holds for all p, qEX (for all p, q with d(p, q) <e). Mappings, as above, satisfying (1) with the strict inequality sign for all p, qEX, pT^q (for all p, q with 0 <d(p, q) <e) are called contractive (e-contractive). A point yE Y EX is said to belong to the/-closure of Y, yE Y1, if /(F) C F and there is a point r¡E Y and a sequence {re,} of positive integers, («i<re2< • • ■ <Ui< ■ • ■ ), so that ...

"!$# !%#& '( )+* # -, /. 0 12 ) 3546 ",27 '( 8* & 79 / '( 0 & #: #;'(#2, ? 0 & 19 # @* #: , @ !%# 'A B #: C -, 0 1D &E; E8'( '(# 7F 3G H 0 2 G < + # -, # 9 #:'( E >I #? JK + < 9 & + JK 0 E JK 7F !$ E & 0 & . / LM < ; #& 1 ; N < D #:)& 0 '( N M < D D 7O #:)& 0 '('( ) E & ): ) :> ,P < 9 ? 9 #F #:'( E 9 #& ?>5 #F #2, Q . #:'( E & # I> #R #: S #2, :> & -,6 #R T &E; ? & '( 2...

Let E be an elliptic curve defined over Q and P ∈ E(Q) a rational point of infinite order. Suppose that E has complex multiplication by an order in the quadratic imaginary field k. Denote by ME,P the set of rational primes ` such that ` splits in k, E has good reduction at `, and P is a primitive point modulo `. Under the generalized Riemann hypothesis, we can determine the positivity of the de...

The concept of perfection of a polytope was introduced by S. A. Robertson. Intuitively speaking, a polytope P is perfect if and only if it cannot be deformed to a polytope of different shape without changing the action of its symmetry group G(P ) on its face-lattice F (P ). By Rostami’s conjecture, the perfect 4-polytopes form a particular set of Wythoffian polytopes. In the present paper first...

Doughnut Method We will calculate the dose deposited at the point P in the middle of the concentric circles. It can be calculated as the difference between the cases : first, the dose deposition by full big circle exposition, and second, the dose deposited when only full small circle is exposed : () () E E E Q f S dS Q f S dS p b s b b b S s s s S b s = − = − ∫ ∫ (A.1.1) where Eb and Es are abs...

Last time we discussed two main categories of linear models, and their combination. Here w t denotes a white noise: a stationary process with E w t = 0, E w 2 t = σ 2 , and E w t w s = 0 for s = t. MA(q) x t = µ + w t + q j=1 θ j w t−j = µ + w t + θ 1 w t−1 + · · · + θ q w t−q. x t is stationary with well-defined variance and covariances for any values of the coefficients θ 0 ,. .. , θ q , if q...