Similarly, in 1844 while working on cubic reciprocity, Eisenstein introduced the ring, where is a cube root of unity.

Similarly roots of cubic and quartic ( third and fourth degree ) equations can be expressed using arithmetic, square roots, and cube roots, or alternatively using arithmetic and trigonometric functions.

Similarly, cubic equations with three real solutions have an algebraic solution that is unhelpful in that it contains cube roots of complex numbers ; again an alternative solution exists in terms of trigonometric functions of real terms.

Similarly, Arizona retains a shall-issue permit process, both for reciprocity purposes and because permit holders are allowed to carry concealed handguns in certain places ( such as bars and restaurants that serve alcohol ) that non-permit holders are not.

Similarly, Herman's theory of doublespeak having an inherent nature to be manipulative and Chomsky's theory of ' dichotomization ' relates directly to the practice of doublespeak and how doublespeak is deliberately deceptive in nature.

Similarly, Slack relates that it is not true that the leg had to be amputated because of gangrene.

Similarly it relates to the infinite series of tilings with the face configurations V3. 2n. 3. 2n, the first in the Euclidean plane, and the rest in the hyperbolic plane.

Similarly, with Critical Infrastructure Protection ( CIP ), which relates to the preparedness and response to serious incidents, the Act gave DHS broad responsibility to minimize damage but only limited authority to share information and to coordinate the development of private sector best practices.

Similarly, the Hebrew Bible relates how, at the time Phineas son of Eleazar appeased God's anger, he merited the divine blessing of God:.

Similarly, Slack relates that it is not true that the leg had to be amputated because of gangrene.

Similarly, as the values of x become smaller and smaller, say. 01,. 001,. 0001, ..., making them infinitesimal relative to the scale shown, the corresponding values of y, 100, 1000, 10, 000 ..., become larger and larger.

Similarly, given any element y of Y, there is a function f < sub > y </ sub >, or f (·, y ), from X to Z, given by f < sub > y </ sub >( x ) := f ( x, y ).

Similarly, 30-06 Springfield can become 308 Winchester, which can become 308 x 1. 5 or any number of specialized benchrest shooting cartridges.

Similarly, if one connects two electrical resistors in parallel, one having resistance x ( e. g. 60Ω ) and one having resistance y ( e. g. 40Ω ), then the effect is the same as if one had used two resistors with the same resistance, both equal to the harmonic mean of x and y ( 48Ω ): the equivalent resistance in either case is 24Ω ( one-half of the harmonic mean ).

Similarly, given points ( x < sub > 1 </ sub >, y < sub > 1 </ sub >, z < sub > 1 </ sub >) and ( x < sub > 2 </ sub >, y < sub > 2 </ sub >, z < sub > 2 </ sub >) in three-space, the distance between them is:

To see this, the three corresponding normal vectors of the corner's perpendicular sides can be considered to form a basis ( a rectangular coordinate system ) ( x, y, z ) in which to represent the direction of an arbitrary incoming ray, b, c. When the ray reflects from the first side, say x, the ray's x component, a, is reversed to-a while the y and z components are unchanged, resulting in a direction of b, c. Similarly, when reflected from side y and finally from side z, the b and c components are reversed.

Similarly, the limit superior of ( x < sub > n </ sub >) is defined by

Similarly they produce 1 for < tt > pow ( x, 0 )</ tt > even when x is 0 or infinity.

# Similarly, any poset P may be regarded as an abstract category with a unique arrow x → y whenever x ≤ y.

Similarly, if g and h are elements of a finite cyclic group G then a solution x of the equation g < sup > x </ sup > = h is called a discrete logarithm to the base g of h in the group G.

Similarly, the 29 minuscule forms ( lowercase or small letters ) are: a, b, c, d, e, f, g, h, i, j, k, l, m, n, o, p, q, r, s, t, u, v, w, x, y, z, å, ä, and ö.

We now say that x ∈ V is an eigenvector of M if x is an eigenvector of T. Similarly, λ ∈ K is an eigenvalue of M if it is an eigenvalue of T and the spectrum of M, written σ < sub > M </ sub >, is the set of all such eigenvalues.

Similarly, as an infinite-dimensional example, the Lebesgue space L < sup > p </ sup > is always a Banach space but is only a Hilbert space when p = 2.

Similarly, relatively little energy is used in producing and combining the raw materials ( although large amounts of CO < sub > 2 </ sub > are produced by the chemical reactions in cement manufacture ).

Similarly, the pentafluoride, DbF < sub > 5 </ sub >, should be even more volatile.

Similarly, the variation in the concentration of gas B is-dC < sub > B </ sub >/ dx.

Similarly, the partial pressure of B changes dP < sub > B </ sub >.

Similarly, for any a in F other than 0, there exists an element a < sup >− 1 </ sup > in F, such that a · a < sup >− 1 </ sup > = 1.

Similarly, < tt >( λx. y ) s → y := s = y </ tt >, demonstrating that < tt > λx. y </ tt > really is a constant function.

Similarly, methyl iodide and methyl triflate are viewed as the equivalent of the methyl cation because they readily undergo S < sub > N </ sub > 2 reactions by weak nucleophiles.

Similarly, for a given vacuum wavelength λ < sub > 0 </ sub >, the wavelength

Similarly, p < sup >-</ sup > would indicate a very lightly doped p-type material.

* Similarly the even-graded element γ = − σ < sub > 1 </ sub > σ < sub > 2 </ sub > corresponds to a vector rotation of 180 °:

Similarly the number for " P " in some countries is actually the weight of an equivalent quantity of P < sub > 2 </ sub > O < sub > 5 </ sub > whereas in others ( including Australia ) it refers to elemental phosphorus.

Similarly, products of the gammas such as γ < sub > μ </ sub > γ < sub > ν </ sub > represent oriented surface elements, and so on.

To avoid this ambiguity, the number could be represented in scientific notation: 8. 0 × 10 < sup > 3 </ sup > m indicates that the first zero is significant ( hence a margin of 50 m ) while 8. 000 × 10 < sup > 3 </ sup > m indicates that all three zeroes are significant, giving a margin of 0. 5 m. Similarly, it is possible to use a multiple of the basic measurement unit: 8. 0 km is equivalent to 8. 0 × 10 < sup > 3 </ sup > m. In fact, it indicates a margin of 0. 05 km ( 50 m ).

Similarly, metal ion hydrolysis causes ions such as < span style =" white-space: nowrap ;">< sup > 3 +</ sup ></ span > to behave as weak acids :< ref name = Burgess > Section 9. 1 " Acidity of Solvated Cations " lists many pK < sub > a </ sub > values .</ ref >

Similarly, division by any power of two 2 < sup > k </ sup > may be performed by right-shifting k positions.

Similarly IF A > B THEN A ← A − B.

Similarly, on the regional channels many class 3, stations have been assigned either to operate daytime only or to operate nighttime with directional antennas and/or lower power.

Similarly, both types of antibodies were found in three regions of the chromatographic eluate, having extremely low, low, and high anionic binding capacity, respectively ( Fig. 3 ).

Similarly, the Soviet Union used the intermediate-range Tu-22M ' Backfire ' in the 1970s, but their Mach 3 bomber project came to naught.

Similarly to other " bounded error " probabilistic classes the choice of 1 / 3 in the definition is arbitrary.

Similarly, since it consists of an estimated 2. 3 million blocks, completing the building in 20 years would involve moving an average of more than 12 of the blocks into place each hour, day and night.

Similarly, an observer near the summit would be unaware of standing on a high mountain, as the slope of the volcano would extend beyond the horizon, a mere 3 kilometers away.

Similarly, some poorly designed earlier trials found that peppermint oil has the ability to reduce colicky abdominal pain due to IBS with an NNT ( number needed to treat ) around 3. 1, but the oil is an irritant to the stomach in the quantity required and therefore needs wrapping for delayed release in the intestine.

Similarly introduced on the last day of the Congressional session four years later, the Act of March 3, 1855 required the prepayment of postage on all mailings.

" Similarly, Article I, Section 3, Clause 6 and Article II, Section 4 both authorize the Senate to serve as a court with the power to remove impeached officials from office, given a two-thirds vote to convict.

Similarly, " cruel, inhuman or degrading punishments " are banned under Article 5 of the Universal Declaration of Human Rights and Article 3 of the European Convention on Human Rights.

Similarly, if n is divisible by 3, and m was already a prime distinct from 3, then n − m would also be coprime to 3 and thus be slightly more likely to be prime than a general number.

Similarly the F < sub > 3 </ sub > of a speaker is proportional to Fs:

Similarly, the benefit of apochromats is not simply that they focus 3 wavelengths sharply, but that their error on other wavelength is also quite small.

Similarly ITV1 is the brand used by ITV plc for the Channel 3 service in these areas.

Similarly, in relation to The True Tragedy of Richard Duke of York, Randall Martin reaches the same conclusion in his 2001 Oxford Shakespeare edition of Henry VI, Part 3.

Similarly, the Vulgate, the Midrash of Esther Rabba, I, 3 and the Josippon identify the King as Artaxerxes.

Similarly, their lower range may extend a few notes below the C < sub > 3 </ sub >.