We give a description of invariants and attractors the critical subcritical Galton–Watson tree measures under operation Horton pruning (cutting leaves with subsequent series reduction). Under regularity condition, class invariant consists binary one-parameter family trees offspring distribution {qk} that has power tail qk∼Ck−(1+1/q0), where q0∈(1/2,1). Each measure non-empty domain attraction c...

Coenzyme Q0 (Q0), a strong electrophile, is toxic to insulin-producing cells. Q0 was incubated with rat and human pancreatic islets and INS-1 insulinoma cells, and its attachment to cellular proteins was studied with Western analysis using antiserum raised against the benzoquinone ring structure of ubiquinone (anti-Q). Q0 covalently bonded to two proteins, one of 50 kDa and another of 70 kDa. B...

4.1.5. The weighted projective space P(q0, . . . ,qn), gcd(q0, . . . ,qn) = 1, is built from a fan in N = Z/Z(q0, . . . ,qn). Let ui ∈ N be the image of ei ∈ Z . The dual lattice is M = {(a0, . . . ,an) ∈ Z n+1 | a0q0+ · · ·+anqn = 0}. Also assume that gcd(q0, . . . , q̂i, . . . ,qn) = 1 for i = 0, . . . ,n. (a) Prove that the ui are the primitive ray generators of the fan giving P(q0, . . . ,qn...

It is shown that the Bochner-Riesz operator on Rn of negative order α is of restricted weak type in the critical points (p0, q0) and (q′ 0, p′0), where 1/q0 = 3/4 + α/2, q0 = p′0/3 for −3/2 < α < 0 in the two-dimensional case and 1/q0 = (n + 1 + 2α)/2n, q0 = (n − 1)p′0/(n + 1), for −(n + 1)/2 < α < −1/2 if n ≥ 3.

Let Ω be a finite symmetric subset of GLn(Z[1/q0]), Γ := 〈Ω〉, and let πm be the group homomorphism induced by the quotient map Z[1/q0] → Z[1/q0]/mZ[1/q0]. Then the family of Cayley graphs {Cay(πm(Γ), πm(Ω))}m is a family of expanders as m ranges over fixed powers of square-free integers and powers of primes that are coprime to q0 if and only if the connected component of the Zariski-closure of ...

The uncertainty of the beam quality conversion factor (k(Q,Q0)) of standard dosimetry of absorbed dose to water in external beam radiotherapy 12 (JSMP12) is determined by combining the uncertainty of each beam quality conversion factor calculated for each type of ionization chamber. However, there is no guarantee that ionization chambers of the same type have the same structure and thickness, s...