We prove two embedding and extension theorems in the context of the constructive theory of metric spaces. The first states that Cantor space embeds in any inhabited complete separable metric space (CSM) without isolated points, X , in such a way that every sequentially continuous function from Cantor space to Z extends to a sequentially continuous function from X to R. The second asserts an ana...

We consider classes of simply connected planar domains which are isophasal, ie, have the same scattering phase s(λ) for all λ > 0. This is a scattering-theoretic analogue of isospectral domains. Using the heat invariants and the determinant of the Laplacian, Osgood, Phillips and Sarnak showed that each isospectral class is sequentially compact in a natural C∞ topology. This followed earlier wor...

We show that the space Lip0(R) is the dual space of L(R;R)/N where N is the subspace of L(R;R) consisting of vector fields whose divergence vanishes identically. We prove that although the quotient space L(R;R)/N is weakly sequentially complete, the subspace N is not nicely placed in other words, its unit ball is not closed for the topology τm of local convergence in measure. We prove that if Ω...

Consider the problem of sequentially testing a null hypothesis vs an alternative hypothesis when the risk function is a linear combination of probability of error in the terminal decision and expected sample size (i.e., constant cost per observation.) Assume that the parameter space is the union of null and alternative, the parameter space is convex, the intersection of null and alternative is ...

THEOREM A. Let (X, τ) be a Hausdorff topological space and Ψ : X →]−∞,+∞], Φ : X → R two functions. Assume that there is r > infX Ψ such that the set Ψ (]−∞, r]) is compact and first-countable. Moreover, suppose that the function Φ is bounded below in Ψ(]−∞, r]) and that the function Ψ+ λΦ is sequentially lower semicontinuous for each λ ≥ 0 small enough. Finally, assume that the set of all glob...

We propose a method for sketching and visualizing geological models by sequentially defining stratigraphic layers, where each layer represents a unique erosion or deposition event. Evolution of rivers and deltas is important for geologists when interpreting the stratigraphy of the subsurface, in particular for hydrocarbon exploration. We illustratively visualize mountains, basins, lakes, rivers...

This study explores the amplitude and phase transmission of audio-frequency sound through a waveguide side-loaded with a series of closely spaced and sequentially tuned Helmholtz resonators. This system exhibits a series of acoustic bandgaps due to the Helmholtz resonance and standing-wave cavity modes. The bandgaps are achieved in a physically compact manner in that the resonators are spaced b...