Crucially, in addition to suction that used an electric fan, a box, and one of his wife's pillowcases, Spangler's design incorporated a rotating brush to loosen debris.

" Crucially, it helps play " in situations which used to be thought of as guesswork.

Crucially, Braille's smaller cells were capable of being recognized as letters with a single touch of a finger.

Crucially, because resisting the Ashikaga required a strong central power and a smooth succession, among them inheritance was no longer shared, but passed on intact to a single heir, who often was not even a blood relative, but a promising man adopted specifically to be heir.

Crucially, his narrator Hythloday embodies the Platonic view that philosophers should not get involved in politics and his character of More has the more pragmatic Ciceronic view ; thus the society Hythloday proposes is the ideal More would want, but without communism, which he saw no possibility of occurring, it was wiser to take a more pragmatic view.

Crucially, people with simultanagnosia are unable to enumerate objects outside the subitizing range, either failing to count certain objects, or alternatively counting the same object several times.

Crucially Kelderek is too afraid of being called a coward to object to this ill treatment.

Crucially, that group does not include the believer.

Crucially, it is connected to a filter cartridge near the mouth either directly, or via a flexible hose.

Crucially, Cannon exercised these powers to maintain discipline within the ranks of his own party: the Republicans were divided into the conservative " Old Guard ," led by Cannon, and the progressives, led by President Theodore Roosevelt.

Crucially, they had absorbed beliefs in the religious superiority of Protestant Christianity, the cultural superiority of European civilization, and the aesthetic superiority of European skin color and hair texture.

Crucially, these numbers were accepted as definitive.

Crucially, ostracism had no relation to the processes of justice.

Crucially the gate, comprising silicon, is heavily p-doped ; and its presence depletes the underlying silicon nanowire thereby preventing carrier flow past the gate.

Crucially, it is designed to run underneath Windows such that the operating system is unaware of its presence.

Crucially, the Birlings must descend from the safety and opulence of their brightly lit Edwardian drawing room and into the dimly lit cobblestoned area to engage with Goole and confess their actions.

Crucially, however, he did not consider composition of permutations.

Crucially, there were more investors in British business.

Crucially, it retains many of the vampire traits popularized by Dracula.

Crucially, Vargas was knocked down in the 1st round and again in the 11th round.

Crucially, Gustavus Adolphus's death enabled the French to gain much firmer control of the anti-Habsburg alliance.

Crucially, the category of primitives is restricted to pure hunter-gatherer societies with no domesticated plants or animals.

Crucially, epiphany cannot be predicted, or controlled.

Crucially, the I Corps did not fight in either battle that day.

Crucially, they retreated not to the east, along their own lines of communication and away from Wellington, but northwards, parallel to Wellington's line of march and still within supporting distance, and remained throughout in communication with Wellington.

Crucially, it improved the overall morale of the troop, as it proved that, despite their early string of defeats, the army was capable to fight extremely well.

Crucially, despite Austrian attempts to trumpet their victory against Napoleon, its political consequences remained limited: there were no signs of a general uprising in Germany, Prussia was still unwilling to enter the war and Great Britain was not ready to launch its promised land expedition in northern Europe, while Russia, France's ally since 1807, was becoming increasingly aggressive against the Austrian forces in Galicia.

Crucially, he had no battle reserves with which to either support his battered line or to launch a counterattack of his own.

Crucially, the British had held the bridge long enough to allow Nijmegen bridge to be captured by the 82nd Airborne.

Sir George Cayley ( 1773 – 1857 ) also used a whirling arm to measure the drag and lift of various airfoils.

Airmen like Otto Lilienthal, who introduced cambered airfoils in 1891, used gliders to analyze aerodynamic forces. The Wright brothers were interested in Lilianthal's work and read several of his publications. They also found inspiration in Octave Chanute, an airman and the author of Progress in Flying Machines ( 1894 ). It was the preliminary work of Cayley, Lilienthal, Chanute, and other early aerospace engineers that brought about the first powered sustained flight at Kitty Hawk, North Carolina on December 17, 1903, by the Wright brothers.

* Cayley diagrams – used for finding cognate linkages in mechanical engineering

The Cayley – Bacharach theorem is also used to prove that the group operation on cubic elliptic curves is associative.

Bienvenu., and was known by Sir George Cayley, but it was the first used of twisted rubber to power a flying model.

While studying compositions of linear transformations, Arthur Cayley was led to define matrix multiplication and inverses.

Cayley table of the symmetric group S < sub > 3 </ sub >( multiplication table of permutation matrix | permutation matrices ) These are the positions of the six matrices: File: Symmetric group 3 ; Cayley table ; positions. svg | 310px Only the unity matrices are arranged symmetrically to the main diagonal-thus the symmetric group is not abelian.

Another method based on the Cayley – Hamilton theorem finds an identity using the matrices ' characteristic polynomial, producing a more effective equation for A < sup > k </ sup > in which a scalar is raised to the required power, rather than an entire matrix.

In linear algebra, the Cayley – Hamilton theorem ( named after the mathematicians Arthur Cayley and William Hamilton ) states that every square matrix over a commutative ring ( such as the real or complex field ) satisfies its own characteristic equation.

The Cayley – Hamilton theorem states that " substituting " the matrix A for λ in this polynomial results in the zero matrix:

When the ring is a field, the Cayley – Hamilton theorem is equivalent to the statement that the minimal polynomial of a square matrix divides its characteristic polynomial.

For a general n × n invertible matrix A, i. e., one with nonzero determinant, A < sup >− 1 </ sup > can thus be written as an ( n − 1 )- th order polynomial expression in A: As indicated, the Cayley – Hamilton theorem amounts to the identity

In fact, this expression, ½ (( trA )< sup > 2 </ sup >− tr ( A < sup > 2 </ sup >)), always gives the coefficient c < sub > n − 2 </ sub > of λ < sup > n − 2 </ sup > in the characteristic polynomial of any n × n matrix ; so, for a 3 × 3 matrix A, the statement of the Cayley – Hamilton theorem can also be written as

As the examples above show, obtaining the statement of the Cayley – Hamilton theorem for an n × n matrix requires two steps: first the coefficients c < sub > i </ sub > of the characteristic polynomial are determined by development as a polynomial in t of the determinant

The left hand side can be worked out to an n × n matrix whose entries are ( enormous ) polynomial expressions in the set of entries of A, so the Cayley – Hamilton theorem states that each of these expressions are equivalent to 0.

While this provides a valid proof ( for matrices over the complex numbers ), the argument is not very satisfactory, since the identities represented by the theorem do not in any way depend on the nature of the matrix ( diagonalizable or not ), nor on the kind of entries allowed ( for matrices with real entries the diagonizable ones do not form a dense set, and it seems strange one would have to consider complex matrices to see that the Cayley – Hamilton theorem holds for them ).

The Cayley – Hamilton theorem states that replacing t by A in the characteristic polynomial ( interpreting the resulting powers as matrix powers, and the constant term c as c times the identity matrix ) yields the zero matrix.

The Cayley – Hamilton theorem states that every square matrix satisfies its own characteristic equation.

He postulated the Cayley – Hamilton theorem — that every square matrix is a root of its own characteristic polynomial, and verified it for matrices of order 2 and 3.

Logical matrix | Binary lower unitriangular Toeplitz matrix | Toeplitz matrices, multiplied using Finite field | F < sub > 2 </ sub > operationsThey form the Cayley table of cyclic group | Z < sub > 4 </ sub > and correspond to v: Gray code permutation powers # 4 bit | powers of the 4-bit Gray code permutation.

A similar situation holds for applying a Cayley transform to the skew-symmetric matrix.