Suppose in an arithmetic unverse we have two predicates φ and ψ for natural numbers, satisfying a base case φ(0) → ψ(0) and an induction step that, for generic n, the hypothesis φ(n) → ψ(n) allows one to deduce φ(n+ 1) → ψ(n+ 1). Then it is already true in that arithmetic universe that (∀n)(φ(n) → ψ(n)). This is substantially harder than in a topos, where cartesian closedness allows one to form...

We study the well-posedness of triply nonlinear degenerate ellipticparabolic-hyperbolic problem b(u)t − div ã(u, ∇φ(u)) + ψ(u) = f, u|t=0 = u0 in a bounded domain with homogeneous Dirichlet boundary conditions. The nonlinearities b, φ and ψ are supposed to be continuous non-decreasing, and the nonlinearity ã falls within the Leray-Lions framework. Some restrictions are imposed on the dependence...

This note is a brief, and somewhat biased, account of the evolution of what people working in P.D.E.’s call Weyl’s Lemma about the regularity of solutions to second order elliptic equations. As distinguished from most modern treatments, which are based on pseudodifferential operator technology, the approach adopted here is, like Weyl’s own, potential theoretic. §1: Where It All Started Given a ...

We investigate several topics on a quadratic form Φ over an algebraic number field including the following three: (A) an equation ξΦ · ξ = Ψ for another form Ψ of a smaller size; (B) classification of Φ over the ring of algebraic integers; (C) ternary forms. In (A) we show that the “class” of such a ξ determines a “class” in the orthogonal group of a form Θ such that Φ ≈ Ψ ⊕Θ. Such was done in ...

In this paper, we present some common fixed point theorems for (ψ,φ)-generalized f -weakly contractive mappings in metric and ordered metric spaces. Our results extend, generalize and improve some well-known results in the literature. Also, we give an example to illustrate our results.

*Correspondence: [email protected]; [email protected] 3Department of Mathematics, Zanjan Branch, Islamic Azad University, Zanjan, Iran Full list of author information is available at the end of the article Abstract In this paper, some new generalization of Darbo’s fixed point theorem is proved by using a F(ψ ,φ)-contraction in terms of a measure of noncompactness. Our result extends to obtai...

Let A and B be unital separable simple amenable C*-algebras which satisfy the Universal Coefficient Theorem. Suppose that A and B are Z-stable and are of rationally tracial rank no more than one. We prove the following: Suppose that φ, ψ : A → B are unital *monomorphisms. There exists a sequence of unitaries {un} ⊂ B such that lim n→∞ unφ(a)un = ψ(a) for all a ∈ A, if and only if [φ] = [ψ] in K...