Let μ = (μ_t)_t>o be a convolution semigroup on ℝ^d. An exit law for μ is a positive measurable function ϕ: ]0,∞[ × ℝ^d → [0,∞] which verifies the functional equation (by putting ϕ_t:=ϕ(t,·))
∀_s,t>0 : μ_s ⚹ ϕ_t = ϕ_s+t   λ.a.e.
where λ is the Lebesgue measure on ℝ^d. Following [1], we prove in this paper that the solutions of this equation are on the form
ϕ_t = d(μ^{^}_t ⚹ β)/dλ   λ.a.e.
where μ^{^} := (μ^{^}_t)_t>o is the reflected convolution semigroup of μ, β is a positive measure on ℝ^d such that μ^{^}_t ⚹ β ≪ λ, for evry t > 0.
Moreover, we study the global solutions and their interpretations in terms of the negative definite function associated to μ.

In the present paper the trace form on the ring of integers of a number field is considered. All quadratic fields are determinated for which the form can be diagonalized, i.e. the quadratic fields with an integral basis orthogonal with respect to the trace. There are also given examples of fields of higher degree with the same property.

In the first part of the present paper we consider Ptolemaic inequality and give some necessary conditions for its solutions. The other part is devoted to solution of some functional equation which, simultaneously, satisfies the Ptolemaic inequality.

The existence and uniqueness of solutions of the two point boundary value problem for ordinary linear differential equations of fourth order in the Colombeau algebra are considered.

In this paper we establish an approximation of approximately quadratic mappings by quadratic mappings, which solves the pertinent Ulam stability problem.

The class of functions of bounded n-th variation, denoted by BV_n[a,b], was introduced by M.T. Popoviciu in 1933. In 1979 A.M. Russell in [6] proved the Jordan-type decomposition theorem for functions from this class, then, applying this result, showed that each BV_n[a,b], with a suitable norm, is a Banach space. For n=0 and n=1 the above facts are well known classical results (cf., for instance, [5], [7]).
However, the proofs given by A . M . Russell for n≥2, based on some properties of divided differences, are rather complicated. The aim of this note, is to give an essentially simpler arguments both, for the Jordan-type decomposition theorem as well as for the completness of the space BV_n[a,b]. In our proofs we apply the Popoviciu theorem and the fact that for every positive integer n≥2, a function f is n-convex iff the derivative f' is (n-1)-convex.

Report of Meeting. The First Katowice-Debrecen Winter Seminar on Functional Equations, February 7-10, 2001, Cieszyn, Poland.