Primitive element

The proof proceeds from the observation that each of the four essential primitive elements for computation - namely, the storage (requiring an internal memory), transmission (requiring an internal clock and wires), and processing (requiring... 

The idea of hydrogen as the ultimate fuel, limitless and powerful, stoked the imagination of nineteenth-century fiction writers. In his 1874 book The Mysterious Island, the ever-prescient writer Jules Verne has his characters discuss what would happen to America s industrial and commercial movement when the world runs out of coal in hundreds of years. The engineer, Cyrus Harding, explains that the world will turn to another fuel, water decomposed into its primitive elements. . . doubtless by electricity, which will then have become a powerful and manageable force. Harding goes on to say ... 

Proof. First of all, assume X is etale over k. Then if U = Spec (R) is an affine piece of X, Qr/u — (0)- By the Problem in 1, this shows that R = A, ki being finite separable/A . Conversely, if K/k is a finite separable extension, then by the theorem of the primitive element,... 

Given the complexity of the multiplication process, it is surprising that the multiplication group of a Galois field has the simple structure of a cyclic group. This means that we can find a primitive element g such that each nonzero mark can be represented by some power of g. Multiplication of nonzero marks in GF p ) is equivalent to addition of powers modulo p — l. If p is an odd prime, we can separate the nonzero marks into squares and not-squares. The squares can result from squaring and are even powers of g, the not-squares cannot result from squaring another mark, and are odd powers of g. The set of all squares is called the quadratic residue set. 

The aim of grammatical inference is to leam models of languages from sample sentences in these languages. A sentence can be any structured composition of primitive elements or symbols, though the most common type of composition is concatenation. So we infer grammar and the corresponding representation is an automaton. 

Diffie-Hellman key exchange is based on the assumed difficulty of the discrete logarithm problem modulo a prime number—that is, that it is difficult to compute z from mod p. Diffie-Hellman allows to parties who have not previously exchanged any keys to agree on a secret key. Alice and Bob agree on a prime modulus p and a primitive element g. Alice picks a random number x and sends... 

SUTURE does not attempt to generate a correct circuit directly, but rather builds it up incrementally by beginning with a skeletal design that will have timing violations and race conditions. The circuit is modifled in a series of local transformations to correct these problems. Only small primitive elements (i.e., logic gates, flip-flops, and latches) are used to assemble the circuit initially so that sequential transformations can later be more easily applied to reduce the overall size of the logic. The four principal steps of the synthesis method are outlined below. 

Water decomposed into its primitive elements... and decomposed doubtless, by electricity. .. will one day be employed as fuel,... hydrogen and oxygen which constitute it, used singly or together, will furnish an inexhaustible source of heat and light, of an intensity of which coal is not capable,... Water will be the coal of the future. 

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