We establish two existence results for elliptic boundary-value problems with discontinuous nonlinearities. One of them concerns implicit elliptic equations of the form ψ(-Δu) = f(x,u). We emphasize that our assumptions permit the nonlinear term f to be discontinuous with respect to the second variable at each point.

We consider iteration of arithmetic and power means and discuss methods for determining their limit. These means appear naturally in connection with some problems in homogenization theory.

Let R be a real closed field, and denote by ${}_{R,n}$ the ring of germs, at the origin of Rⁿ, of $C^{\infty }$ functions in a neighborhood of 0 ∈ Rⁿ. For each n ∈ ℕ, we construct a quasianalytic subring ${}_{R,n}{\subset }_{R,n}$ with some natural properties. We prove that, for each n ∈ ℕ, ${}_{R,n}$ is a noetherian ring and if R = ℝ (the field of real numbers), then ${}_{ℝ,n}=ₙ$, where ₙ is the ring of germs, at the origin of ℝⁿ, of real analytic functions. Finally, we prove the Real Nullstellensatz and solve Hilbert’s 17th Problem for the ring ${}_{R,n}$.

We say that a function f from [0,1] to a Banach space X is increasing with respect to E ⊂ X* if x* ∘ f is increasing for every x* ∈ E. A function $f:{[0,1]}^m\to X$ is separately increasing if it is increasing in each variable separately. We show that if X is a Banach space that does not contain any isomorphic copy of c₀ or such that X* is separable, then for every separately increasing function $f:{[0,1]}^m\to X$ with respect to any norming subset there exists a separately increasing function $g:{[0,1]}^m\to ℝ$ such that the sets of points of discontinuity...

It is shown that product weakly measurable lower weak semi-Carathéodory multifunction is superpositionally measurable.

We prove the existence of an effectively computable integer polynomial P(x,t₀,...,t₅) having the following property. Every continuous function $f:{ℝ}^s\to ℝ$ can be approximated with arbitrary accuracy by an infinite sum ${\sum }_{r=1}^{\infty }{H}_r(x₁,...,x_s)\in C^{\infty }\left({ℝ}^s\right)$ of analytic functions $H_r$, each solving the same system of universal partial differential equations, namely $P(x_{\sigma };H_r,\partial H_r/\partial x_{\sigma },...,\partial ⁵H_r/\partial x{⁵}_{\sigma }⁵)=0$ (σ = 1,..., s).

The purpose of this paper is to carry over to the o-minimal settings some results about the Euler characteristic of algebraic and analytic sets. Consider a polynomially bounded o-minimal structure on the field ℝ of reals. A ($C^{\infty }$) smooth definable function φ: U → ℝ on an open set U in ℝⁿ determines two closed subsets W := u ∈ U: φ(u) ≤ 0, Z := u ∈ U: φ(u) = 0. We shall investigate the links of the sets W and Z at the points u ∈ U, which are well defined up to a definable homeomorphism. It is proven...

We introduce the real valued real analytic function κ(t) implicitly defined by $e^{2\pi i\kappa \left(t\right)}=-e^{-2i\vartheta \left(t\right)}\left({\zeta }^{\text{'}}(1/2-it)\right)/\left({\zeta }^{\text{'}}(1/2+it)\right)$ (κ(0) = -1/2). By studying the equation κ(t) = n (without making any unproved hypotheses), we show that (and how) this function is closely related to the (exact) position of the zeros of Riemann’s ζ(s) and ζ’(s). Assuming the Riemann hypothesis and the simplicity of the zeros of ζ(s), it follows that the ordinate of the zero 1/2 + iγₙ of ζ(s) is the unique solution to the equation κ(t) = n.

We present an existence theorem for integral equations of Urysohn-Volterra type involving fuzzy set valued mappings. A fixed point theorem due to Schauder is the main tool in our analysis.