During the second half of the twentieth century, mathematicians such as Chevalley and Steinberg also increased our understanding of finite analogs of classical groups, and other related groups.

Arnold's trinities begin with R / C / H ( the real numbers, complex numbers, and quaternions ), which he remarks " everyone knows ", and proceeds to imagine the other trinities as " complexifications " and " quaternionifications " of classical ( real ) mathematics, by analogy with finding symplectic analogs of classic Riemannian geometry, which he had previously proposed in the 1970s.

Following an incident in which a healthy young man suffered a myocardial infarction ( heart attack ) linked to bitter orange, a case study found that dietary supplement manufacturers who replaced ephedra with its analogs from " bitter orange " had in effect found a loophole in the ephedra ban, substituting a similarly dangerous substance while labeling the products as " ephedra-free ".

Note that, unlike the case on a plane, spherical " biangles " ( two-sided analogs to triangle ) are possible ( such as a slice cut out of an orange ).

Additionally, legal ambiguities regarding the legality of certain analogs of scheduled substances had been established in the aforementioned court case ( In particular, the similarities of AET and DMT were debated ).

In the case of some analogs, photochromic behavior can even be carried out in single crystals without disrupting the crystal structure.

Following the presentation of a healthy young man with a myocardial infarction, a case study and subsequent literature review found that the makers of " nutritional supplements " who replaced ephedrine with its analogs p-synephrine and / or p-octopamine from " bitter orange " had in effect simply found a loophole in the FDA's April 2004 regulation banning ephedra in those supplements by substituting a similar substance the regulation did not address, while permitting them to label the products as " ephedra-free ".

The active principle responsible for the laxative effect is emodin, present in its natural form as a complex of its analogs.

This remarkably systematic correspondence is due to the identity of the commutators of the umbral quantities to their continuum analogs ( h → 0 limits ),

Sheet music is a handwritten or printed form of music notation that uses modern musical symbols ; like its analogs — books, pamphlets, etc.

These studies draw on the tenets of behavioral psychology, where the main goal is to discover analogs to human behavior in experimentally-tractable non-human animals.

In mathematics, the exterior product or wedge product of vectors is an algebraic construction used in Euclidean geometry to study areas, volumes, and their higher-dimensional analogs.

Each alternate Abenobashi is a surreal manifestation of Sasshi's otaku interests, populated by analogs of the protagonist's relatives and acquaintances and a blue-haired stranger known as Eutus.

In western countries, it is more common to use a pure culture containing only Rhizopus oligosporus, which makes very little B < sub > 12 </ sub > and could be missing Klebsiella pneumoniae, which has been shown to produce significant levels of B < sub > 12 </ sub > analogs in tempeh when present.

Transcendental function is a term often used to describe the trigonometric functions ( sine, cosine, tangent, their reciprocals cotangent, secant, and cosecant, the now little-used versine, haversine, and coversine, their analogs the hyperbolic functions and so forth ).

Formally, a Lie superalgebra is a ( nonassociative ) Z < sub > 2 </ sub >- graded algebra, or superalgebra, over a commutative ring ( typically R or C ) whose product, called the Lie superbracket or supercommutator, satisfies the two conditions ( analogs of the usual Lie algebra axioms, with grading ):

* 53A5-optimised by first fitting to reproduce the thermodynamic properties of pure liquids of a range of small polar molecules and the solvation free enthalpies of amino acid analogs in cyclohexane, is an expansion and renumbering of 45A3.

A newer hormonal intervention used in Europe is use of GnRH analogs such as nafarelin or buserelin ; the success rates and putative mechanism of action are similar to hCG, but some surgeons have combined the two treatments and reported higher descent rates.

It is known in mathematics as the diffusion equation, and has many analogs in other fields.

One strategy for a habitable planet search is to do a ' deeper ' search ( i. e. to lower mass sensitivity in the habitable zone ) of a smaller number of targets if Earth analogs are common.

Iodopsins are very close analogs of the visual purple rhodopsin that is used in night vision.

Collaborative intelligence relates to whether prediction is defined as active, how heuristics are used, and whether analogs to developmental processes for facilitated variation enable systems to evolve non-randomly toward increased functional effectiveness.

Long-term treatment of hypoparathyroidism is with vitamin D analogs and calcium supplementation may be ineffective in some due to potential renal damage.

The basic garment known in English most often as a " sarong ", sewn or unsewn, has analogs in many regions, where it shows variations in style and is known by different names.

* LUC is a public-key cryptosystem based on Lucas sequences that implements the analogs of ElGamal ( LUCELG ), Diffie-Hellman ( LUCDIF ), and RSA ( LUCRSA ).

Erawa's main difference from the Soviet analogs is that Erawa's containers fit almost without gaps while on the Soviet modernised T-72 gaps reach 10 – 15 mm noticeably decreasing defense effectiveness.

Since it can be concluded that the estrogenic activity of butylparaben is negligible under normal use, the same should be concluded for shorter analogs.

Despite the fat solubility of aniracetam, its half-life is much shorter than that of common racetam analogs such as piracetam.

The basic structure of the stairs and platforms is modeled on the Penrose triangle, or " impossible triangle ", which has parallels with higher-dimensional analogs of the Möbius strip.

Another cause of steatorrhea is due to the adverse effect of octreotide or lanreotide, which are analogs of somatostatin, used clinically to treat acromegaly.

These drugs, AVI-6002 and AVI-6003 are novel analogs based on AVI's PMO antisense chemistry in which anti-viral potency is enhanced by the addition of positively-charged components to the morpholino oligomer chain.

This orthogonality relation can then be used to extract the coefficients in the Fourier – Bessel series, where a function is expanded in the basis of the functions J < sub > α </ sub >( x u < sub > α, m </ sub >) for fixed α and varying m.

The coefficients in the upper figure are computed by the Fourier series integral.

* It is the discrete analogy of the formula for the coefficients of a Fourier series:

Conversely, if the data are sparse — that is, if only K out of N Fourier coefficients are nonzero — then the complexity can be reduced to O ( K log N log ( N / K )), and this has been demonstrated to lead to practical speedups compared to an ordinary FFT for N / K > 32 in a large-N example ( N = 2 < sup > 22 </ sup >) using a probabilistic approximate algorithm ( which estimates the largest K coefficients to several decimal places ).

The coefficients in the upper and lower figures are respectively computed by the Fourier series integral and the discrete Fourier transform | DFT summation.

The Fourier transform of a periodic function, s < sub > P </ sub >( t ), with period P, becomes a Dirac comb function, modulated by a sequence of complex coefficients:

The inverse transform, known as Fourier series, is a representation of s < sub > P </ sub >( t ) in terms of a summation of a potentially infinite number of harmonically related sinusoids or complex exponential functions, each with an amplitude and phase specified by one of the coefficients:

Thus, any periodic summation in the frequency domain can be represented by a Fourier series, whose coefficients are samples of a related continuous time function:

The Fourier series coefficients, defined by:

The DTFT of a periodic sequence, s < sub > N </ sub >, with period N, becomes another Dirac comb function, modulated by the coefficients of a Fourier series.

which is a periodic function and its equivalent representation as a Fourier series, whose coefficients are x.

and hence Fourier sums of continuum functions are readily mapped to umbral Fourier sums faithfully, i. e., involving the same Fourier coefficients multiplying these umbral basis exponentials.

Additionally, when using Fourier analysis techniques for waves in a lossy medium, the resulting attenuation is easier to deal with using complex Fourier coefficients.

The " weights ", or coefficients, of the terms in the Fourier expansion of a function can be thought of as components of a vector in an infinite dimensional space known as a Hilbert space.

In the case of a periodic function, such as a continuous, but not necessarily sinusoidal, musical tone, the Fourier transform can be simplified to the calculation of a discrete set of complex amplitudes, called Fourier series coefficients.

Re-writing sines and cosines as complex exponentials makes it necessary for the Fourier coefficients to be complex valued.

As we increase the length of the interval on which we calculate the Fourier series, then the Fourier series coefficients begin to look like the Fourier transform and the sum of the Fourier series of ƒ begins to look like the inverse Fourier transform.