If the Af bond is linear then there are three reasonable positions for the hydrogen atoms: ( 1 ) The hydrogen atoms are centered and hence all lie on a sheet midway between the oxygen sheets ; ;

This curve is of symbol Af since it meets **yl, and hence every line of Af in the Af invariant points on **yl and since it obviously meets every line of Af in a single point.

Settled phosphors, as generally used in image intensifiers, have low optical contact with the substrate surface, hence Af shall be assumed.

The cladding thickness is about 0.5 M, hence, Af and Af.

The minimal polynomial for x ( hence ) is a factor of.

For flight dynamics applications the earth axes are generally of minimal use, and hence will be ignored.

It has also been argued that the play could not have been staged at the Globe, and hence, Shakespeare's involvement may be minimal.

A true minimal surface is more properly illustrated by a soap film, which has equal pressure on inside as outside, hence is a surface with zero mean curvature.

After about traction loss is minimal due to the combined effect of the car moving faster and the downforce, hence the car continues accelerating at a very high rate.

So z < sub > 0 </ sub > is a root of the polynomial which has degree n. But, the minimal polynomial is unique, hence

Being also non-periodic, it is then of " minimal complexity ", and hence a Sturmian word, with slope.

Many conditions are categorized as nephrotic syndromes — minimal change disease is unique, because it is the only one lacking any evidence of pathology on light microscopy ( hence the name ).

For years pathologists found no changes when viewing specimens under light microscopy ; hence the name minimal change disease.

), necessary to manufacture a car, are much larger than resources necessary for growing a carrot, explaining why the cost ( and, hence, minimal price ) of a car is larger than the cost of a carrot.

The time between maternal death and the start of fetal blood harvesting is relatively long and therefore means that there is minimal possibility of brain activity in the fetus at the time of blood harvesting and hence minimal possibility of discomfort.

It is not possible to directly execute a DLL, since it requires an EXE for the operating system to load it through an entry point, hence the existence of utilities like RUNDLL. EXE or RUNDLL32. EXE which provide the entry point and minimal framework for DLLs that contain enough functionality to execute without much support.

* If there is a polynomial algorithm for any NP-hard problem, then there are polynomial algorithms for all problems in NP, and hence P

The determinant is a polynomial map, and hence GL ( n, R ) is an open affine subvariety of M < sub > n </ sub >( R ) ( a non-empty open subset of M < sub > n </ sub >( R ) in the Zariski topology ), and therefore

Note that the above construction of the Galois group for a fifth degree polynomial only applies to the general polynomial, specific polynomials of the fifth degree may have different Galois groups with quite different properties, e. g. has a splitting field generated by a primitive 5th root of unity, and hence its Galois group is abelian and the equation itself solvable by radicals.

It can be parameterized by drawing a line with slope t through the rational point, and intersection with the plane quadratic curve ; this gives a polynomial with F-rational coefficients and one F-rational root, hence the other root is F-rational ( i. e., belongs to F ) also.

The solutions to monic polynomial equations over an integral domain are important in the theory of integral extensions and integrally closed domains, and hence for algebraic number theory.

For instance, the polynomial g ( X )= X < sup > 2 </ sup >+ 1 with real coefficients has precisely deg ( g )= 2 roots in the complex plane ; namely the imaginary unit i, and its additive inverse − i, and hence does have distinct roots.

On the other hand, the polynomial h ( X )=( X − 2 )< sup > 2 </ sup > with real coefficients does not have distinct roots ; only 2 can be a root of this polynomial in the complex plane and hence it has only one, and not deg ( f )= 2 roots.

In general, it can be shown that the polynomial f with coefficients in F has distinct roots if and only if for any extension field, and any, does not divide f in E. For instance, in the above paragraph, one observes that g has distinct roots and indeed g ( X )=( X + i )( X − i ) in the complex plane ( and hence cannot have any factor of the form for any in the complex plane ).

In fact, it is true that there is no irreducible polynomial with rational or real coefficients that does not have distinct roots ; in the language of field theory, every algebraic extension of or is separable and hence both of these fields are perfect.

The separability of a polynomial depends on the field in which its coefficients are considered to lie ; for instance, if g is an inseparable polynomial in F, and one considers a splitting field, E, for g over F, g is necessarily separable in E since an arbitrary irreducible factor of g in E is linear and hence has distinct roots.

It is an algebraic surface in three-dimensional projective space defined by a single quaternary cubic polynomial which is homogeneous of degree 3 ( hence, cubic ).

that is, until I became an editor, hence, in his eyes, a rival.

A road block to desirable local or borough improvements, heretofore dependent on the pocketbook vote of taxpayers and hence a drag on progress, is removed by making these a charge against the whole city instead of an assessment paid by those immediately affected.

It is distinctly possible, therefore, that simultaneous pressures in all three vessels would have rendered the shunts inoperable and hence, uninjectable.

The latter is useful for modifying information about some or all forms of a word, hence reducing the work required to improve dictionary contents.

Consequently, the system giving area coverage ( if such coverage is less than world wide ) must be flexible and hence at least partially mobile.

For, in the process of decanting, the bottle is only tilted once instead of several or more times at the table: hence, a minimum of the undesirable mixture of wine and dregs.

If this is true, then the universe today looks just as it did millions of years ago and as it will look millions of years hence, even though the universe is expanding.

A possible etymology is a derivation from the Greek word – aiges = " waves " ( Hesychius of Alexandria ; metaphorical use of ( aix ) " goat "), hence " wavy sea ", cf.

In order for one to achieve proper, firm and healthy Iman one must practice righteous deeds or else his level of Iman chokes and shrinks and eventually can wither away if one does not practice Islam long enough, hence the depth of practicing Islam is good deeds.

The applet is written in a compiled language, whereas the scripting language of the container is an interpreted language, hence the greater performance or functionality of the applet.

It receives its name from its soil ( weathered from the weak underlying limestone ), which is black in colour, almost destitute of sand and loam, and rich in limestone and marl formations, especially adapted to the production of cotton ; hence the region is also called the Cotton Belt.

There is, however, speculation that local criminal gangs barter abalone illegally with Chinese nationals in exchange for chemicals used in the production of drugs, reducing the need for the use of money and hence avoiding money laundering difficulties.

It was inspired by the English garden city movement ; hence the original English name Park ( in the Catalan language spoken in Catalonia where Barcelona is located, the word for " Park " is " Parc ", and the name of the place is " Parc Güell " in its original language ).

Baltic amber is distinguished by its yield of succinic acid, hence the name succinite.

As king, Afonso IV is remembered as a soldier and a valiant general, hence the nickname the Brave.