In this paper we consider an abstract Wiener space (X,γ,H) and open subset O⊆X which satisfies suitable assumptions. For every p∈(1,+∞) define the Sobolev W01,p(O,γ) as closure of Lipschitz continuous functions have support with positive distance from ∂O respect to natural norm, show that under assumptions on O can be characterized in W1,p(O,γ) null trace at boundary ∂O, or, equivalently, defin...

Formulae for the levels and sublevels of certain quaternion and octonion algebras are established. Corollaries concerning the equality of levels and sublevels of quaternion algebras with those of associated octonion algebras are presented. Let R be a not necessarily associative ring with unity. The level and sublevel of R, respectively denoted by s(R) and s(R), are defined as follows: s(R) = in...

In the setting adopted by Edmond and Thibault [Mathematical Programming 104 (2005), 347--373], we study a class of perturbed sweeping processes. Under suitable assumptions, obtain two solution existence theorems for processes with constraint sets being prox-regular sublevel sets. The results are applied to analyzing behavior some concrete mechanical processes, which appear first time in this pa...

In this paper we study the boundedness of composition operators on weighted Bergman spaces and Hardy space over polydisc Dn. Studying volume sublevel sets show for which n necessary conditions obtained by Bayart are sufficient. For arbitrary prove rank sufficiency theorem which, in particular, provides us with a simple criterion describing bidisc. Such consistent characterization is classical t...

In this work we analyze magnetic sublevel populations in a neutral beam penetrating a fusion plasma. The collisional-radiative model NOMAD was extended to include magnetic parabolic sublevels with principal quantum numbers n ≤ 10. The collisional parameters were calculated with the advanced atomic-orbital close coupling method and the Glauber approximation. The ionization by the induced electri...

A new paradigm for point cloud data analysis has emerged recently, where point clouds are no longer treated as mere compact sets but rather as empirical measures. A notion of distance to such measures has been defined and shown to be stable with respect to perturbations of the measure. This distance can easily be computed pointwise in the case of a point cloud, but its sublevel-sets, which carr...

1 Proof of the Proposition 1 We now prove the Proposition 1 that gives the condition of compactness of sublevel set. Proof. Let B(r) and S(r) denote the ball and sphere of radius r, centered at the origin. By affine transformation, we can assume that X∗ contains the origin O, X∗ ⊂ B(1), and X∗ ∩ S(1) = φ. Then, we have that for ∀x ∈ S(1), (∇f(x), x) ≥ f(x)− f(O) > 0, where we use convexity for ...

Let M be a smooth, compact manifold of dimension n ≥ 5 and sectional curvature |K| ≤ 1. Let Met(M) = Riem(M)/Diff(M) be the space of Riemannian metrics on M modulo isometries. Nabutovsky and Weinberger studied the connected components of sublevel sets (and local minima) for certain functions on Met(M) such as the diameter. They showed that for every Turing machine Te, e ∈ ω, there is a sequence...

Abstract Entanglement between a spin-wave qubit (memory qubit) and photonic is basic building block for quantum repeaters. Duan-Lukin-Cirac-Zoller (DLCZ) scheme, which generates spin waves via spontaneous Raman scattering (SRS) of Stokes photons in atomic ensemble, provides promising way to generate such entanglement. In recent work [arXiv: 2006.05631, accepted by communications physics], DLCZ-...

We prove limit theorems for the greatest common divisor and least multiple of random integers. While case integers uniformly distributed on a hypercube with growing size is classical, we look at uniform distribution sublevel sets multivariate symmetric polynomials, which call hyperbolic regions. Along way deriving our main results, obtain some asymptotic estimates number integer points in these...