors for the NLS, escorting the protein to a nuclear pore complex to be transported into the nucleoplasm. Proteins in the nucleoplasm are mainly tasked with participating in and regulating cellular functions that are DNA-dependent, including transcription, RNA splicing, DNA repair, DNA replication, and a variety of metabolic processes.
|
https://en.wikipedia.org/wiki/Nucleoplasm
|
passage: Partial sums of series sometimes have simpler closed form expressions, for instance an arithmetic series has partial sums
$$
s_n = \sum_{k=0}^{n} \left(a + kd\right)
= a + (a + d) + (a + 2d) + \cdots + (a + nd)
= (n+1)\bigl(a + \tfrac12 n d\bigr),
$$
and a geometric series has partial sums
$$
s_n = \sum_{k=0}^{n} ar^k = a + ar + ar^2 + \cdots + ar^n = a\frac{1 - r^{n+1}}{1 - r}
$$
if or simply if .
### Sum of a series
Strictly speaking, a series is said to converge, to be convergent, or to be summable when the sequence of its partial sums has a limit. When the limit of the sequence of partial sums does not exist, the series diverges or is divergent. When the limit of the partial sums exists, it is called the sum of the series or value of the series:
$$
\sum_{k = 0}^\infty a_k = \lim_{n\to\infty} \sum_{k=0}^n a_k = \lim_{n\to\infty} s_n.
$$
A series with only a finite number of nonzero terms is always convergent. Such series are useful for considering finite sums without taking care of the numbers of terms.
|
https://en.wikipedia.org/wiki/Series_%28mathematics%29
|
passage: As a reaction to their bleak reality, they then created a community separate from mainstream society to enjoy each other's company and their favourite loud music. The NWOBHM was criticised as being local media hype for mostly talentless musicians. Nonetheless, it generated a renewal in the genre of heavy metal music and furthered the progress of the heavy metal subculture, whose updated behavioural and visual codes were quickly adopted by metal fans worldwide after the spread of the music to continental Europe, North America and Japan.
By some estimates, the movement spawned as many as a thousand heavy metal bands. Only a few survived the advent of MTV and the rise of the more commercial glam metal in the second half of the 1980s. Iron Maiden and Def Leppard became superstars;
### Motörhead
and Saxon also had considerable success. Other groups, such as Diamond Head, Venom, and Raven, had more limited chart success, but influenced the successful extreme metal subgenres of the mid-to-late 1980s and 1990s. Many bands from the NWOBHM reunited in the 2000s and remained active through live performances and new studio albums.
## Background
### Social unrest
In the second half of the 1970s, the United Kingdom was in a state of social unrest and widespread poverty as a result of the ineffective social politics of both Conservative and Labour Party governments during a three-year period of economic recession. As a consequence of deindustrialisation, the unemployment rate was exceptionally high, especially among working class youth. It continued to rise in the early 1980s, peaking in February 1983.
|
https://en.wikipedia.org/wiki/New_wave_of_British_heavy_metal
|
passage: Not checking assertions avoids the cost of evaluating the assertions while (assuming the assertions are free of side effects) still producing the same result under normal conditions. Under abnormal conditions, disabling assertion checking can mean that a program that would have aborted will continue to run. This is sometimes preferable.
Some languages, including C, YASS and C++, can completely remove assertions at compile time using the preprocessor.
Similarly, launching the Python interpreter with "" (for "optimize") as an argument will cause the Python code generator to not emit any bytecode for asserts.
Java requires an option to be passed to the run-time engine in order to enable assertions. Absent the option, assertions are bypassed, but they always remain in the code unless optimised away by a JIT compiler at run-time or excluded at compile time via the programmer manually placing each assertion behind an `if (false)` clause.
Programmers can build checks into their code that are always active by bypassing or manipulating the language's normal assertion-checking mechanisms.
## Comparison with error handling
Assertions are distinct from routine error-handling. Assertions document logically impossible situations and discover programming errors: if the impossible occurs, then something fundamental is clearly wrong with the program. This is distinct from error handling: most error conditions are possible, although some may be extremely unlikely to occur in practice.
|
https://en.wikipedia.org/wiki/Assertion_%28software_development%29
|
passage: Drug discovery is related to pharmacoeconomics, which is the sub-discipline of health economics that considers the value of drugs. Pharmacoeconomics evaluates the cost and benefits of drugs in order to guide optimal healthcare resource allocation. The techniques used for the discovery, formulation, manufacturing and quality control of drugs discovery is studied by pharmaceutical engineering, a branch of engineering. Safety pharmacology specialises in detecting and investigating potential undesirable effects of drugs.
Development of medication is a vital concern to medicine, but also has strong economical and political implications. To protect the consumer and prevent abuse, many governments regulate the manufacture, sale, and administration of medication. In the United States, the main body that regulates pharmaceuticals is the Food and Drug Administration; they enforce standards set by the United States Pharmacopoeia. In the European Union, the main body that regulates pharmaceuticals is the European Medicines Agency (EMA), and they enforce standards set by the European Pharmacopoeia.
The metabolic stability and the reactivity of a library of candidate drug compounds have to be assessed for drug metabolism and toxicological studies. Many methods have been proposed for quantitative predictions in drug metabolism; one example of a recent computational method is SPORCalc. A slight alteration to the chemical structure of a medicinal compound could alter its medicinal properties, depending on how the alteration relates to the structure of the substrate or receptor site on which it acts: this is called the structural activity relationship (SAR).
|
https://en.wikipedia.org/wiki/Pharmacology
|
passage: The additive inverse of any
$$
\overline x
$$
in is
$$
-\overline x=\overline{-x}.
$$
For example,
$$
-\overline{3} = \overline{-3} = \overline{1}.
$$
has a subring , and if
$$
p
$$
is prime, then has no subrings.
### Example: 2-by-2 matrices
The set of 2-by-2 square matrices with entries in a field is
$$
\operatorname{M}_2(F) = \left\{ \left.\begin{pmatrix} a & b \\ c & d \end{pmatrix} \right|\ a, b, c, d \in F \right\}.
$$
With the operations of matrix addition and matrix multiplication,
$$
\operatorname{M}_2(F)
$$
satisfies the above ring axioms. The element
$$
\left( \begin{smallmatrix} 1 & 0 \\ 0 & 1 \end{smallmatrix}\right)
$$
is the multiplicative identity of the ring.
|
https://en.wikipedia.org/wiki/Ring_%28mathematics%29
|
passage: Classical mechanics is unable to determine this distance because of the enormous number of bombardments a Brownian particle will undergo, roughly of the order of 1014 collisions per second.
He regarded the increment of particle positions in time
$$
\tau
$$
in a one-dimensional (x) space (with the coordinates chosen so that the origin lies at the initial position of the particle) as a random variable (
$$
q
$$
) with some probability density function
$$
\varphi(q)
$$
(i.e.,
$$
\varphi(q)
$$
is the probability density for a jump of magnitude
$$
q
$$
, i.e., the probability density of the particle incrementing its position from
$$
x
$$
to
$$
x + q
$$
in the time interval
$$
\tau
$$
). Further, assuming conservation of particle number, he expanded the number density
$$
\rho(x,t+\tau)
$$
(number of particles per unit volume around
$$
x
$$
) at time
$$
t + \tau
$$
in a Taylor series,
$$
\begin{align}
\rho(x, t+\tau)
={}& \rho(x,t) + \tau \frac{\partial\rho(x,t)}{\partial t} + \cdots
\\[2ex]
={}& \int_{-\infty}^{\infty} \rho(x - q, t) \, \varphi(q) \, dq
|
https://en.wikipedia.org/wiki/Brownian_motion
|
passage: Under normal data processing, scouring all the patient’s medical data to ensure they are getting the best treatment could take too long and risk the patients’ health or even life. However, using semantically processed ontologies, the first responders could save the patient’s life. Tools like a semantic reasoner can use ontology to infer the what best medicine to administer to the patient is based on their medical history, such as if they have a certain cancer or other conditions, simply by examining the natural language used in the patient's medical records. This would allow the first responders to quickly and efficiently search for medicine without having worry about the patient’s medical history themselves, as the semantic reasoner would already have analyzed this data and found solutions. In general, this illustrates the incredible strength of using semantic data mining and ontologies. They allow for quicker and more efficient data extraction on the user side, as the user has fewer variables to account for, since the semantically pre-processed data and ontology built for the data have already accounted for many of these variables. However, there are some drawbacks to this approach. Namely, it requires a high amount of computational power and complexity, even with relatively small data sets. This could result in higher costs and increased difficulties in building and maintaining semantic data processing systems. This can be mitigated somewhat if the data set is already well organized and formatted, but even then, the complexity is still higher when compared to standard data processing.
Below is a simple a diagram combining some of the processes, in particular semantic data mining and their use in ontology.
|
https://en.wikipedia.org/wiki/Data_preprocessing
|
passage: ### The transcendence of
implies that geometric constructions involving compass and straightedge only cannot produce certain results, for example squaring the circle.
In 1900 David Hilbert posed a question about transcendental numbers, Hilbert's seventh problem: If is an algebraic number that is not 0 or 1, and is an irrational algebraic number, is necessarily transcendental? The affirmative answer was provided in 1934 by the Gelfond–Schneider theorem. This work was extended by Alan Baker in the 1960s in his work on lower bounds for linear forms in any number of logarithms (of algebraic numbers).
## Properties
A transcendental number is a (possibly complex) number that is not the root of any integer polynomial. Every real transcendental number must also be irrational, since every rational number is the root of some integer polynomial of degree one. The set of transcendental numbers is uncountably infinite. Since the polynomials with rational coefficients are countable, and since each such polynomial has a finite number of zeroes, the algebraic numbers must also be countable. However, Cantor's diagonal argument proves that the real numbers (and therefore also the complex numbers) are uncountable. Since the real numbers are the union of algebraic and transcendental numbers, it is impossible for both subsets to be countable. This makes the transcendental numbers uncountable.
No rational number is transcendental and all real transcendental numbers are irrational.
|
https://en.wikipedia.org/wiki/Transcendental_number
|
passage: A macromolecule is a very large molecule important to biological processes, such as a protein or nucleic acid. It is composed of thousands of covalently bonded atoms. Many macromolecules are polymers of smaller molecules called monomers. The most common macromolecules in biochemistry are biopolymers (nucleic acids, proteins, and carbohydrates) and large non-polymeric molecules such as lipids, nanogels and macrocycles. Synthetic fibers and experimental materials such as carbon nanotubes are also examples of macromolecules.
## Definition
The term macromolecule (macro- + molecule) was coined by Nobel laureate Hermann Staudinger in the 1920s, although his first relevant publication on this field only mentions high molecular compounds (in excess of 1,000 atoms). At that time the term polymer, as introduced by Berzelius in 1832, had a different meaning from that of today: it simply was another form of isomerism for example with benzene and acetylene and had little to do with size.
Usage of the term to describe large molecules varies among the disciplines. For example, while biology refers to macromolecules as the four large molecules comprising living things, in chemistry, the term may refer to aggregates of two or more molecules held together by intermolecular forces rather than covalent bonds but which do not readily dissociate.
According to the standard IUPAC definition, the term macromolecule as used in polymer science refers only to a single molecule.
|
https://en.wikipedia.org/wiki/Macromolecule
|
passage: The role the Pfaffian plays can be understood from a geometric viewpoint by developing Clifford algebra from simplices. This derivation provides a better connection between Pascal's triangle and simplices because it provides an interpretation of the first column of ones.
### Blades, grades, and basis
A multivector that is the exterior product of
$$
r
$$
linearly independent vectors is called a blade, and is said to be of grade . A multivector that is the sum of blades of grade
$$
r
$$
is called a (homogeneous) multivector of grade . From the axioms, with closure, every multivector of the geometric algebra is a sum of blades.
Consider a set of
$$
r
$$
linearly independent vectors
$$
\{a_1,\ldots,a_r\}
$$
spanning an -dimensional subspace of the vector space.
|
https://en.wikipedia.org/wiki/Geometric_algebra
|
passage: - The Bessel function of the first kind , its first derivative, and the quotient
$$
\tfrac{J'_\nu (x)}{J_\nu (x)}
$$
are transcendental when is rational and is algebraic and nonzero, and all nonzero roots of and are transcendental when is rational.
- The number
$$
\tfrac{\pi}{2} \tfrac{Y_0 (2)}{J_0 (2)} - \gamma
$$
, where and are Bessel functions and is the Euler–Mascheroni constant.
- Values of the Fibonacci zeta function at the positive even argument.
- Any Liouville number, in particular: Liouville's constant
$$
\sum_{k=1}^\infty\frac1{10^{k!}}
$$
.
- Numbers with irrationality measure larger than 2, such as the Champernowne constant
$$
C_{10}
$$
(by Roth's theorem).
- Numbers artificially constructed not to be algebraic periods.
- Any non-computable number, in particular: Chaitin's constant.
- Constructed irrational numbers which are not simply normal in any base.
-
|
https://en.wikipedia.org/wiki/Transcendental_number
|
passage: In fact, for every nonzero degree a there is a degree b incomparable with a.
- There is a set of
$$
2^{\aleph_0}
$$
pairwise incomparable Turing degrees.
- There are pairs of degrees with no greatest lower bound. Thus
$$
\mathcal{D}
$$
is not a lattice.
- Every countable partially ordered set can be embedded in the Turing degrees.
- An infinite strictly increasing sequence a1, a2, ... of Turing degrees cannot have a least upper bound, but it always has an exact pair c, d such that ∀e (e<c∧e<d ⇔ ∃i e≤ai), and thus it has (non-unique) minimal upper bounds.
- Assuming the axiom of constructibility, it can be shown there is a maximal chain of degrees of order type
$$
\omega_1
$$
.
### Properties involving the jump
- For every degree a there is a degree strictly between a and a′. In fact, there is a countable family of pairwise incomparable degrees between a and a′.
- Jump inversion: a degree a is of the form b′ if and only
|
https://en.wikipedia.org/wiki/Turing_degree
|
passage: Otlet not only imagined that all the world's knowledge should be interlinked and made available remotely to anyone, but he also proceeded to build a structured document collection. This collection involved standardized paper sheets and cards filed in custom-designed cabinets according to a hierarchical index (which culled information worldwide from diverse sources) and a commercial information retrieval service (which answered written requests by copying relevant information from index cards). Users of this service were even warned if their query was likely to produce more than 50 results per search.
By 1937 documentation had formally been institutionalized, as evidenced by the founding of the American Documentation Institute (ADI), later called the American Society for Information Science and Technology.
### Transition to modern information science
With the 1950s came increasing awareness of the potential of automatic devices for literature searching and information storage and retrieval. As these concepts grew in magnitude and potential, so did the variety of information science interests. By the 1960s and 70s, there was a move from batch processing to online modes, from mainframe to mini and microcomputers. Additionally, traditional boundaries among disciplines began to fade and many information science scholars joined with other programs. They further made themselves multidisciplinary by incorporating disciplines in the sciences, humanities and social sciences, as well as other professional programs, such as law and medicine in their curriculum.
Among the individuals who had distinct opportunities to facilitate interdisciplinary activity targeted at scientific communication was Foster E. Mohrhardt, director of the National Agricultural Library from 1954 to 1968.
|
https://en.wikipedia.org/wiki/Information_science
|
passage: It is frequently combined with other intraocular procedures for the treatment of giant retinal tears, tractional retinal detachments, and posterior vitreous detachments.
- Pan retinal photocoagulation is a type of photocoagulation therapy used in the treatment of diabetic retinopathy.
- Retinal detachment repair
- Ignipuncture is an obsolete procedure that involves cauterization of the retina with a very hot, pointed instrument.
- A scleral buckle is used in the repair of a retinal detachment to indent or "buckle" the sclera inward, usually by sewing a piece of preserved sclera or silicone rubber to its surface.
- Laser photocoagulation, or photocoagulation therapy, is the use of a laser to seal a retinal tear.
- Pneumatic retinopexy
- Retinal cryopexy, or retinal cryotherapy, is a procedure that uses intense cold to induce a chorioretinal scar and to destroy retinal or choroidal tissue.
- Macular hole repair
- Partial lamellar sclerouvectomy
- Partial lamellar sclerocyclochoroidectomy
- Partial lamellar sclerochoroidectomy
- Posterior sclerotomy is an opening made into the vitreous through the sclera, as for detached retina or the removal of a foreign body.
- Radial optic neurotomy
- Macular translocation surgery
- through 360° retinotomy
- through scleral imbrication technique
|
https://en.wikipedia.org/wiki/Eye_surgery
|
passage: When the isoperimetric inequality becomes an equality, there is only one such triangle, which is equilateral.
### Angle bisector length
If the two equal sides have length
$$
a
$$
and the other side has length
$$
b
$$
, then the internal angle bisector
$$
t
$$
from one of the two equal-angled vertices satisfies
$$
\frac{2ab}{a+b} > t > \frac{ab\sqrt{2}}{a+b}
$$
as well as
$$
t<\frac{4a}{3};
$$
and conversely, if the latter condition holds, an isosceles triangle parametrized by
$$
a
$$
and
$$
t
$$
exists.
The Steiner–Lehmus theorem states that every triangle with two angle bisectors of equal lengths is isosceles. It was formulated in 1840 by C. L. Lehmus. Its other namesake, Jakob Steiner, was one of the first to provide a solution.
Although originally formulated only for internal angle bisectors, it works for many (but not all) cases when, instead, two external angle bisectors are equal.
The 30-30-120 isosceles triangle makes a boundary case for this variation of the theorem, as it has four equal angle bisectors (two internal, two external).
### Radii
The inradius and circumradius formulas for an isosceles triangle may be derived from their formulas for arbitrary triangles.
|
https://en.wikipedia.org/wiki/Isosceles_triangle
|
passage: For example, an elliptic Lorentz transformation can have any two distinct fixed points on the celestial sphere, but points still flow along circular arcs from one fixed point toward the other. The other cases are similar.
Elliptic
An elliptic element of is
$$
P_1 = \begin{bmatrix}
\exp\left(\frac{i}{2}\theta\right) & 0 \\
0 & \exp\left(-\frac{i}{2}\theta\right)
\end{bmatrix}
$$
and has fixed points = 0, ∞. Writing the action as and collecting terms, the spinor map converts this to the (restricted) Lorentz transformation
$$
Q_1 = \begin{bmatrix}
BLOCK0 \end{bmatrix} =
\exp\left(\theta\begin{bmatrix}
BLOCK1 \end{bmatrix}\right) ~.
$$
This transformation then represents a rotation about the axis, exp(). The one-parameter subgroup it generates is obtained by taking to be a real variable, the rotation angle, instead of a constant.
The corresponding continuous transformations of the celestial sphere (except for the identity) all share the same two fixed points, the North and South poles. The transformations move all other points around latitude circles so that this group yields a continuous counter-clockwise rotation about the axis as increases. The angle doubling evident in the spinor map is a characteristic feature of spinorial double coverings.
|
https://en.wikipedia.org/wiki/Lorentz_group
|
passage: Maschke's theorem implies that
$$
\mathbb{C}[G]
$$
is a semisimple ring, so by the Artin–Wedderburn theorem it decomposes as a direct product of matrix rings. The Fourier transform on finite groups explicitly exhibits this decomposition, with a matrix ring of dimension
$$
d_\varrho
$$
for each irreducible representation.
More specifically, the Peter-Weyl theorem (for finite groups) states that there is an isomorphism
$$
\mathbb C[G] \cong \bigoplus_{i} \mathrm{End}(V_i)
$$
given by
$$
\sum_{g \in G} a_g g \mapsto \left(\sum a_g \rho_i(g): V_i \to V_i\right)
$$
The left hand side is the group algebra of G. The direct sum is over a complete set of inequivalent irreducible G-representations
$$
\varrho_i : G \to \mathrm{GL}(V_i)
$$
.
The Fourier transform for a finite group is just this isomorphism. The product formula mentioned above is equivalent to saying that this map is a ring isomorphism.
|
https://en.wikipedia.org/wiki/Fourier_transform_on_finite_groups
|
passage: The assignment
$$
[x] \mapsto \hat{f}([x])
$$
defines a bijection
$$
\hat{f} : X /{\sim} \;\to\; Y
$$
between the fibers of
$$
f
$$
and points in
$$
Y.
$$
Define the map
$$
q : X \to X / {\sim}
$$
as above (by
$$
q(x) := [x]
$$
) and give
$$
X / {\sim}
$$
the quotient topology induced by
$$
q
$$
(which makes
$$
q
$$
a quotient map). These maps are related by:
$$
f = \hat{f} \circ q \quad \text{ and } \quad q = \hat{f}^{-1} \circ f.
$$
From this and the fact that
$$
q : X \to X / {\sim}
$$
is a quotient map, it follows that
$$
f : X \to Y
$$
is continuous if and only if this is true of
$$
\hat{f} : X / {\sim} \;\to\; Y.
$$
Furthermore,
$$
f : X \to Y
$$
is a quotient map if and only if
$$
\hat{f} : X / {\sim} \;\to\; Y
$$
is a homeomorphism (or equivalently, if and only if both
$$
\hat{f}
$$
and its inverse are continuous).
|
https://en.wikipedia.org/wiki/Quotient_space_%28topology%29
|
passage: Thus every set that is recursively enumerable by an oracle machine with an oracle for
$$
\emptyset ^{(1)}
$$
, is in
$$
\Sigma^0_2
$$
.
The converse is true as well: Suppose
$$
\varphi(n)
$$
is a formula in
$$
\Sigma^0_2
$$
with
$$
k_1
$$
existential quantifiers followed by
$$
k_2
$$
universal quantifiers. Equivalently,
$$
\varphi(n)
$$
has
$$
k_1
$$
> existential quantifiers followed by a negation of a formula in
$$
\Sigma^0_1
$$
; the latter formula can be enumerated by a Turing machine and can thus be checked immediately by an oracle for
$$
\emptyset ^{(1)}
$$
.
We may thus enumerate the
$$
k_1
$$
–tuples of natural numbers and run an oracle machine with an oracle for
$$
\emptyset ^{(1)}
$$
that goes through all of them until it finds a satisfaction for the formula. This oracle machine halts on precisely the set of natural numbers satisfying
$$
\varphi(n)
$$
, and thus enumerates its corresponding set.
### Higher Turing jumps
More generally, suppose every set that is recursively enumerable by an oracle machine with an oracle for
$$
\emptyset ^{(p)}
$$
is in
$$
\Sigma^0_{p+1}
$$
.
|
https://en.wikipedia.org/wiki/Post%27s_theorem
|
passage: Whereas prior releases of Windows supported per-file encryption using Encrypting File System, the Enterprise and Ultimate editions of Vista include BitLocker Drive Encryption, which can protect entire volumes, notably the operating system volume. However, BitLocker requires approximately a 1.5-gigabyte partition to be permanently not encrypted and to contain system files for Windows to boot. In normal circumstances, the only time this partition is accessed is when the computer is booting, or when there is a Windows update that changes files in this area, which is a legitimate reason to access this section of the drive. The area can be a potential security issue, because a hexadecimal editor (such as dskprobe.exe), or malicious software running with administrator and/or kernel level privileges would be able to write to this "Ghost Partition" and allow a piece of malicious software to compromise the system, or disable the encryption. BitLocker can work in conjunction with a Trusted Platform Module (TPM) cryptoprocessor (version 1.2) embedded in a computer's motherboard, or with a USB key. However, as with other full disk encryption technologies, BitLocker is vulnerable to a cold boot attack, especially where TPM is used as a key protector without a boot PIN being required too.
A variety of other privilege-restriction techniques are also built into Vista. An example is the concept of "integrity levels" in user processes, whereby a process with a lower integrity level cannot interact with processes of a higher integrity level and cannot perform DLL–injection to processes of a higher integrity level.
|
https://en.wikipedia.org/wiki/Windows_Vista
|
passage: Because of the Markov assumption, the true state is conditionally independent of all earlier states given the immediately previous state.
$$
p(\mathbf{x}_k\mid \mathbf{x}_0,\dots,\mathbf{x}_{k-1}) = p(\mathbf{x}_k\mid \mathbf{x}_{k-1})
$$
Similarly, the measurement at the k-th timestep is dependent only upon the current state and is conditionally independent of all other states given the current state.
$$
p(\mathbf{z}_k\mid\mathbf{x}_0,\dots,\mathbf{x}_k) = p(\mathbf{z}_k\mid \mathbf{x}_k )
$$
Using these assumptions the probability distribution over all states of the hidden Markov model can be written simply as:
$$
p\left(\mathbf{x}_0, \dots, \mathbf{x}_k, \mathbf{z}_1, \dots, \mathbf{z}_k\right) = p\left(\mathbf{x}_0\right)\prod_{i=1}^k p\left(\mathbf{z}_i \mid \mathbf{x}_i\right)p\left(\mathbf{x}_i \mid \mathbf{x}_{i-1}\right)
$$
However, when a Kalman filter is used to estimate the state x, the probability distribution of interest is that associated with the current states conditioned on the measurements up to the current timestep.
|
https://en.wikipedia.org/wiki/Kalman_filter
|
passage: In 2022, Costa et al. established morphologic, genetic, and evolutionary divergence between the two ecotypes in the western North Atlantic, resurrecting Tursiops erebennus for the coastal form while the offshore form was retained in T. truncatus.
The Society for Marine Mammalogy's Committee on Taxonomy presently recognizes three species of bottlenose dolphin: T. truncatus, T. aduncus, and T. erebennus. They also recognize three subspecies of common bottlenose dolphin in addition to the nominotypical subspecies: the Black Sea bottlenose dolphin (T. t. ponticus), Lahille's bottlenose dolphin (T. t. gephyreus), and the Eastern Tropical Pacific bottlenose dolphin (T. t. nuuanu). The IUCN, on their Red List of endangered species, currently recognises only two species of bottlenose dolphins. The American Society of Mammalogists also recognizes only two species. While acknowledging the studies describing T. australis, it classifies it within T. aduncus.
Some recent genetic evidence suggests the Indo-Pacific bottlenose dolphin belongs in the genus Stenella, since it is more like the Atlantic spotted dolphin (Stenella frontalis) than the common bottlenose dolphin.
|
https://en.wikipedia.org/wiki/Bottlenose_dolphin
|
passage: Gauss's Theorema Egregium (Latin for "Remarkable Theorem") is a major result of differential geometry, proved by Carl Friedrich Gauss in 1827, that concerns the curvature of surfaces. The theorem says that Gaussian curvature can be determined entirely by measuring angles, distances and their rates on a surface, without reference to the particular manner in which the surface is embedded in the ambient 3-dimensional Euclidean space. In other words, the Gaussian curvature of a surface does not change if one bends the surface without stretching it. Thus the Gaussian curvature is an intrinsic invariant of a surface.
Gauss presented the theorem in this manner (translated from Latin):
Thus the formula of the preceding article leads itself to the remarkable Theorem. If a curved surface is developed upon any other surface whatever, the measure of curvature in each point remains unchanged.
The theorem is "remarkable" because the definition of Gaussian curvature makes ample reference to the specific way the surface is embedded in 3-dimensional space, and it is quite surprising that the result does not depend on its embedding.
In modern mathematical terminology, the theorem may be stated as follows:
## Elementary applications
A sphere of radius R has constant Gaussian curvature which is equal to 1/R2. At the same time, a plane has zero Gaussian curvature. As a corollary of Theorema Egregium, a piece of paper cannot be bent onto a sphere without crumpling.
|
https://en.wikipedia.org/wiki/Theorema_Egregium
|
passage: Dirac's approach is also called second quantization or quantum field theory; earlier quantum mechanical treatments only treat material particles as quantum mechanical, not the electromagnetic field.
Einstein was troubled by the fact that his theory seemed incomplete, since it did not determine the direction of a spontaneously emitted photon. A probabilistic nature of light-particle motion was first considered by Newton in his treatment of birefringence and, more generally, of the splitting of light beams at interfaces into a transmitted beam and a reflected beam. Newton hypothesized that hidden variables in the light particle determined which of the two paths a single photon would take. Similarly, Einstein hoped for a more complete theory that would leave nothing to chance, beginning his separation from quantum mechanics. Ironically, Max Born's probabilistic interpretation of the wave function was inspired by Einstein's later work searching for a more complete theory.
## Quantum field theory
### Quantization of the electromagnetic field
In 1910, Peter Debye derived Planck's law of black-body radiation from a relatively simple assumption. He decomposed the electromagnetic field in a cavity into its Fourier modes, and assumed that the energy in any mode was an integer multiple of
$$
h\nu
$$
, where
$$
\nu
$$
is the frequency of the electromagnetic mode. Planck's law of black-body radiation follows immediately as a geometric sum. However, Debye's approach failed to give the correct formula for the energy fluctuations of black-body radiation, which were derived by Einstein in 1909.
|
https://en.wikipedia.org/wiki/Photon
|
passage: The string field
$$
\Phi
$$
is taken to be in the NS sector of the large Hilbert space, i.e. including the zero mode of
$$
\xi
$$
. It is not known how to incorporate the R sector, although some preliminary ideas exist.
The equations of motion take the form
$$
\eta_0 \left(e^{-\Phi} Q_B e^{\Phi} \right) = 0 .
$$
The action is invariant under the gauge transformation
$$
e^{\Phi} \to e^{Q_B \Lambda} e^{\Phi} e^{\eta_0 \Lambda'} .
$$
The principal advantage of this action is that it free from any insertions of picture-changing operators. It has been shown to reproduce correctly tree level amplitudes and has been found, numerically, to have a tachyon vacuum with appropriate energy. The known analytic solutions to the classical equations of motion include the tachyon vacuum and marginal deformations.
### Other formulations of covariant open superstring field theory
A formulation of superstring field theory using the non-minimal pure-spinor variables was introduced by Berkovits. The action is cubic and includes a midpoint insertion whose kernel is trivial. As always within the pure-spinor formulation, the Ramond sector can be easily treated. However, it is not known how to incorporate the GSO- sectors into the formalism.
|
https://en.wikipedia.org/wiki/String_field_theory
|
passage: An election is usually a competition between different parties.
A political system is a framework which defines acceptable political methods within a society. The history of political thought can be traced back to early antiquity, with seminal works such as Plato's Republic, Aristotle's Politics, Confucius's political manuscripts and Chanakya's Arthashastra.
## Etymology
The English word politics has its roots in the name of Aristotle's classic work, Politiká, which introduced the Ancient Greek term (). In the mid-15th century, Aristotle's composition was rendered in Early Modern English as ,Buhler, C. F., ed. 1961 [1941]. The Dictes and Sayings of the Philosophers. London: Early English Text Society, Original Series No. 211 . which became Politics in Modern English.
The singular politic first attested in English in 1430, coming from Middle French —itself taking from , a Latinization of the Greek () from () and ().
### Definitions
- Harold Lasswell: "who gets what, when, how"
- David Easton: "the authoritative allocation of values for a society"
- Vladimir Lenin: "the most concentrated expression of economics"
- Otto von Bismarck: "the capacity of always choosing at each instant, in constantly changing situations, the least harmful, the most useful"
- Bernard Crick: "a distinctive form of rule whereby people act together through institutionalized procedures to resolve differences"
- Adrian Leftwich: "comprises all the activities of co-operation, negotiation and conflict within and between societies"
|
https://en.wikipedia.org/wiki/Politics
|
passage: The distortion created thereby is usually attenuated by aligning one plane of the imaged object to be parallel with the plane of projection, creating a truly-formed, full-size image of the chosen plane. Special types of oblique projections include military, cavalier and cabinet projection.
## Analytic representation
If the image plane is given by equation
$$
\Pi:~\vec{n}\cdot\vec{x}-d=0
$$
and the direction of projection by
$$
\vec v
$$
, then the projection line through the point
$$
P:~\vec p
$$
is parametrized by
$$
g:~\vec x = \vec p+t\vec{v}
$$
with
$$
t\in\mathbb{R}
$$
.
The image
$$
P'
$$
of
$$
P
$$
is the intersection of line
$$
g
$$
with plane
$$
\Pi
$$
; it is given by the equation
$$
P':~\vec{p}' = \vec p + \frac{d-\vec p\cdot\vec{n}}{\vec{n}\cdot\vec{v}}\;\vec{v} \ .
$$
In several cases, these formulas can be simplified.
(S1)
|
https://en.wikipedia.org/wiki/Parallel_projection
|
passage: From this perspective, Compton scattering could be considered elastic because the internal state of the electron does not change during the scattering process. In the latter perspective, the atom's state is changed, constituting an inelastic collision. Whether Compton scattering is considered elastic or inelastic depends on which perspective is being used, as well as the context.
Compton scattering is one of four competing processes when photons interact with matter. At energies of a few eV to a few keV, corresponding to visible light through soft X-rays, a photon can be completely absorbed and its energy can eject an electron from its host atom, a process known as the photoelectric effect. High-energy photons of and above may bombard the nucleus and cause an electron and a positron to be formed, a process called pair production; even-higher-energy photons (beyond a threshold energy of at least , depending on the nuclei involved), can eject a nucleon or alpha particle from the nucleus in a process called photodisintegration. Compton scattering is the most important interaction in the intervening energy region, at photon energies greater than those typical of the photoelectric effect but less than the pair-production threshold.
## Description of the phenomenon
By the early 20th century, research into the interaction of X-rays with matter was well under way.
|
https://en.wikipedia.org/wiki/Compton_scattering
|
passage: At the origin, after the separation of hot and cold, a ball of flame appeared that surrounded Earth like bark on a tree. This ball broke apart to form the rest of the Universe. It resembled a system of hollow concentric wheels, filled with fire, with the rims pierced by holes like those of a flute; no heavenly bodies as such, only light through the holes. Three wheels, in order outwards from Earth: stars (including planets), moon, and a large Sun. Most of Anaximander's model of the Universe comes from pseudo-Plutarch (II, 20–28):
"[The Sun] is a circle twenty-eight times as big as the Earth, with the outline similar to that of a fire-filled chariot wheel, on which appears a mouth in certain places and through which it exposes its fire, as through the hole on a flute. [...] the Sun is equal to the Earth, but the circle on which it breathes and on which it's borne is twenty-seven times as big as the whole earth. [...] [The eclipse] is when the mouth from which comes the fire heat is closed. [...] [The Moon] is a circle nineteen times as big as the whole earth, all filled with fire, like that of the Sun". Atomist universe Anaxagoras (500–428 BCE) and later Epicurus Infinite in extent The universe contains only two things: an infinite number of tiny seeds (atoms) and the void of infinite extent. All atoms are made of the same substance, but differ in size and shape.
|
https://en.wikipedia.org/wiki/Cosmology
|
passage: For instance, take , then is certainly an upper bound of , since is positive and ; that is, no element of is larger than . However, we can choose a smaller upper bound, say ; this is also an upper bound of for the same reasons, but it is smaller than , so is not a least-upper-bound of . We can proceed similarly to find an upper bound of that is smaller than , say , etc., such that we never find a least-upper-bound of in .
The least upper bound property can be generalized to the setting of partially ordered sets. See completeness (order theory).
Dedekind completeness
Dedekind completeness is the property that every Dedekind cut of the real numbers is generated by a real number. In a synthetic approach to the real numbers, this is the version of completeness that is most often included as an axiom.
The rational number line Q is not Dedekind complete. An example is the Dedekind cut
$$
L = \{ x \in \Q \mid x^2 \le 2 \vee x < 0\}.
$$
$$
R = \{ x \in \Q \mid x^2 \ge 2 \wedge x > 0 \}.
$$
L does not have a maximum and R does not have a minimum, so this cut is not generated by a rational number.
|
https://en.wikipedia.org/wiki/Completeness_of_the_real_numbers
|
passage: Similar to early regular landline telephones, operators of sound-powered telephones generally alert the receiver of a call using a hand-cranked generator (magneto), which generates an electrical current which activates a buzzer at the receiver's end, sometimes known as a howler or growler. Some telephone systems can use external electrical power to operate ringers or amplifiers, but will revert to sound-powered communications in the event of failure of the external power supply. Stations are usually connected via twisted pair wires to reduce electrical interference, and can be positioned at considerable distances from each other in the order of several kilometers. Using 1mm core diameter twisted-pair wiring, some sound-powered telephone systems can operate a pair of handsets positioned up to 48 km (30 miles) apart.
### Applications
Because sound-powered telephones do not require external electrical power, they are used where reliable communications are vital even in event of loss of power. They are often used for communications in airports, railways and public utilities, mining, ski slopes, bridges, sporting arenas and shipyards. Because they operate at low voltages, they are suitable for use in situations where there is a risk of explosions or fire, such as chemical plants, oil and gas works, arsenals, mines and quarries. They are frequently used aboard ships, especially naval vessels, and in land military communications.
Aboard naval vessels, sound-powered telephones generally have auxiliary wiring circuits routed through the ship, to reduce the likelihood that all circuits will be rendered inoperable by battle damage.
|
https://en.wikipedia.org/wiki/Telephone
|
passage: The resource space model (RSM) is a non-relational data model based on multi-dimensional classification.
### Graph model
Graph databases allow even more general structure than a network database; any node may be connected to any other node.
Multivalue model
Multivalue databases are "lumpy" data, in that they can store exactly the same way as relational databases, but they also permit a level of depth which the relational model can only approximate using sub-tables. This is nearly identical to the way XML expresses data, where a given field/attribute can have multiple right answers at the same time. Multivalue can be thought of as a compressed form of XML.
An example is an invoice, which in either multivalue or relational data could be seen as (A) Invoice Header Table - one entry per invoice, and (B) Invoice Detail Table - one entry per line item. In the multivalue model, we have the option of storing the data as on table, with an embedded table to represent the detail: (A) Invoice Table - one entry per invoice, no other tables needed.
The advantage is that the atomicity of the Invoice (conceptual) and the Invoice (data representation) are one-to-one. This also results in fewer reads, less referential integrity issues, and a dramatic decrease in the hardware needed to support a given transaction volume.
### Object-oriented database models
In the 1990s, the object-oriented programming paradigm was applied to database technology, creating a new database model known as object databases.
|
https://en.wikipedia.org/wiki/Database_model
|
passage: This tensor is significant as part of the Ricci decomposition; it is orthogonal to the difference between the Riemann tensor and itself. The other two parts of the Ricci decomposition correspond to the components of the Ricci curvature which do not contribute to scalar curvature, and to the Weyl tensor, which is the part of the Riemann tensor which does not contribute to the Ricci curvature. Put differently, the above tensor field is the only part of the Riemann curvature tensor which contributes to the scalar curvature; the other parts are orthogonal to it and make no such contribution. There is also a Ricci decomposition for the curvature of a Kähler metric.
### Basic formulas
The scalar curvature of a conformally changed metric can be computed:
$$
R(e^{2f}g)=e^{-2f}\Big(R(g)-2(n-1)\Delta^gf-(n-2)(n-1)g(df,df)\Big),
$$
using the convention for the Laplace–Beltrami operator.
|
https://en.wikipedia.org/wiki/Scalar_curvature
|
passage: A common alternative is Lagrange's notation
$$
\frac{dy}{dx}\, = y' = f'(x).
$$
Another alternative is Newton's notation, often used for derivatives with respect to time (like velocity), which requires placing a dot over the dependent variable (in this case, ):
$$
\frac{dx}{dt} = \dot{x}.
$$
Lagrange's "prime" notation is especially useful in discussions of derived functions and has the advantage of having a natural way of denoting the value of the derived function at a specific value. However, the Leibniz notation has other virtues that have kept it popular through the years.
In its modern interpretation, the expression should not be read as the division of two quantities and (as Leibniz had envisioned it); rather, the whole expression should be seen as a single symbol that is shorthand for
$$
\lim_{\Delta x \to 0} \frac{\Delta y}{\Delta x}
$$
(note vs. , where indicates a finite difference).
The expression may also be thought of as the application of the differential operator (again, a single symbol) to , regarded as a function of . This operator is written in Euler's notation. Leibniz did not use this form, but his use of the symbol corresponds fairly closely to this modern concept.
While there is traditionally no division implied by the notation (but see Nonstandard analysis), the division-like notation is useful since in many situations, the derivative operator does behave like a division, making some results about derivatives easy to obtain and remember.
|
https://en.wikipedia.org/wiki/Leibniz%27s_notation
|
passage: This example is developed further in the article on the Kleene star.
###
### General case
In the general case, the algebraic relations need not be associative, in which case the starting point is not the set of all words, but rather, strings punctuated with parentheses, which are used to indicate the non-associative groupings of letters. Such a string may equivalently be represented by a binary tree or a free magma; the leaves of the tree are the letters from the alphabet.
The algebraic relations may then be general arities or finitary relations on the leaves of the tree. Rather than starting with the collection of all possible parenthesized strings, it can be more convenient to start with the Herbrand universe. Properly describing or enumerating the contents of a free object can be easy or difficult, depending on the particular algebraic object in question. For example, the free group in two generators is easily described. By contrast, little or nothing is known about the structure of free Heyting algebras in more than one generator. The problem of determining if two different strings belong to the same equivalence class is known as the word problem.
As the examples suggest, free objects look like constructions from syntax; one may reverse that to some extent by saying that major uses of syntax can be explained and characterised as free objects, in a way that makes apparently heavy 'punctuation' explicable (and more memorable).
|
https://en.wikipedia.org/wiki/Free_object
|
passage: Start with the interval [0, 2] and glue, for each positive integer n, an interval of length 1/n to the point 1 − 1/n in the original interval. The set of singular points is discrete, but fails to be closed since 1 is an ordinary point in this
$$
\mathbb R
$$
-tree. Gluing an interval to 1 would result in a closed set of singular points at the expense of discreteness.
- The Paris metric makes the plane into a real tree. It is defined as follows: one fixes an origin
$$
P
$$
, and if two points are on the same ray from
$$
P
$$
, their distance is defined as the Euclidean distance. Otherwise, their distance is defined to be the sum of the Euclidean distances of these two points to the origin
$$
P
$$
.
- The plane under the Paris metric is an example of a hedgehog space, a collection of line segments joined at a common endpoint. Any such space is a real tree.
## Characterizations
Here are equivalent characterizations of real trees which can be used as definitions:
1) (similar to trees as graphs) A real tree is a geodesic metric space which contains no subset homeomorphic to a circle.
2) A real tree is a connected metric space
$$
(X,d)
$$
which has the four points condition (see figure):
|
https://en.wikipedia.org/wiki/Real_tree
|
passage: However, the study of this class of graphs is significantly older than their name. Peter G. Tait initiated the study of snarks in 1880, when he proved that the four color theorem is equivalent to the statement that no snark is planar. The first graph known to be a snark was the Petersen graph; it was proved to be a snark by Julius Petersen in 1898, although it had already been studied for a different purpose by Alfred Kempe in 1886.
The next four known snarks were
- the Blanuša snarks (two with 18 vertices), discovered by Danilo Blanuša in 1946,
- the Descartes snark (210 vertices), discovered by Bill Tutte in 1948, and
- the Szekeres snark (50 vertices), discovered by George Szekeres in 1973.
In 1975, Rufus Isaacs generalized Blanuša's method to construct two infinite families of snarks: the flower snarks and the Blanuša–Descartes–Szekeres snarks, a family that includes the two Blanuša snarks, the Descartes snark and the Szekeres snark. Isaacs also discovered a 30-vertex snark that does not belong to the Blanuša–Descartes–Szekeres family and that is not a flower snark: the double-star snark. Another infinite family, the Loupekine snarks, was published by Isaacs in 1976, credited to F. Loupekine. It includes two 22-vertex snarks derived from the Petersen graph.
The 50-vertex Watkins snark was discovered in 1989.
|
https://en.wikipedia.org/wiki/Snark_%28graph_theory%29
|
passage: In these cases, one is often interested in the half-planes defined by three consecutive points, and the dihedral angle between two consecutive such half-planes. If , and are three consecutive bond vectors, the intersection of the half-planes is oriented, which allows defining a dihedral angle that belongs to the interval . This dihedral angle is defined by
$$
\begin{align}
\cos \varphi&=\frac{ (\mathbf{u}_1 \times \mathbf{u}_2) \cdot (\mathbf{u}_2 \times \mathbf{u}_3)}{|\mathbf{u}_1 \times \mathbf{u}_2|\, |\mathbf{u}_2 \times \mathbf{u}_3|}\\
\sin \varphi&=\frac{ \mathbf{u}_2 \cdot((\mathbf{u}_1 \times \mathbf{u}_2) \times (\mathbf{u}_2 \times \mathbf{u}_3))}{|\mathbf{u}_2|\, |\mathbf{u}_1 \times \mathbf{u}_2|\, |\mathbf{u}_2 \times \mathbf{u}_3|},
\end{align}
$$
or, using the function atan2,
$$
|
https://en.wikipedia.org/wiki/Dihedral_angle
|
passage: - Td includes diagonal mirror planes (each diagonal plane contains only one twofold axis and passes between two other twofold axes, as in D2d). This addition of diagonal planes results in three improper rotation operations S4.
- Th includes three horizontal mirror planes. Each plane contains two twofold axes and is perpendicular to the third twofold axis, which results in inversion center i.
- O (the chiral octahedral group) has the rotation axes of an octahedron or cube (three 4-fold axes, four 3-fold axes, and six diagonal 2-fold axes).
- Oh includes horizontal mirror planes and, as a consequence, vertical mirror planes. It contains also inversion center and improper rotation operations.
- I (the chiral icosahedral group) indicates that the group has the rotation axes of an icosahedron or dodecahedron (six 5-fold axes, ten 3-fold axes, and 15 2-fold axes).
- Ih includes horizontal mirror planes and contains also inversion center and improper rotation operations.
All groups that do not contain more than one higher-order axis (order 3 or more) can be arranged as shown in a table below; symbols in red are rarely used.
n = 12345678... ∞ Cn C1 C2 C3 C4 C5 C6 C7 C8 C∞ Cnv C1v = C1h C2v C3v C4v C5v C6v C7v C8v C∞v Cnh C1h = Cs C2h C3h C4h
|
https://en.wikipedia.org/wiki/Schoenflies_notation
|
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