Edge quantisation of elliptic operators

The ellipticity of operators on a manifold with edge is defined as the bijectivity of the components of a principal symbolic hierarchy , where the second component takes values in operators on the infinite model cone of the local wedges. In the general understanding of edge problems there are two basic aspects: Quantisation of edge-degenerate operators in weighted Sobolev spaces, and verifying the ellipticity of the principal edge symbol which includes the (in general not explicitly known) number of additional conditions of trace and potential type on the edge. We focus here on these questions and give explicit answers for a wide class of elliptic operators that are connected with the ellipticity of edge boundary value problems and reductions to the boundary. In particular, we study the edge quantisation and ellipticity for Dirichlet–Neumann operators with respect to interfaces of some codimension on a boundary. We show analogues of the Agranovich–Dynin formula for edge boundary value problems. Nicoleta Dines and Bert-Wolfgang Schulze were supported by Chinese-German Cooperation Program “Partial Differential Equations”, NNSF of China and DFG of Germany. Xiaochun Liu was supported by NNSF of China through Grant No. 10501034, and Chinese-German Cooperation Program “Partial Differential Equations”, NNSF of China and DFG of Germany.     

Edge-boundary problems with singular trace conditions

The ellipticity of boundary value problems on a smooth manifold with boundary relies on a two-component principal symbolic structure , consisting of interior and boundary symbols. In the case of a smooth edge on manifolds with boundary, we have a third symbolic component, namely, the edge symbol , referring to extra conditions on the edge, analogously as boundary conditions. Apart from such conditions ‘in integral form’ there may exist singular trace conditions, investigated in Kapanadze et al., Internal Equations and Operator Theory, 61, 241–279, 2008 on ‘closed’ manifolds with edge. Here, we concentrate on the phenomena in combination with boundary conditions and edge problem.     

Runge–Kutta convolution quadrature for operators arising in wave propagation

An error analysis of Runge–Kutta convolution quadrature is presented for a class of non-sectorial operators whose Laplace transform satisfies, besides the standard assumptions of analyticity in a half-plane Re s > σ ₀ and a polynomial bound \operatornameO(|s|^m₁){\operatorname{O}(|s|^{\mu_1})} there, the stronger polynomial bound \operatornameO(s^m₂){\operatorname{O}(s^{\mu_2})} in convex sectors of the form |\operatorname*arg s| ￡ p/2-q{|\operatorname*{arg} s| \leq \pi/2-\theta} for θ > 0. The order of convergence of the Runge–Kutta convolution quadrature is determined by μ ₂ and the underlying Runge–Kutta method, but is independent of μ ₁. Time domain boundary integral operators for wave propagation problems have Laplace transforms that satisfy bounds of the above type. Numerical examples from acoustic scattering show that the theory describes accurately the convergence behaviour of Runge–Kutta convolution quadrature for this class of applications. Our results show in particular that the full classical order of the Runge–Kutta method is attained away from the scattering boundary.     

On the Oscillation of Second Order Neutral Differential Equations of Emden-Fowler Type

Some oscillation criteria are established by the averaging technique for the second order neutral delay differential equation of Emden-Fowler type (a(t)x￠(t))￠+q₁(t)| y(t-s₁)|^a sgn y(t-s₁) +q₂(t)| y(t-s₂)|^b sgn y(t-s₂)=0,    t 3 t₀,(a(t)x'(t))'+q_1(t)| y(t-\sigma_1)|^{\alpha}\,{\rm sgn}\,y(t-\sigma_1) +q_2(t)| y(t-\sigma_2)|^{\beta}\,{\rm sgn}\,y(t-\sigma_2)=0,\quad t \ge t_0, where x(t) = y(t) + p(t)y(t − τ), τ, σ₁ and σ₂ are nonnegative constants, α > 0, β > 0, and a, p, q ₁, q₂ ? C([t₀, ￥), \Bbb R)q_2\in C([t_0, \infty), {\Bbb R}) . The results of this paper extend and improve some known results. In particular, two interesting examples that point out the importance of our theorems are also included.     

Vertex-Distinguishing Edge Colorings of Graphs with Degree Sum Conditions

An edge coloring is called vertex-distinguishing if every two distinct vertices are incident to different sets of colored edges. The minimum number of colors required for a vertex-distinguishing proper edge coloring of a simple graph G is denoted by c￠_vd(G){\chi'_{vd}(G)}. It is proved that c￠_vd(G) ￡ D(G)+5{\chi'_{vd}(G)\leq\Delta(G)+5} if G is a connected graph of order n ≥ 3 and s₂(G) 3 \frac2n3{\sigma_{2}(G)\geq\frac{2n}{3}}, where σ ₂(G) denotes the minimum degree sum of two nonadjacent vertices in G.     

A Local Functional Calculus and Related Results on the Single-Valued Extension Property

We study the local functional calculus of an operator T having the single-valued extension property. We consider a vector f(T, v) for an analytic function f on a neighborhood of the local spectrum of a vector v with respect to T and show that the local spectrum of v and the local spectrum of f(T, v)are equal with the possible exception of points of the local spectrum of v that are zeros of f, that is, we show that s_T \sigma_{T} (v) is equal to s_T \sigma_{T} (f(T,v)) union the set of zeros of f on s_T \sigma_{T} (v). This local functional calculus extends the Riesz functional calculus for operators. For an analytic function f on a neighborhood of s \sigma (T), we use the above mentioned proposition to obtain proofs of the results that if T has the single-valued extension property, then f(T) also has the single-valued extension property, and conversely if f is not constant on each connected component of a neighborhood of s \sigma (T) and f(T) has the singlevalued extension property, then T also does.     

The geometry of whips

In this article, we study geometric aspects of the space of arcs parameterized by unit speed in the L ² metric. Physically, this corresponds to the motion of a whip, and it also arises in studying shape recognition. The geodesic equation is the nonlinear, nonlocal wave equation η _tt = ∂ _s (σ η _s ), with \lvert h_s\rvert o 1{\lvert \eta_{s}\rvert\equiv 1} and σ given by s_ss- \lvert h_ss\rvert² s = -\lvert h_st\rvert²{\sigma_{ss}- \lvert \eta_{ss}\rvert^2 \sigma = -\lvert \eta_{st}\rvert^2}, with boundary conditions σ(t, 1) = σ(t, −1) = 0 and η(t, 0) = 0. We prove that the space of arcs is a submanifold of the space of all curves, that the orthogonal projection exists but is not smooth, and as a consequence we get a Riemannian exponential map that is continuous and even differentiable but not C ¹. This is related to the fact that the curvature is positive but unbounded above, so that there are conjugate points at arbitrarily short times along any geodesic.     

Propagation Properties for Schrödinger Operators Affiliated with Certain C*-Algebras

We consider anisotropic Schrödinger operators H = -D + V H = -{\Delta} + V in L²(\mathbbRⁿ) L^{2}(\mathbb{R}^n) . To certain asymptotic regions F we assign asymptotic Hamiltonians H_F such that (a) s(H_F) ì s_ess(H) \sigma(H_F) \subset \sigma_{\textrm{ess}}(H) , (b) states with energies not belonging to s(H_F) \sigma(H_F) do not propagate into a neighbourhood of F under the evolution group defined by H. The proof relies on C*-algebra techniques. We can treat in particular potentials that tend asymptotically to different periodic functions in different cones, potentials with oscillation that decays at infinity, as well as some examples considered before by Davies and Simon in [4].     

Geometry in preduals of spaces of 2-homogeneous polynomials on Hilbert spaces

Let H be a (real or complex) Hilbert space. Using spectral theory and properties of the Schatten–Von Neumann operators, we prove that every symmetric tensor of unit norm in H [^(?)] _s,psH{H \hat{\otimes} _{s,\pi _{s}}H} is an infinite absolute convex combination of points of the form x?x{x\otimes x} with x in the unit sphere of the Hilbert space. We use this to obtain explicit characterizations of the smooth points of the unit ball of H [^(?)] _s,psH{H \hat{\otimes} _{s,\pi _{s}}H} .     

Quantum Diffusion and Delocalization for Band Matrices with General Distribution

We consider Hermitian and symmetric random band matrices H in d \geqslant 1{d \geqslant 1} dimensions. The matrix elements H _xy , indexed by x,y ? L ì \mathbbZ^d{x,y \in \Lambda \subset \mathbb{Z}^d}, are independent and their variances satisfy s_xy²:=\mathbbE |H_xy|² = W^-d f((x - y)/W){\sigma_{xy}^2:=\mathbb{E} |{H_{xy}}|^2 = W^{-d} f((x - y)/W)} for some probability density f. We assume that the law of each matrix element H _xy is symmetric and exhibits subexponential decay. We prove that the time evolution of a quantum particle subject to the Hamiltonian H is diffusive on time scales t << W^d/3{t\ll W^{d/3}} . We also show that the localization length of the eigenvectors of H is larger than a factor W^d/6{W^{d/6}} times the band width W. All results are uniform in the size |Λ| of the matrix. This extends our recent result (Erdős and Knowles in Commun. Math. Phys., 2011) to general band matrices. As another consequence of our proof we show that, for a larger class of random matrices satisfying ?_xs_xy²=1{\sum_x\sigma_{xy}^2=1} for all y, the largest eigenvalue of H is bounded with high probability by 2 + M^{-2/3 + e}{2 + M^{-2/3 + \varepsilon}} for any ${\varepsilon > 0}${\varepsilon > 0}, where M : = 1 / (max_x,ys_xy²){M := 1 / (\max_{x,y}\sigma_{xy}^2)} .     

Dung’s Argumentation is Essentially Equivalent to Classical Propositional Logic with the Peirce–Quine Dagger

In this paper we show that some versions of Dung’s abstract argumentation frames are equivalent to classical propositional logic. In fact, Dung’s attack relation is none other than the generalised Peirce–Quine dagger connective of classical logic which can generate the other connectives ?, ù, ú, ?{\neg, \wedge, \vee, \to} of classical logic. After establishing the above correspondence we offer variations of the Dung argumentation frames in parallel to variations of classical logic, such as resource logics, predicate logic, etc., etc., and create resource argumentation frames, predicate argumentation frames, etc., etc. We also offer the notion of logic proof as a geometrical walk along the nodes of a Dung network and thus we are able to offer a geometrical abstraction of the notion of inference based argumentation. Thus our paper is also a contribution to the question:     

Infinite T?plitz–Lipschitz matrices and operators

We introduce a class of infinite matrices (A_ss￠, s, s￠ ? \mathbbZ^d){(A_{ss\prime}, s, s\prime \in \mathbb{Z}^d)} , which are asymptotically (as |s| + |s′| → ∞) close to Hankel–T?plitz matrices. We prove that this class forms an algebra, and that flow-maps of nonautonomous linear equations with coefficients from the class also belong to it.     

Boundedness of maximal,potential type,and singular integral operators in the generalized variable exponent Morrey type spaces

We consider generalized Morrey type spaces M^{p( ·),q( ·),w( ·)}( W) {\mathcal{M}^{p\left( \cdot \right),\theta \left( \cdot \right),\omega \left( \cdot \right)}}\left( \Omega \right) with variable exponents p(x), θ(r) and a general function ω(x, r) defining a Morrey type norm. In the case of bounded sets W ì \mathbbRⁿ \Omega \subset {\mathbb{R}^n} , we prove the boundedness of the Hardy–Littlewood maximal operator and Calderón–Zygmund singular integral operators with standard kernel. We prove a Sobolev–Adams type embedding theorem M^{p( ·),q₁( ·),w₁( ·)}( W) ? M^{q( ·),q₂( ·),w₂( ·)}( W) {\mathcal{M}^{p\left( \cdot \right),{\theta_1}\left( \cdot \right),{\omega_1}\left( \cdot \right)}}\left( \Omega \right) \to {\mathcal{M}^{q\left( \cdot \right),{\theta_2}\left( \cdot \right),{\omega_2}\left( \cdot \right)}}\left( \Omega \right) for the potential type operator I ^α(·) of variable order. In all the cases, we do not impose any monotonicity type conditions on ω(x, r) with respect to r. Bibliography: 40 titles.     

Spectral Density and Sobolev Inequalities for Pure and Mixed States

We prove some general Sobolev-type and related inequalities for positive operators A of given ultracontractive spectral decay F(l) = ||_cA(]0, l])||_1,￥{F(\lambda) = \vert\vert_{\chi_A}(\left]0, \lambda \right])\vert\vert_{1,\infty}}, without assuming e ^−tA is sub-Markovian. These inequalities hold on functions, or pure states, as usual, but also on mixed states, or density operators in the quantum-mechanical sense. As an illustration, one can relate the Novikov−Shubin numbers of coverings of finite simplicial complexes to the vanishing of the torsion of the ℓ ^p,2-cohomology for some p ≥ 2.     

Summation of <Emphasis Type="Italic">p</Emphasis>-Faber series by the Abel–poisson method in the integral metric

We establish conditions on the boundary G \Gamma of a bounded simply connected domain W ì \mathbbC \Omega \subset \mathbb{C} under which the p-Faber series of an arbitrary function from the Smirnov space E_p( W),1 \leqslant p < ￥ {E_p}\left( \Omega \right),1 \leqslant p < \infty , can be summed by the Abel–Poisson method on the boundary of the domain up to the limit values of the function itself in the metric of the space L_p( G) {L_p}\left( \Gamma \right) .     

On expansions of numbers in alternating <Emphasis Type="Italic">s</Emphasis>-adic series and Ostrogradskii series of the first and second kind

We present expansions of real numbers in alternating s-adic series (1 < s ∈ N), in particular, s-adic Ostrogradskii series of the first and second kind. We study the “geometry” of this representation of numbers and solve metric and probability problems, including the problem of structure and metric-topological and fractal properties of the distribution of the random variable

Toeplitz Operators on Generalized Bergman Spaces

We consider the weighted Bergman spaces HL²(\mathbb B^d, m_l){\mathcal {H}L^{2}(\mathbb {B}^{d}, \mu_{\lambda})}, where we set dm_l(z) = c_l(1-|z|²)^l dt(z){d\mu_{\lambda}(z) = c_{\lambda}(1-|z|^2)^{\lambda} d\tau(z)}, with τ being the hyperbolic volume measure. These spaces are nonzero if and only if λ > d. For 0 < λ ≤ d, spaces with the same formula for the reproducing kernel can be defined using a Sobolev-type norm. We define Toeplitz operators on these generalized Bergman spaces and investigate their properties. Specifically, we describe classes of symbols for which the corresponding Toeplitz operators can be defined as bounded operators or as a Hilbert–Schmidt operators on the generalized Bergman spaces.     

On Fundamental Solutions in Clifford Analysis

Euclidean Clifford analysis is a higher dimensional function theory offering a refinement of classical harmonic analysis. The theory is centred around the concept of monogenic functions, which constitute the kernel of a first order vector valued, rotation invariant, differential operator ?{\underline{\partial}} called the Dirac operator, which factorizes the Laplacian. More recently, Hermitean Clifford analysis has emerged as a new branch of Clifford analysis, offering yet a refinement of the Euclidean case; it focusses on a subclass of monogenic functions, i.e. the simultaneous null solutions, called Hermitean (or h−) monogenic functions, of two Hermitean Dirac operators ?_z{\partial_{\underline{z}}} and ?_z^f{\partial_{\underline{z}^\dagger}} which are invariant under the action of the unitary group, and constitute a splitting of the original Euclidean Dirac operator. In Euclidean Clifford analysis, the Clifford–Cauchy integral formula has proven to be a corner stone of the function theory, as is the case for the traditional Cauchy formula for holomorphic functions in the complex plane. Also a Hermitean Clifford–Cauchy integral formula has been established by means of a matrix approach. Naturally Cauchy integral formulae rely upon the existence of fundamental solutions of the Dirac operators under consideration. The aim of this paper is twofold. We want to reveal the underlying structure of these fundamental solutions and to show the particular results hidden behind a formula such as, e.g. ?E = d{\underline{\partial}E = \delta}. Moreover we will refine these relations by constructing fundamental solutions for the differential operators issuing from the Euclidean and Hermitean Dirac operators by splitting the Clifford algebra product into its dot and wedge parts.     

Mellin‐edge representations of elliptic operators

We construct a class of elliptic operators in the edge algebra on a manifold M with an embedded submanifold Y interpreted as an edge. The ellipticity refers to a principal symbolic structure consisting of the standard interior symbol and an operator‐valued edge symbol. Given a differential operator A on M for every (sufficiently large) s we construct an associated operator ??_s in the edge calculus. We show that ellipticity of A in the usual sense entails ellipticity of ??_s as an edge operator (up to a discrete set of reals s). Parametrices P of A then correspond to parametrices ??_s of ??_s, interpreted as Mellin‐edge representations of P. Copyright © 2005 John Wiley & Sons, Ltd.