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...

Abstract In disordered systems, the amplitudes of localized states will decrease exponentially away from their centers and localization lengths are characterizing such decreasing. this article, we find a model in which each eigenstate is decreasing at two distinct rates. The one-dimensional system with constrained Hilbert space: all eigenstates |Ψ〉s should be orthogonal to state |Φ〉, 〈Φ|Ψ〉 = 0, ...

which governs a classical motion in the space of link variables. φ is the pseudofermion field, whereas P is the canonical momenta, which is randomized at the beginning of each trajectory. For the classical motion we need the gauge derivative of pseudofermionic action, namely the fermion force. For the fermion force one needs to invert the overlap operator on the φ field: force = −ψ dDD dU ψ, wh...

Let M be a compact manifold with smooth boundary. We study the heat content asymptotics on M defined by a time dependent heat source and time dependent boundary conditions. We adopt the following notational conventions: let M be a compact Riemannian manifold with smooth boundary ∂M , let ∆ = δd be the scalar Laplacian on M , let x be a point of the interior of M , let y be a point of the bounda...

Let (Mn,g) be a closed Riemannian manifold and N a warped product manifold of two space forms. We investigate geometric properties by the spectra of the Jacobi operator of a harmonic map φ : M → N . In particular, we show if N is a warped product manifold of Euclidean space with a space form and φ,ψ : M → N are two projectively harmonic maps, then the energy of φ and ψ are equal up to constant ...

and Applied Analysis 3 As application, we consider the holomorphic case and we prove the following result: Theorem 1.3. We suppose that P is aC0-contraction holomorphic semigroup on B and β be a Bochner subordinator satisfying ∫1 0 1 s βt ds < ∞, t > 0. 1.9 Then each zero-integrable P-exit law ψ is subordinated to a unique P-exit law φ. Moreover, φ is given by φt ( q a ) PtVq ( ψ ) − bAPtVq(ψ) ...

A substitution φ is strong Pisot if its abelianization matrix is non-singular and all eigenvalues except the Perron-Frobenius eigenvalue have modulus less than one. For strong Pisot φ that satisfies a no cycle condition and for which the translation flow on the tiling space Tφ has pure discrete spectrum, we describe the collection T P φ of pairs of proximal tilings in Tφ in a natural way as a s...

We fix a set AP of atomic propositions containing {p, q} and its associated alphabet Σ = 2. [1] a) Consider the formulae φ1 = G(p→ q) and φ2 = G(p→ ((¬p) R q)). Prove or disprove that φ1 implies φ2, i.e., that φ1 → φ2 is valid. Prove or disprove that φ2 implies φ1. [5] b) Consider the formulae α = pU q, β = q U p, ψ = (p→ α)∧ (q → β) and φ = Gψ. The aim is to compute the generalized Büchi autom...

We consider the critical focusing NLS in 1-d of the form (1.1) i∂ t ψ + ∂ 2 x ψ = −|ψ| 4 ψ, i = √ −1, ψ = ψ(t, x), and ψ complex valued. It is well-known that this equation permits standing wave solutions of the form φ(t, x) = e iαt φ 0 (x, α), α > 0 Indeed, requiring positivity and evenness in x for φ 0 (x, α) implies for example φ 0 (x, α) = α 1 2 (3 2) 1 4 cosh 1 2 (α 2 x) Another remarkable...