it is shown that a commutative reduced ring r is a baer ring if and only if it is a cs-ring; if and only if every dense subset of spec (r) containing max (r) is an extremally disconnected space; if and only if every non-zero ideal of r is essential in a principal ideal generated by an idempotent.

For a non-commutative domain R and a multiplicatively closed set S the (left) Ore localization of R at S exists if and only if S satisfies the (left) Ore property. Since the concept has been introduced by Ore back in the 1930’s, Ore localizations have been widely used in theory and in applications. We investigate the arithmetics of the localized ring SR from both theoretical and practical point...

The spherical means Radon transform $\mathcal{M}f(x,r)$ is defined by the integral of a function $f$ in $\mathbb{R}^{n}$ over sphere $S(x,r)$ radius $r$ centered at $x$, normalized area sphere. problem reconstructing from data where $x$ belongs to hypersurface $\Gamma\subset\mathbb{R}^{n}$ and $r \in(0,\infty)$ has important applications modern imaging modalities, such as photo- thermo- acousti...

We consider the ordinary differential equation u′(t) = f(t, u(t)), where f : [a, b] × D → Rn is a given function, while D is an open subset in Rn. We prove that, if K ⊂ D is locally closed and there exists a comparison function ω : [a, b]× R+ → R such that

where g : R 7→ R strictly monotone increasing and differentiable, Ω open set with compact closure in R , and D convex closed subset of R. Under the assumption that ∇ū ∈ D a.e. in Ω, there is a unique solution u to (1.1) and we can actually give an explicit representation of u is terms of a Lax-type formula. The solution is clearly Lipschitz continuous because ∇u ∈ ∂D a.e. in Ω. The Euler-Lagran...

Is this really evidence (as is often cited) that the Continuum Hypothesis has no answer? Another prominent problem from the early 20th century concerns the projective sets, [8]; these are the subsets of Rn which are generated from the closed sets in finitely many steps taking images by continuous functions, f : Rn → Rn, and complements. A function, f : R→ R, is projective if the graph of f is a...

Theorem 1. Let K = 〈K,+, ·, . . .〉 be an expansion of K all of whose atomic relations are definable in R. If K is a proper expansion of 〈K,+, ·〉 then the field R is definable in K. By “proper expansion of 〈K,+, ·〉”, we mean that there is a definable set in K which is not definable in 〈K,+, ·〉 (even with parameters). In particular, the theorem implies that there are no proper expansions of an al...