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Competing theories, that the observed filament was actually an illusion created by an axiconic (bessel) or moving focus instead of a "waveguided" concentration of the optical energy, were put to rest by workers at Los Alamos National Laboratory in 1997. Though sophisticated models have been developed to describe the filamentation process, a model proposed by Akozbek et al. provides a semi-analytical and easy to understand solution for the propagation of strong laser pulses in the air. Filament propagation in a semiconductor medium can also be observed in large aperture vertical cavity surface emitting lasers. | https://en.wikipedia.org/wiki/Filament_propagation |
In nonlinear optimization, quasiconvex programming studies iterative methods that converge to a minimum (if one exists) for quasiconvex functions. Quasiconvex programming is a generalization of convex programming. Quasiconvex programming is used in the solution of "surrogate" dual problems, whose biduals provide quasiconvex closures of the primal problem, which therefore provide tighter bounds than do the convex closures provided by Lagrangian dual problems. In theory, quasiconvex programming and convex programming problems can be solved in reasonable amount of time, where the number of iterations grows like a polynomial in the dimension of the problem (and in the reciprocal of the approximation error tolerated); however, such theoretically "efficient" methods use "divergent-series" stepsize rules, which were first developed for classical subgradient methods. Classical subgradient methods using divergent-series rules are much slower than modern methods of convex minimization, such as subgradient projection methods, bundle methods of descent, and nonsmooth filter methods. | https://en.wikipedia.org/wiki/Quasi-convex_function |
In nonlinear programming, the constraints are not necessarily linear. Nonetheless, many of the same principles apply. To ensure that the global maximum of a non-linear problem can be identified easily, the problem formulation often requires that the functions be convex and have compact lower level sets. This is the significance of the Karush–Kuhn–Tucker conditions. | https://en.wikipedia.org/wiki/Lagrange_duality |
They provide necessary conditions for identifying local optima of non-linear programming problems. There are additional conditions (constraint qualifications) that are necessary so that it will be possible to define the direction to an optimal solution. An optimal solution is one that is a local optimum, but possibly not a global optimum. | https://en.wikipedia.org/wiki/Lagrange_duality |
In nonlinear regression, a statistical model of the form, y ∼ f ( x , β ) {\displaystyle \mathbf {y} \sim f(\mathbf {x} ,{\boldsymbol {\beta }})} relates a vector of independent variables, x {\displaystyle \mathbf {x} } , and its associated observed dependent variables, y {\displaystyle \mathbf {y} } . The function f {\displaystyle f} is nonlinear in the components of the vector of parameters β {\displaystyle \beta } , but otherwise arbitrary. For example, the Michaelis–Menten model for enzyme kinetics has two parameters and one independent variable, related by f {\displaystyle f} by: f ( x , β ) = β 1 x β 2 + x {\displaystyle f(x,{\boldsymbol {\beta }})={\frac {\beta _{1}x}{\beta _{2}+x}}} This function is nonlinear because it cannot be expressed as a linear combination of the two β {\displaystyle \beta } s. Systematic error may be present in the independent variables but its treatment is outside the scope of regression analysis. If the independent variables are not error-free, this is an errors-in-variables model, also outside this scope. | https://en.wikipedia.org/wiki/Nonlinear_regression |
Other examples of nonlinear functions include exponential functions, logarithmic functions, trigonometric functions, power functions, Gaussian function, and Lorentz distributions. Some functions, such as the exponential or logarithmic functions, can be transformed so that they are linear. When so transformed, standard linear regression can be performed but must be applied with caution. | https://en.wikipedia.org/wiki/Nonlinear_regression |
See Linearization§Transformation, below, for more details. In general, there is no closed-form expression for the best-fitting parameters, as there is in linear regression. Usually numerical optimization algorithms are applied to determine the best-fitting parameters. | https://en.wikipedia.org/wiki/Nonlinear_regression |
Again in contrast to linear regression, there may be many local minima of the function to be optimized and even the global minimum may produce a biased estimate. In practice, estimated values of the parameters are used, in conjunction with the optimization algorithm, to attempt to find the global minimum of a sum of squares. For details concerning nonlinear data modeling see least squares and non-linear least squares. | https://en.wikipedia.org/wiki/Nonlinear_regression |
In nonlinear systems, a resonant interaction is the interaction of three or more waves, usually but not always of small amplitude. Resonant interactions occur when a simple set of criteria coupling wave-vectors and the dispersion equation are met. The simplicity of the criteria make technique popular in multiple fields. Its most prominent and well-developed forms appear in the study of gravity waves, but also finds numerous applications from astrophysics and biology to engineering and medicine. | https://en.wikipedia.org/wiki/Resonant_interaction |
Theoretical work on partial differential equations provides insights into chaos theory; there are curious links to number theory. Resonant interactions allow waves to (elastically) scatter, diffuse or to become unstable. Diffusion processes are responsible for the eventual thermalization of most nonlinear systems; instabilities offer insight into high-dimensional chaos and turbulence. | https://en.wikipedia.org/wiki/Resonant_interaction |
In nonlinear systems, the formalism of input-output stability is an important tool in studying the stability of interconnected systems since the gain of a system directly relates to how the norm of a signal increases or decreases as it passes through the system. The small-gain theorem gives a sufficient condition for finite-gain L {\displaystyle {\mathcal {L}}} stability of the feedback connection. The small gain theorem was proved by George Zames in 1966. It can be seen as a generalization of the Nyquist criterion to non-linear time-varying MIMO systems (systems with multiple inputs and multiple outputs). | https://en.wikipedia.org/wiki/Small-gain_theorem |
Theorem. Assume two stable systems S 1 {\displaystyle S_{1}} and S 2 {\displaystyle S_{2}} are connected in a feedback loop, then the closed loop system is input-output stable if ‖ S 1 ‖ ⋅ ‖ S 2 ‖ < 1 {\displaystyle \|S_{1}\|\cdot \|S_{2}\|<1} and both S 1 {\displaystyle S_{1}} and S 2 {\displaystyle S_{2}} are stable by themselves. (This norm is typically the H ∞ {\displaystyle {\mathcal {H}}_{\infty }} -norm, the size of the largest singular value of the transfer function over all frequencies. Any induced Norm will also lead to the same results). | https://en.wikipedia.org/wiki/Small-gain_theorem |
In nonlinear systems, the three-wave equations, sometimes called the three-wave resonant interaction equations or triad resonances, describe small-amplitude waves in a variety of non-linear media, including electrical circuits and non-linear optics. They are a set of completely integrable nonlinear partial differential equations. Because they provide the simplest, most direct example of a resonant interaction, have broad applicability in the sciences, and are completely integrable, they have been intensively studied since the 1970s. | https://en.wikipedia.org/wiki/Three-wave_equation |
In nonparametric regression, we have random variables X {\displaystyle X} and Y {\displaystyle Y} and assume the following relationship: E = m ( x ) , {\displaystyle \mathbb {E} =m(x),} where m ( x ) {\displaystyle m(x)} is some deterministic function. Linear regression is a restricted case of nonparametric regression where m ( x ) {\displaystyle m(x)} is assumed to be affine. Some authors use a slightly stronger assumption of additive noise: Y = m ( X ) + U , {\displaystyle Y=m(X)+U,} where the random variable U {\displaystyle U} is the `noise term', with mean 0. Without the assumption that m {\displaystyle m} belongs to a specific parametric family of functions it is impossible to get an unbiased estimate for m {\displaystyle m} , however most estimators are consistent under suitable conditions. | https://en.wikipedia.org/wiki/Nonparametric_regression |
In nonparametric statistics, a 1977 paper by Persi Diaconis and Graham studied the statistical properties of Spearman's footrule, a measure of rank correlation that compares two permutations by summing, over each item, the distance between the positions of the item in the two permutations. They compared this measure to other rank correlation methods, resulting in the "Diaconis–Graham inequalities" I + E ≤ D ≤ 2 I {\displaystyle I+E\leq D\leq 2I} where D {\displaystyle D} is Spearman's footrule, I {\displaystyle I} is the number of inversions between the two permutations (a non-normalized version of the Kendall rank correlation coefficient), and E {\displaystyle E} is the minimum number of two-element swaps needed to obtain one permutation from the other.The Chung–Diaconis–Graham random process is a random walk on the integers modulo an odd integer p {\displaystyle p} , in which at each step one doubles the previous number and then randomly adds zero, 1 {\displaystyle 1} , or − 1 {\displaystyle -1} (modulo p {\displaystyle p} ). In a 1987 paper, Chung, Diaconis, and Graham studied the mixing time of this process, motivated by the study of pseudorandom number generators. | https://en.wikipedia.org/wiki/Ronald_Graham |
In nonparametric statistics, a kernel is a weighting function used in non-parametric estimation techniques. Kernels are used in kernel density estimation to estimate random variables' density functions, or in kernel regression to estimate the conditional expectation of a random variable. Kernels are also used in time-series, in the use of the periodogram to estimate the spectral density where they are known as window functions. An additional use is in the estimation of a time-varying intensity for a point process where window functions (kernels) are convolved with time-series data. Commonly, kernel widths must also be specified when running a non-parametric estimation. | https://en.wikipedia.org/wiki/Epanechnikov_kernel |
In nonpartisan elections, each candidate for office is eligible based on his or her own merits rather than as a member of a political party. No political affiliation (if one exists) is shown on the ballot next to a candidate. Generally, the winner is chosen from a runoff election where the candidates are the top two vote-getters from a primary election. | https://en.wikipedia.org/wiki/Non-partisan_democracy |
In some elections the candidates might be members of a national party but do not run as party members for local office. Nonpartisan democracies may possess indirect elections whereby an electorate are chosen who in turn vote for the representative(s). (This is sometimes known as a 2-tier election, such as an electoral college.) | https://en.wikipedia.org/wiki/Non-partisan_democracy |
The system can work with a first past the post electoral system but is incompatible with (partisan) proportional representation systems other than single transferable vote or reweighted cardinal voting systems, or semi proportional systems such as cumulative voting and single non transferable vote.Nonpartisan elections are generally held for municipal and county offices, especially school boards, and are also common in the election of judges. In some nonpartisan elections it is common knowledge which candidates are members of and backed by which parties; in others, parties are almost wholly uninvolved and voters make choices with little or no regard to partisan considerations. While nonpartisan democracies can allow for a wide selection of candidates (especially within a no-nomination system whereby voters can choose any non-restricted person in their area), such systems are not incompatible with indirect elections (such as for large geographical areas), whereby delegates may be chosen who in turn elect the representatives. | https://en.wikipedia.org/wiki/Non-partisan_democracy |
In nonpartisan legislatures, there are no typically formal party alignments within the legislature; even if there are caucuses for specific issues. Alliances and causes with a nonpartisan body are often temporary and fluid since legislators who oppose each other on some issues may agree on other issues. Despite being nonpartisan, legislators typically have consistent and identifiable voting patterns. Decisions to investigate and enforce ethics violations by government officials are generally done on the basis of evidence instead of party affiliation. Committee chairs and other leaders within the legislature are often chosen for seniority and expertise, unlike the leaders in a partisan legislature who are often chosen because of loyalty to a party. | https://en.wikipedia.org/wiki/Non-partisan_democracy |
In nonprofit organizations, the term "program expense ratio" refers to program expenses divided by total expenses. This is one of the primary financial indicators of concern to charities and their donors. The sum of the program expense ratio and the "support service expense ratio" is by definition 100% for a non-profit organization. Charities having a higher program expense ratio (and thus a lower support service expense ratio) are often considered to be more efficient.The support service expense ratio is also commonly called the "overhead". Leading sources of information about charities, including GuideStar, Charity Navigator and the Wise Giving Alliance, say that the support service expense ratio (i.e. "overhead") can be an important indicator, especially if it is at one extreme or another, but generally speaking it is just as important to look at other factors including transparency, governance, leadership, and results.According to Charity Navigator (as of 2009), the national median for the support service expense ratio was 10 percent, and that expense ratio was less than 30 percent for more than three-fourths of the charities ranked on its website. | https://en.wikipedia.org/wiki/Expense_ratio |
In nonrelativistic classical mechanics, a closed system is a physical system that doesn't exchange any matter with its surroundings, and isn't subject to any net force whose source is external to the system. A closed system in classical mechanics would be equivalent to an isolated system in thermodynamics. Closed systems are often used to limit the factors that can affect the results of a specific problem or experiment. | https://en.wikipedia.org/wiki/Closed_systems |
In nonrelativistic quantum mechanics, an account can be given of the existence of mass and spin (normally explained in Wigner's classification of relativistic mechanics) in terms of the representation theory of the Galilean group, which is the spacetime symmetry group of nonrelativistic quantum mechanics. In 3 + 1 dimensions, this is the subgroup of the affine group on (t, x, y, z), whose linear part leaves invariant both the metric (gμν = diag(1, 0, 0, 0)) and the (independent) dual metric (gμν = diag(0, 1, 1, 1)). A similar definition applies for n + 1 dimensions. We are interested in projective representations of this group, which are equivalent to unitary representations of the nontrivial central extension of the universal covering group of the Galilean group by the one-dimensional Lie group R, cf. the article Galilean group for the central extension of its Lie algebra. | https://en.wikipedia.org/wiki/Representation_theory_of_the_Galilean_group |
The method of induced representations will be used to survey these. We focus on the (centrally extended, Bargmann) Lie algebra here, because it is simpler to analyze and we can always extend the results to the full Lie group through the Frobenius theorem. = 0 {\displaystyle =0} = 0 {\displaystyle =0} = 0 {\displaystyle =0} = 0 {\displaystyle =0} = i ℏ {\displaystyle =i\hbar } = i ℏ {\displaystyle =i\hbar } = i ℏ {\displaystyle =i\hbar } = i ℏ P i {\displaystyle =i\hbar P_{i}} = i ℏ M δ i j . | https://en.wikipedia.org/wiki/Representation_theory_of_the_Galilean_group |
{\displaystyle =i\hbar M\delta _{ij}~.} E is the generator of time translations (Hamiltonian), Pi is the generator of translations (momentum operator), Ci is the generator of Galilean boosts, and Lij stands for a generator of rotations (angular momentum operator). The central charge M is a Casimir invariant. | https://en.wikipedia.org/wiki/Representation_theory_of_the_Galilean_group |
The mass-shell invariant M E − P 2 2 {\displaystyle ME-{P^{2} \over 2}} is an additional Casimir invariant. In 3 + 1 dimensions, a third Casimir invariant is W2, where W → ≡ M L → + P → × C → , {\displaystyle {\vec {W}}\equiv M{\vec {L}}+{\vec {P}}\times {\vec {C}}~,} somewhat analogous to the Pauli–Lubanski pseudovector of relativistic mechanics. More generally, in n + 1 dimensions, invariants will be a function of W i j = M L i j + P i C j − P j C i {\displaystyle W_{ij}=ML_{ij}+P_{i}C_{j}-P_{j}C_{i}} and W i j k = P i L j k + P j L k i + P k L i j , {\displaystyle W_{ijk}=P_{i}L_{jk}+P_{j}L_{ki}+P_{k}L_{ij}~,} as well as of the above mass-shell invariant and central charge. | https://en.wikipedia.org/wiki/Representation_theory_of_the_Galilean_group |
Using Schur's lemma, in an irreducible unitary representation, all these Casimir invariants are multiples of the identity. Call these coefficients m and mE0 and (in the case of 3 + 1 dimensions) w, respectively. Recalling that we are considering unitary representations here, we see that these eigenvalues have to be real numbers. | https://en.wikipedia.org/wiki/Representation_theory_of_the_Galilean_group |
Thus, m > 0, m = 0 and m < 0. (The last case is similar to the first.) In 3 + 1 dimensions, when In m > 0, we can write, w = ms for the third invariant, where s represents the spin, or intrinsic angular momentum. | https://en.wikipedia.org/wiki/Representation_theory_of_the_Galilean_group |
More generally, in n + 1 dimensions, the generators L and C will be related, respectively, to the total angular momentum and center-of-mass moment by W i j = M S i j {\displaystyle W_{ij}=MS_{ij}} L i j = S i j + X i P j − X j P i {\displaystyle L_{ij}=S_{ij}+X_{i}P_{j}-X_{j}P_{i}} C i = M X i − P i t . {\displaystyle C_{i}=MX_{i}-P_{i}t~.} | https://en.wikipedia.org/wiki/Representation_theory_of_the_Galilean_group |
From a purely representation-theoretic point of view, one would have to study all of the representations; but, here, we are only interested in applications to quantum mechanics. There, E represents the energy, which has to be bounded below, if thermodynamic stability is required. Consider first the case where m is nonzero. | https://en.wikipedia.org/wiki/Representation_theory_of_the_Galilean_group |
Considering the (E, P→) space with the constraint we see that the Galilean boosts act transitively on this hypersurface. In fact, treating the energy E as the Hamiltonian, differentiating with respect to P, and applying Hamilton's equations, we obtain the mass-velocity relation m v→ = P→. The hypersurface is parametrized by this velocity In v→. | https://en.wikipedia.org/wiki/Representation_theory_of_the_Galilean_group |
Consider the stabilizer of a point on the orbit, (E0, 0), where the velocity is 0. Because of transitivity, we know the unitary irrep contains a nontrivial linear subspace with these energy-momentum eigenvalues. (This subspace only exists in a rigged Hilbert space, because the momentum spectrum is continuous.) | https://en.wikipedia.org/wiki/Representation_theory_of_the_Galilean_group |
The subspace is spanned by E, P→, M and Lij. We already know how the subspace of the irrep transforms under all operators but the angular momentum. Note that the rotation subgroup is Spin(3). | https://en.wikipedia.org/wiki/Representation_theory_of_the_Galilean_group |
We have to look at its double cover, because we are considering projective representations. This is called the little group, a name given by Eugene Wigner. His method of induced representations specifies that the irrep is given by the direct sum of all the fibers in a vector bundle over the mE = mE0 + P2/2 hypersurface, whose fibers are a unitary irrep of Spin(3). | https://en.wikipedia.org/wiki/Representation_theory_of_the_Galilean_group |
Spin(3) is none other than SU(2). (See representation theory of SU(2), where it is shown that the unitary irreps of SU(2) are labeled by s, a non-negative integer multiple of one half. This is called spin, for historical reasons.) | https://en.wikipedia.org/wiki/Representation_theory_of_the_Galilean_group |
Consequently, for m ≠ 0, the unitary irreps are classified by m, E0 and a spin s. Looking at the spectrum of E, it is evident that if m is negative, the spectrum of E is not bounded below. Hence, only the case with a positive mass is physical. Now, consider the case m = 0. | https://en.wikipedia.org/wiki/Representation_theory_of_the_Galilean_group |
By unitarity, is nonpositive. Suppose it is zero. Here, it is also the boosts as well as the rotations that constitute the little group. | https://en.wikipedia.org/wiki/Representation_theory_of_the_Galilean_group |
Any unitary irrep of this little group also gives rise to a projective irrep of the Galilean group. As far as we can tell, only the case which transforms trivially under the little group has any physical interpretation, and it corresponds to the no-particle state, the vacuum. The case where the invariant is negative requires additional comment. | https://en.wikipedia.org/wiki/Representation_theory_of_the_Galilean_group |
This corresponds to the representation class for m = 0 and non-zero P→. Extending the bradyon, luxon, tachyon classification from the representation theory of the Poincaré group to an analogous classification, here, one may term these states as synchrons. They represent an instantaneous transfer of non-zero momentum across a (possibly large) distance. | https://en.wikipedia.org/wiki/Representation_theory_of_the_Galilean_group |
Associated with them, by above, is a "time" operator t = − P → ⋅ C → P 2 , {\displaystyle t=-{{\vec {P}}\cdot {\vec {C}} \over P^{2}}~,} which may be identified with the time of transfer. These states are naturally interpreted as the carriers of instantaneous action-at-a-distance forces. N.B. In the 3 + 1-dimensional Galilei group, the boost generator may be decomposed into C → = W → × P → P 2 − P → t , {\displaystyle {\vec {C}}={{\vec {W}}\times {\vec {P}} \over P^{2}}-{\vec {P}}t~,} with W→ playing a role analogous to helicity. | https://en.wikipedia.org/wiki/Representation_theory_of_the_Galilean_group |
In nonstandard analysis, a branch of mathematics, a hyperfinite set or *-finite set is a type of internal set. An internal set H of internal cardinality g ∈ *N (the hypernaturals) is hyperfinite if and only if there exists an internal bijection between G = {1,2,3,...,g} and H. Hyperfinite sets share the properties of finite sets: A hyperfinite set has minimal and maximal elements, and a hyperfinite union of a hyperfinite collection of hyperfinite sets may be derived. The sum of the elements of any hyperfinite subset of *R always exists, leading to the possibility of well-defined integration.Hyperfinite sets can be used to approximate other sets. If a hyperfinite set approximates an interval, it is called a near interval with respect to that interval. Consider a hyperfinite set K = k 1 , k 2 , … , k n {\displaystyle K={k_{1},k_{2},\dots ,k_{n}}} with a hypernatural n. K is a near interval for if k1 = a and kn = b, and if the difference between successive elements of K is infinitesimal. Phrased otherwise, the requirement is that for every r ∈ there is a ki ∈ K such that ki ≈ r. This, for example, allows for an approximation to the unit circle, considered as the set e i θ {\displaystyle e^{i\theta }} for θ in the interval .In general, subsets of hyperfinite sets are not hyperfinite, often because they do not contain the extreme elements of the parent set. | https://en.wikipedia.org/wiki/Hyperfinite_set |
In nonstandard analysis, a branch of mathematics, overspill (referred to as overflow by Goldblatt (1998, p. 129)) is a widely used proof technique. It is based on the fact that the set of standard natural numbers N is not an internal subset of the internal set *N of hypernatural numbers. By applying the induction principle for the standard integers N and the transfer principle we get the principle of internal induction: For any internal subset A of *N, if 1 is an element of A, and for every element n of A, n + 1 also belongs to A,then A = *NIf N were an internal set, then instantiating the internal induction principle with N, it would follow N = *N which is known not to be the case. | https://en.wikipedia.org/wiki/Overspill |
The overspill principle has a number of useful consequences: The set of standard hyperreals is not internal. The set of bounded hyperreals is not internal. The set of infinitesimal hyperreals is not internal.In particular: If an internal set contains all infinitesimal non-negative hyperreals, it contains a positive non-infinitesimal (or appreciable) hyperreal. If an internal set contains N it contains an unlimited (infinite) element of *N. | https://en.wikipedia.org/wiki/Overspill |
In nonstandard analysis, a discipline within classical mathematics, microcontinuity (or S-continuity) of an internal function f at a point a is defined as follows: for all x infinitely close to a, the value f(x) is infinitely close to f(a).Here x runs through the domain of f. In formulas, this can be expressed as follows: if x ≈ a {\displaystyle x\approx a} then f ( x ) ≈ f ( a ) {\displaystyle f(x)\approx f(a)} .For a function f defined on R {\displaystyle \mathbb {R} } , the definition can be expressed in terms of the halo as follows: f is microcontinuous at c ∈ R {\displaystyle c\in \mathbb {R} } if and only if f ( h a l ( c ) ) ⊆ h a l ( f ( c ) ) {\displaystyle f(hal(c))\subseteq hal(f(c))} , where the natural extension of f to the hyperreals is still denoted f. Alternatively, the property of microcontinuity at c can be expressed by stating that the composition st ∘ f {\displaystyle {\text{st}}\circ f} is constant on the halo of c, where "st" is the standard part function. | https://en.wikipedia.org/wiki/Microcontinuity |
In nonstandard analysis, a field of mathematics, the increment theorem states the following: Suppose a function y = f(x) is differentiable at x and that Δx is infinitesimal. Then for some infinitesimal ε, where If Δ x ≠ 0 {\textstyle \Delta x\neq 0} then we may write which implies that Δ y Δ x ≈ f ′ ( x ) {\textstyle {\frac {\Delta y}{\Delta x}}\approx f'(x)} , or in other words that Δ y Δ x {\textstyle {\frac {\Delta y}{\Delta x}}} is infinitely close to f ′ ( x ) {\textstyle f'(x)} , or f ′ ( x ) {\textstyle f'(x)} is the standard part of Δ y Δ x {\textstyle {\frac {\Delta y}{\Delta x}}} . A similar theorem exists in standard Calculus. Again assume that y = f(x) is differentiable, but now let Δx be a nonzero standard real number. Then the same equation holds with the same definition of Δy, but instead of ε being infinitesimal, we have (treating x and f as given so that ε is a function of Δx alone). | https://en.wikipedia.org/wiki/Increment_theorem |
In nonstandard analysis, a hyperinteger n is a hyperreal number that is equal to its own integer part. A hyperinteger may be either finite or infinite. A finite hyperinteger is an ordinary integer. An example of an infinite hyperinteger is given by the class of the sequence (1, 2, 3, ...) in the ultrapower construction of the hyperreals. | https://en.wikipedia.org/wiki/Hyperinteger |
In nonstandard analysis, a monad or also a halo is the set of points infinitesimally close to a given point.Given a hyperreal number x in R∗, the monad of x is the set monad ( x ) = { y ∈ R ∗ ∣ x − y is infinitesimal } . {\displaystyle {\text{monad}}(x)=\{y\in \mathbb {R} ^{*}\mid x-y{\text{ is infinitesimal}}\}.} If x is finite (limited), the unique real number in the monad of x is called the standard part of x. == References == | https://en.wikipedia.org/wiki/Monad_(nonstandard_analysis) |
In nonstandard analysis, the standard part function is a function from the limited (finite) hyperreal numbers to the real numbers. Briefly, the standard part function "rounds off" a finite hyperreal to the nearest real. It associates to every such hyperreal x {\displaystyle x} , the unique real x 0 {\displaystyle x_{0}} infinitely close to it, i.e. x − x 0 {\displaystyle x-x_{0}} is infinitesimal. As such, it is a mathematical implementation of the historical concept of adequality introduced by Pierre de Fermat, as well as Leibniz's Transcendental law of homogeneity. | https://en.wikipedia.org/wiki/Standard_part_function |
The standard part function was first defined by Abraham Robinson who used the notation ∘ x {\displaystyle {}^{\circ }x} for the standard part of a hyperreal x {\displaystyle x} (see Robinson 1974). This concept plays a key role in defining the concepts of the calculus, such as continuity, the derivative, and the integral, in nonstandard analysis. The latter theory is a rigorous formalization of calculations with infinitesimals. The standard part of x is sometimes referred to as its shadow. | https://en.wikipedia.org/wiki/Standard_part_function |
In normal anatomy, the LRV travels between the SMA and the AA. Occasionally, the LRV travels behind the AA and in front of the spinal column. NCS is divided based on how the LRV travels, with anterior NCS being entrapment by the SMA and AA and posterior NCS being compression by the AA and spinal column. | https://en.wikipedia.org/wiki/Nutcracker_syndrome |
NCS can also be due to other causes such as compression by pancreatic cancer, retroperitoneal tumors, and abdominal aortic aneurysms. Although other subtypes exist, these causes are more uncommon in comparison to entrapment by the SMA and the AA. Patients with NCS have a tendency to have a tall and lean stature, as this can lead to a narrower gap between the SMA and the AA for the LRV. | https://en.wikipedia.org/wiki/Nutcracker_syndrome |
In normal binocular vision there is an effect of parallax, and therefore the dominant eye is the one that is primarily relied on for precise positional information. This may be extremely important in sports which require aim, such as archery, darts or shooting sports. It has been asserted that cross-dominance (in which the dominant eye is on one side and the dominant hand is on the other) is advantageous in sports requiring side-on stances (e.g. baseball, cricket, golf); and tennis, however, studies within the last 20 years have shown this not to be the case. In a 1998 study of professional baseball players, hand–ocular dominance patterns did not show an effect on batting average or ERA. | https://en.wikipedia.org/wiki/Ocular_dominance |
Similarly, in 2005, a South African study found that "cricketers were not more likely to have crossed dominance" than the normal population.Ocular dominance is an important consideration in predicting patient satisfaction with monovision correction in cataract surgery refractive surgery, also laser eye surgery, and contact lens wear. The dominant eye has more neural connections to the brain than the other eye does. | https://en.wikipedia.org/wiki/Ocular_dominance |
According to a sixty-person study in the Proceedings of the Royal Society B, in non-dyslexic people, the blue cone-free spot in the dominant eye tends to be round and the same spot in the non-dominant eye tends to be unevenly shaped; in dyslexic people both eyes tend to have round areas. The study suggests this difference may be a potential, and possibly treatable, cause of dyslexia; however, further tests are required to confirm this. At least 700 million people worldwide have dyslexia. In response to the study, John Stein of the University of Oxford cautions that while the study is "really interesting", there is no one single cause of dyslexia. | https://en.wikipedia.org/wiki/Ocular_dominance |
In normal bone, fractures occur when there is significant force applied or repetitive trauma over a long time. Fractures can also occur when a bone is weakened, such as with osteoporosis, or when there is a structural problem, such as when the bone remodels excessively (such as Paget's disease) or is the site of the growth of cancer. Common fractures include wrist fractures and hip fractures, associated with osteoporosis, vertebral fractures associated with high-energy trauma and cancer, and fractures of long-bones. Not all fractures are painful. | https://en.wikipedia.org/wiki/Compact_bone |
When serious, depending on the fractures type and location, complications may include flail chest, compartment syndromes or fat embolism. Compound fractures involve the bone's penetration through the skin. Some complex fractures can be treated by the use of bone grafting procedures that replace missing bone portions. | https://en.wikipedia.org/wiki/Compact_bone |
Fractures and their underlying causes can be investigated by X-rays, CT scans and MRIs. Fractures are described by their location and shape, and several classification systems exist, depending on the location of the fracture. A common long bone fracture in children is a Salter–Harris fracture. | https://en.wikipedia.org/wiki/Compact_bone |
When fractures are managed, pain relief is often given, and the fractured area is often immobilised. This is to promote bone healing. In addition, surgical measures such as internal fixation may be used. Because of the immobilisation, people with fractures are often advised to undergo rehabilitation. | https://en.wikipedia.org/wiki/Compact_bone |
In normal brain cells, PLA2 regulation accounts for a balance between arachidonic acid's conversion into proinflammatory mediators and its reincorporation into the membrane. In the absence of strict regulation of PLA2 activity, a disproportionate amount of proinflammatory mediators are produced. The resulting induced oxidative stress and neuroinflammation is analogous to neurological diseases such as Alzheimer's disease, epilepsy, multiple sclerosis, ischemia. Lysophospholipids are another class of molecules released from the membrane that are upstream predecessors of platelet activating factors (PAF). | https://en.wikipedia.org/wiki/Phospholipase_A2 |
Abnormal levels of potent PAF are also associated with neurological damage. An optimal enzyme inhibitor would specifically target PLA2 activity on neural cell membranes already under oxidative stress and potent inflammation. Thus, specific inhibitors of brain PLA2 could be a pharmaceutical approach to treatment of several disorders associated with neural trauma.Increase in phospholipase A2 activity is an acute-phase reaction that rises during inflammation, which is also seen to be exponentially higher in low back disc herniations compared to rheumatoid arthritis. It is a mixture of inflammation and substance P that are responsible for pain.Increased phospholipase A2 has also been associated with neuropsychiatric disorders such as schizophrenia and pervasive developmental disorders (such as autism), though the mechanisms involved are not known. | https://en.wikipedia.org/wiki/Phospholipase_A2 |
In normal calcium regulation, a decrease in plasma calcium levels causes the parathyroid glands to secrete parathyroid hormone (PTH), which regulates the activation of Vitamin D3 in the kidney. These two compounds act to increase blood calcium levels by increasing absorption of dietary calcium from the intestine, increasing renal tubular reabsorption of calcium in the kidney, and increasing resorption of calcium from bones.It has been found that tissue is less responsive to parathyroid hormone prepartum, compared to postpartum. It is believed that hypocalcemia causing milk fever is due to a lower level of responsiveness of the cow's tissues to circulating parathyroid hormone.The resultant decreased plasma calcium causes hyperexcitability of the nervous system and weakened muscle contractions, which result in both tetany and paresis. | https://en.wikipedia.org/wiki/Milk_fever |
In normal cells, TGF-β, acting through its signaling pathway, stops the cell cycle at the G1 stage to stop proliferation, induce differentiation, or promote apoptosis. In many cancer cells, parts of the TGF-β signaling pathway are mutated, and TGF-β no longer controls the cell. These cancer cells proliferate. The surrounding stromal cells (fibroblasts) also proliferate. | https://en.wikipedia.org/wiki/Tumor_growth_factor_(TGF)_beta |
Both cells increase their production of TGF-β. This TGF-β acts on the surrounding stromal cells, immune cells, endothelial and smooth-muscle cells. It causes immunosuppression and angiogenesis, which makes the cancer more invasive. | https://en.wikipedia.org/wiki/Tumor_growth_factor_(TGF)_beta |
TGF-β1 has been implicated in the process of activating Hepatic Stellate Cells (HSCs) with the magnitude of hepatic fibrosis being in proportion to increase in TGF-β levels. Studies have shown that ACTA2 is associated with TGF-β pathway that enhances contractile properties of HSCs leading to Liver fibrosis. TGF-β also converts effector T-cells, which normally attack cancer with an inflammatory (immune) reaction, into regulatory (suppressor) T-cells, which turn off the inflammatory reaction. | https://en.wikipedia.org/wiki/Tumor_growth_factor_(TGF)_beta |
Normal tissue integrity is preserved by feedback interactions between different cell types that express adhesion molecules and secrete cytokines. Disruption of these feedback mechanisms in cancer damages a tissue. When TGF-β signaling fails to control NF-κB activity in cancer cells, this has at least two potential effects: first, it enables the malignant tumor to persist in the presence of activated immune cells, and second, the cancer cell outlasts immune cells because it survives in the presence of apoptotic, and anti-inflammatory mediators.Furthermore, forkhead box protein 3 (FOXP3) as a transcription factor is an essential molecular marker of regulatory T (Treg) cells. FOXP3 polymorphism (rs3761548) might be involved in cancer progression like gastric cancer through influencing Tregs function and the secretion of immunomodulatory cytokines such as IL-10, IL-35, and TGF-β. | https://en.wikipedia.org/wiki/Tumor_growth_factor_(TGF)_beta |
In normal cellular operations, there is a balance between the production of lipids, and their oxidation or transport. In lipotoxic cells, there is an imbalance between the amount of lipids produced and the amount used. Upon entrance of the cell, fatty acids can be converted to different types of lipids for storage. Triacylglycerol consists of three fatty acids bound to a glycerol molecule and is considered the most neutral and harmless type of intracellular lipid storage. | https://en.wikipedia.org/wiki/Lipotoxicity |
Alternatively, fatty acids can be converted to lipid intermediates like diacylglycerol, ceramides and fatty acyl-CoAs. These lipid intermediates can impair cellular function, which is referred to as lipotoxicity.Adipocytes, the cells that normally function as lipid store of the body, are well equipped to handle the excess lipids. Yet, too great of an excess will overburden these cells and cause a spillover into non-adipose cells, which do not have the necessary storage space. | https://en.wikipedia.org/wiki/Lipotoxicity |
When the storage capacity of non-adipose cells is exceeded, cellular dysfunction and/or death result. The mechanism by which lipotoxicity causes death and dysfunction is not well understood. The cause of apoptosis and extent of cellular dysfunction is related to the type of cell affected, as well as the type and quantity of excess lipids. | https://en.wikipedia.org/wiki/Lipotoxicity |
A theory has been put forward by Cambridge researchers relating the development of lipotoxicity to the perturbation of membrane glycerophospholipid/sphingolipid homeostasis and their associated signalling events.Currently, there is no universally accepted theory for why certain individuals are afflicted with lipotoxicity. Research is ongoing into a genetic cause, but no individual gene has been named as the causative agent. The causative role of obesity in lipotoxicity is controversial. | https://en.wikipedia.org/wiki/Lipotoxicity |
Some researchers claim that obesity has protective effects against lipotoxicity as it results in extra adipose tissue in which excess lipids can be stored. Others claim obesity is a risk factor for lipotoxicity. | https://en.wikipedia.org/wiki/Lipotoxicity |
Both sides accept that high fat diets put patients at increased risk for lipotoxic cells. Individuals with high numbers of lipotoxic cells usually experience both leptin and insulin resistance. However, no causative mechanism has been found for this correlation. | https://en.wikipedia.org/wiki/Lipotoxicity |
In normal circumstances after injury HIF-1a is degraded by prolyl hydroxylases (PHDs). In June 2015, scientists found that the continued up-regulation of HIF-1a via PHD inhibitors regenerates lost or damaged tissue in mammals that have a repair response; and the continued down-regulation of Hif-1a results in healing with a scarring response in mammals with a previous regenerative response to the loss of tissue. The act of regulating HIF-1a can either turn off, or turn on the key process of mammalian regeneration. One such regenerative process in which HIF1A is involved is skin healing. | https://en.wikipedia.org/wiki/Hypoxia_inducible_factor |
Researchers at the Stanford University School of Medicine demonstrated that HIF1A activation was able to prevent and treat chronic wounds in diabetic and aged mice. Not only did the wounds in the mice heal more quickly, but the quality of the new skin was even better than the original. Additionally the regenerative effect of HIF-1A modulation on aged skin cells was described and a rejuvenating effect on aged facial skin was demonstrated in patients. | https://en.wikipedia.org/wiki/Hypoxia_inducible_factor |
HIF modulation has also been linked to a beneficial effect on hair loss. The biotech company Tomorrowlabs GmbH, founded in Vienna in 2016 by the physician Dominik Duscher and pharmacologist Dominik Thor, makes use of this mechanism. Based on the patent-pending HSF ("HIF strengthening factor") active ingredient, products have been developed that are supposed to promote skin and hair regeneration. | https://en.wikipedia.org/wiki/Hypoxia_inducible_factor |
In normal circumstances after injury HIF1A is degraded by prolyl hydroxylases (PHDs). In June 2015, scientists found that the continued up-regulation of HIF1A via PHD inhibitors regenerates lost or damaged tissue in mammals that have a repair response; and the continued down-regulation of HIF1A results in healing with a scarring response in mammals with a previous regenerative response to the loss of tissue. The act of regulating HIF1A can either turn off, or turn on the key processes of mammalian regeneration. One such regenerative process in which HIF1A is involved is peripheral nerve regeneration. | https://en.wikipedia.org/wiki/Hypoxia-inducible_factor_1,_alpha_subunit |
Following axon injury, HIF1A activates VEGFA to promote regeneration and functional recovery. HIF1A also controls skin healing. Researchers at the Stanford University School of Medicine demonstrated that HIF1A activation was able to prevent and treat chronic wounds in diabetic and aged mice. | https://en.wikipedia.org/wiki/Hypoxia-inducible_factor_1,_alpha_subunit |
Not only did the wounds in the mice heal more quickly, but the quality of the new skin was even better than the original. Additionally the regenerative effect of HIF-1A modulation on aged skin cells was described and a rejuvenating effect on aged facial skin was demonstrated in patients. HIF modulation has also been linked to a beneficial effect on hair loss. The biotech company Tomorrowlabs GmbH, founded in Vienna in 2016 by the physician Dominik Duscher and pharmacologist Dominik Thor, makes use of this mechanism. Based on the patent-pending HSF ("HIF strengthening factor") active ingredient, products have been developed that are supposed to promote skin and hair regeneration. | https://en.wikipedia.org/wiki/Hypoxia-inducible_factor_1,_alpha_subunit |
In normal circumstances, average capital is positive. When an intra-period outflow is large and early enough, average capital can be negative or zero. Negative average capital causes the Modified Dietz return to be negative when there is a profit, and positive when there is a loss. This resembles the behaviour of a liability or short position, even if the investment is not actually a liability or a short position. | https://en.wikipedia.org/wiki/Modified_Dietz_method |
In cases where the average capital is zero, no Modified Dietz return can be calculated. If the average capital is close to zero, the Modified Dietz return will be large (large and positive, or large and negative). One partial workaround solution involves as a first step, to capture the exception, detecting for example when the start value (or first inflow) is positive, and the average capital is negative. Then in this case, use the simple return method, adjusting the end value for outflows. This is equivalent to the sum of constituent contributions, where the contributions are based on simple returns and weights depending on start values. | https://en.wikipedia.org/wiki/Modified_Dietz_method |
In normal conditions, alveolar macrophages adhere closely to alveolar epithelial cells, thus inducing the expression of the αvβ6 integrin. Integrins are dimeric cell-surface receptors composed of alpha and beta subunits, which activates TGF-β.< TGF-β is a multifunctional cytokine that modulates a variety of biological processes such as cell growth, apoptosis, extracellular matrix synthesis, inflammation, and immune responses. TGF-β tightly regulates anti-inflammatory activity by suppressing pro-inflammatory cytokine production, thereby inhibiting T-lymphocyte function. Integrins avβ6 and avβ8 sequester latent TGF-β to the cell surface, where activation can be tightly coupled to cellular responses to environmental stress in the maintenance of homeostasis; integrins also localize activated TGFβ in the vicinity of the macrophages. | https://en.wikipedia.org/wiki/Alveolar_macrophage |
Normally mature TGFβ is secreted as a latent complex with its N-terminal fragment, latency-associated peptide (LAP), which inhibits its activity. The latent complex is covalently linked to the extracellular matrix by binding to latent TGF-β-binding proteins. TGF-β is activated by diverse mechanisms in the lung, ultimately involving either proteolysis or conformational alteration of the LAP. | https://en.wikipedia.org/wiki/Alveolar_macrophage |
αvβ6 integrin is able to mediate activation of TGF-β by binding to TGF-β1 LAP, which serves as a ligand binding site for the integrin, and is an essential component of the TGF-β activation apparatus. Once activated, TGFβ leads to the suppression of macrophage functionality (cytokine production and phagocytosis). Binding of activated TGF-β to its receptors expressed on alveolar macrophages induces a downstream signaling cascade, including phosphorylation of receptor-regulated Small Mothers Against Decapentaplegic (R-SMAD)homologs 2 and 3. | https://en.wikipedia.org/wiki/Alveolar_macrophage |
Phosphorylated SMAD-2 and -3 then form heteromeric complexes with common-mediator SMAD 4 (co-SMAD-4). Once assembled, the complexes translocates into the nucleus via the nuclear pore with the assistance of importins alpha/beta. Once in the nucleus, these complexes accumulate and eventually act as a transcription factors, regulating the expression of TGF-β target genes. Thus TGF-β signaling involves a direct pathway from the receptors on the surface of a cell to the nucleus. | https://en.wikipedia.org/wiki/Alveolar_macrophage |
In normal conditions, common centrifugal pumps are unable to evacuate the air from an inlet line leading to a fluid level whose geodetic altitude is below that of the pump. Self-priming pumps have to be capable of evacuating air (see Venting) from the pump suction line without any external auxiliary devices. Centrifugal pumps with an internal suction stage such as water-jet pumps or side-channel pumps are also classified as self-priming pumps. Self-Priming centrifugal pumps were invented in 1935. | https://en.wikipedia.org/wiki/Centrifugal_Pump |
One of the first companies to market a self-priming centrifugal pump was American Marsh in 1938.Centrifugal pumps that are not designed with an internal or external self-priming stage can only start to pump the fluid after the pump has initially been primed with the fluid. Sturdier but slower, their impellers are designed to move liquid, which is far denser than air, leaving them unable to operate when air is present. In addition, a suction-side swing check valve or a vent valve must be fitted to prevent any siphon action and ensure that the fluid remains in the casing when the pump has been stopped. | https://en.wikipedia.org/wiki/Centrifugal_Pump |
In self-priming centrifugal pumps with a separation chamber the fluid pumped and the entrained air bubbles are pumped into the separation chamber by the impeller action. The air escapes through the pump discharge nozzle whilst the fluid drops back down and is once more entrained by the impeller. The suction line is thus continuously evacuated. | https://en.wikipedia.org/wiki/Centrifugal_Pump |
The design required for such a self-priming feature has an adverse effect on pump efficiency. Also, the dimensions of the separating chamber are relatively large. For these reasons this solution is only adopted for small pumps, e.g. garden pumps. | https://en.wikipedia.org/wiki/Centrifugal_Pump |
More frequently used types of self-priming pumps are side-channel and water-ring pumps. Another type of self-priming pump is a centrifugal pump with two casing chambers and an open impeller. This design is not only used for its self-priming capabilities but also for its degassing effects when pumping two-phase mixtures (air/gas and liquid) for a short time in process engineering or when handling polluted fluids, for example, when draining water from construction pits. | https://en.wikipedia.org/wiki/Centrifugal_Pump |
This pump type operates without a foot valve and without an evacuation device on the suction side. The pump has to be primed with the fluid to be handled prior to commissioning. Two-phase mixture is pumped until the suction line has been evacuated and the fluid level has been pushed into the front suction intake chamber by atmospheric pressure. During normal pumping operation this pump works like an ordinary centrifugal pump. | https://en.wikipedia.org/wiki/Centrifugal_Pump |
In normal conditions, the vascular endothelial nitric oxide synthase produces nitric oxide from L-arginine in the presence of oxygen.This nitric oxide diffuses into neighboring cells (including vascular smooth muscle cells and platelets), where it increases the activity of the enzyme soluble guanylate cyclase, leading to increased formation of cyclic guanosine monophosphate (cGMP) from guanosine triphosphate (GTP). The cGMP then activates cGMP-dependent kinase or PKG (protein kinase G). Activated PKG promotes vasorelaxation (via a reduction of intracellular calcium levels), alters the expression of genes involved in smooth muscle cell contraction, migration and differentiation, and inhibits platelet activation. Nitric oxide–soluble guanylate cyclase signaling also leads to anti-inflammatory effects.Phosphodiesterase type 5 (PDE5), which is abundant in the pulmonary tissue, hydrolyzes the cyclic bond of cGMP. Consequently, the concentration of cGMP (and thus PKG activity) decreases. | https://en.wikipedia.org/wiki/Pulmonary_hypertension |
In normal configuration, the light beam samples perpendicularly the electrode surface. Normal configuration provides optical information related to the changes that take place in the solution adjacent to the electrode and on the electrode surface. The optical path length coincides with the diffusion layer thickness, which is usually in the order of micrometers. This arrangement is the most suitable when the compound of interest is deposited or adsorbed on the working electrode, because it provides information about all processes occurring on the electrode surface.UV-Vis absorption SEC in normal arrangement can be performed using both transmission and reflection phenomena. | https://en.wikipedia.org/wiki/UV-Vis_absorption_spectroelectrochemistry |
Normal transmissionIn normal transmission, the light beam passes through a optically transparent working electrode, collecting information about the phenomena that take place on the surface of the electrode and on the solution adjacent to it. Electrodes in this configuration must be composed of materials that have great electrical conductivity and adequate optical transparency in the spectral region of interest.The external reflection mode was proposed to improve the sensitivity and to use non-transparent electrodes. Normal reflectionIn normal reflection, the light beam travels in a perpendicular direction to the working electrode surface on which the reflection occurs. | https://en.wikipedia.org/wiki/UV-Vis_absorption_spectroelectrochemistry |
The reflected beam is collected to be analyzed in the spectrometer. It is also possible to work with other incidence and collection angles. This configuration is an alternative when the working electrode is non-transparent. | https://en.wikipedia.org/wiki/UV-Vis_absorption_spectroelectrochemistry |
In this configuration, the optical path-length in solution is on the order of twice the diffusion layer thickness. It should be noticed that growth of films on the electrode surface could cause optical interference phenomena. As it is based on reflection phenomenon, in many cases reflectance is used as unit of measurement instead of absorbance. | https://en.wikipedia.org/wiki/UV-Vis_absorption_spectroelectrochemistry |
In normal development, endogenous Sonic hedgehog signaling stimulates rapid proliferation of cerebellar granule neuron progenitors (CGNPs) in the external granule layer (EGL). Cerebellum development occurs during late embryogenesis and the early postnatal period, with CGNP proliferation in the EGL peaking during early development (P7, postnatal day 7, in the mouse). As CGNPs terminally differentiate into cerebellum granule cells (also called cerebellar granule neurons, CGNs), they migrate to the internal granule layer (IGL), forming the mature cerebellum (by P20, post-natal day 20 in the mouse). Mutations that abnormally activate Sonic hedgehog signaling predispose to cancer of the cerebellum (medulloblastoma) in humans with Gorlin syndrome and in genetically engineered mouse models. | https://en.wikipedia.org/wiki/Parallel_fiber |
In normal dopamine and serotonin (5-HT) neurotransmitter synthesis, AADC is not the rate-limiting step in either reaction. However, AADC becomes the rate-limiting step of dopamine synthesis in patients treated with L-DOPA (such as in Parkinson's disease), and the rate-limiting step of serotonin synthesis in people treated with 5-HTP (such as in mild depression or dysthymia). AADC is inhibited by carbidopa outside of the blood brain barrier to inhibit the premature conversion of L-DOPA to dopamine in the treatment of Parkinson's. In humans, AADC is also the rate-limiting enzyme in the formation of trace amines. | https://en.wikipedia.org/wiki/DOPA_decarboxylase |
Aromatic l-amino acid decarboxylase deficiency is associated with various symptoms as severe developmental delay, oculogyric crises and autonomic dysfunction. The molecular and clinical spectrum of AAAC deficiency is heterogeneous. The first case of AADC deficiency was described in twin brothers 1990. | https://en.wikipedia.org/wiki/DOPA_decarboxylase |
Patients can be treated with dopamine agonists, MAO inhibitors, and pyridoxine (vitamin B6). Clinical phenotype and response to treatment is variable and the long-term and functional outcome is unknown. To provide a basis for improving the understanding of the epidemiology, genotype–phenotype correlation and outcome of these diseases their impact on the quality of life of patients, and for evaluating diagnostic and therapeutic strategies a patient registry was established by the noncommercial International Working Group on Neurotransmitter Related Disorders (iNTD).Immunohistochemical studies have revealed that AADC is expressed in various neuronal cell types such as serotonergic and catecholaminergic neurons. | https://en.wikipedia.org/wiki/DOPA_decarboxylase |
Neurons that express AADC but are not considered classical monoaminergic cell neurons are termed D cells. Cells that are immunoreactive for AADC have also been found in the human brainstem. These cells include melanin-pigmented cells that are typically designated as catecholaminergic and may also be serotonergic. Significant localization of dopaminergic cells that are also immunoreactive for AADC is reported in the substantia nigra, ventral tegmental area, and the mesencephalic reticular formation. Unlike previous reports on animal models, nonaminergic (D cells) are unlikely to be observed in the human brain. | https://en.wikipedia.org/wiki/DOPA_decarboxylase |
In normal electron demand Diels-Alder reactions, the Z-substituted dienophiles react with 1-substituted butadienes to give 3,4-disubstituted cyclohexenes, independent of the nature of diene substituents. This is also known as ortho effect. | https://en.wikipedia.org/wiki/Ortho_effect |
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