Systems of equations with sets of integers as unknowns are considered. It isshown that the class of sets representable by unique solutions of equations using the oper-ations of union and addition S + T = {m+ n |m ∈ S, n ∈ T} and with ultimately periodicconstants is exactly the class of hyper-arithmetical sets. Equations using addition only canrepresent every hyper-arithmetical s...

It is proved that, for any positive integer m, the weight of the unionclosure of any m distinct 2-sets is at least as large as the weight of the union-closure of the first m 2-sets in squashed (antilexicographic) order, where all i-sets have the same non-negative weight wi with wi ≤ wi+1 for all i, and the weight of a family of sets is the sum of the weights of its members. As special cases, so...

In this paper, we establish sharp two-sided estimates for the transition densities of relativistic stable processes [i.e., for the heat kernels of the operators m− (m2/α − )α/2] in C1,1 open sets. Here m> 0 and α ∈ (0,2). The estimates are uniform in m ∈ (0,M] for each fixed M > 0. Letting m ↓ 0, we recover the Dirichlet heat kernel estimates for α/2 := −(− )α/2 in C1,1 open sets obtained in [1...

In this paper, sharp two-sided estimates for the transition densities of relativistic α-stable processes with mass m ∈ (0,1] in C1,1 exterior open sets are established for all time t > 0. These transition densities are also the Dirichlet heat kernels of m− (m2/α − )α/2 with m ∈ (0,1] in C1,1 exterior open sets. The estimates are uniform in m in the sense that the constants are independent of m ...

6. Suppose X is a metric space, let BX be the σ-algebra generated by the open sets of X, we call BX the Borel σ-algebra and its elements Borel sets. 7. A countable intersection of open sets is called a Gδ set ; a countable union of closed sets is called an Fσ set . 8. We call a collection of subsets of X an elementary family if (i) ∅ ∈ E ; (ii) if E,F ∈ E , then E ∩ F ∈ E ; and (iii) if E ∈ E ,...