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Let β > 1, and Q(β) be the smallest field extension of the rationals containing β. Then any real number in [0,1) having a periodic β-expansion must lie in Q(β). On the other hand, the converse need not be true. The converse does hold if β is a Pisot number, [8] although necessary and sufficient conditions are not known.
In analytic number theory and related branches of mathematics, a complex-valued arithmetic function: is a Dirichlet character of modulus (where is a positive integer) if for all integers and : [1]
Euler's identity is also a special case of the more general identity that the n th roots of unity, for n > 1, add up to 0: = = Euler's identity is the case where n = 2. A similar identity also applies to quaternion exponential: let {i, j, k} be the basis quaternions; then,
He wrote that "as 5 is to 8 so is 8 to 13, practically, and as 8 is to 13, so is 13 to 21 almost", and concluded that these ratios approach the golden ratio : [28] [29] + =. This convergence holds regardless of the starting values U 0 {\displaystyle U_{0}} and U 1 {\displaystyle U_{1}} , unless U 1 = − U 0 / φ {\displaystyle U_{1}=-U_{0 ...
In 1987, however, W. Dale Brownawell gave an upper bound for the effective Nullstellensatz that is simply exponential in the number of variables. [8] Brownawell's proof relied on analytic techniques valid only in characteristic 0, but, one year later, János Kollár gave a purely algebraic proof, valid in any characteristic, of a slightly ...
In dosimetry, linear energy transfer (LET) is the amount of energy that an ionizing particle transfers to the material traversed per unit distance. It describes the action of radiation into matter. It is identical to the retarding force acting on a charged ionizing particle travelling through the matter. [ 1 ]
Let's take a closer look at IonQ's most recent results and outlook to see if this is a good opportunity to buy the dip. ... or $0.93 a share, compared to a loss of $41.9 million, or $0.20 a share ...
Let (,,) be a probability space and let be an index set with a total order (often , +, or a subset of +).. For every let be a sub-σ-algebra of .Then := is called a filtration, if for all .