We explain how to use results from Iwasawa theory to obtain information about p-parts of Tate-Shafarevich groups of specific elliptic curves over Q. Our method provides a practical way to compute #X(E/Q)(p) in many cases when traditional p-descent methods are completely impractical and also in situations where results of Kolyvagin do not apply, e.g., when the rank of the Mordell-Weil group is g...

Let G be an infinite compact group and G its dual. For 1< p < oom P(G) is a module over 21(G) A(G), the Fourier algebra of G. For 1 < p, q < o0, let Ip q = HomA(G)( P(G), ?'(G)). If G is abelian, then Jll is the space of LP(G)-multipliers. For 1 < p < 2 and p the conjugate index of p, A(G) -1,1 p,p p, p 2, 2 (G). Further, the space ll is the dual of a space called d p a subspace of eo(G). Using...

In this appendix we show that 1 2 ∆ F (θ)∆ is a second order Taylor approximation of D KL (p(θ)p(θ + ∆)). First, let g q (θ) :=D KL (qp(θ)) = ω∈Ω q(ω) ln q(ω) p(ω|θ). We begin by deriving equations for the Jacobian and Hessian of g q at θ: ∂g q (θ) ∂θ = ω∈Ω q(ω) p(ω|θ) q(ω) ∂ ∂θ q(ω) p(ω|θ) = ω∈Ω q(ω) p(ω|θ) q(ω) −q(ω) ∂p(ω|θ) ∂θ p(ω|θ) 2 = ω∈Ω − q(ω) p(ω|θ) ∂p(ω|θ) ∂θ , (4) and so: ∂ 2 g q (θ)...

In this paper, we define bi-periodic (p,q)-Fibonacci and (p,q)-Lucas sequences, which generalize Fibonacci type, Lucas type using recurrence relations of sequences. Generating functions Binet formulas that allow us to calculate the nth terms these sequences are given convergence properties their consecutive examined. Also, prove some fundamental identities conform well-known

T h i s paper d i s c u s s e s e x p l a n a t o r y schema a c q u i s i t i o n , a l e a r n i n g t e c h n i q u e w i t h s e v e r a l i n t e r e s t i n g p r o p e r t i e s . I t does n o t r e q u i r e a t e a c h e r o r concep t m a t c h i n g p r e d i c a t e t o b e p r o v i d e d . I t does n o t r e l y o n s e a r c h i n g a concep t space t o p roduce g e n e r a l i z...

ν = q · P M = E − E′ is the lepton’s energy loss in the nucleon rest frame (in earlier literature sometimes ν = q · P ). Here, E and E′ are the initial and final lepton energies in the nucleon rest frame. Q = −q2 = 2(EE′−−→k · −→k )−ml −m2l′ where ml(ml′) is the initial (final) lepton mass. If EE′ sin(θ/2) ≫ m2l , m2l′ , then ≈ 4EE′ sin(θ/2), where θ is the lepton’s scattering angle with respec...