Hardy spaces associated to the Schrödinger operator on strongly Lipschitz domains of {\mathbb{R}^d}

Let L = ?Δ + V be a Schrödinger operator and Ω be a strongly Lipschitz domain of ${\mathbb R^{d}}Let L = −Δ + V be a Schr?dinger operator and Ω be a strongly Lipschitz domain of \mathbb R^d{\mathbb R^{d}} , where Δ is the Laplacian on \mathbb R^d{\mathbb R^{d}} and the potential V is a nonnegative polynomial on \mathbb R^d{\mathbb R^{d}} . In this paper, we investigate the Hardy spaces on Ω associated to the Schr?dinger operator L.     

Operators Commuting with the Volterra Operator are not Weakly Supercyclic

We prove that any bounded linear operator on L _p [0, 1] for 1 ≤ p < ∞, commuting with the Volterra operator V, is not weakly supercyclic, which answers affirmatively a question raised by Léon-Saavedra and Piqueras-Lerena. It is achieved by providing an algebraic type condition on an operator which prevents it from being weakly supercyclic and is satisfied for any operator commuting with V.     

Independence Distribution Preserving Covariance Structures for the Multivariate Linear Model

Consider the multivariate linear model for the random matrixY_n×pMN(XB, VΣ), whereBis the parameter matrix,Xis a model matrix, not necessarily of full rank, andVΣ is annp×nppositive-definite dispersion matrix. This paper presents sufficient conditions on the positive-definite matrixVsuch that the statistics for testingH₀: CB=0vsH_a: CB≠0have the same distribution as under the i.i.d. covariance structureIΣ.     

Splitting Off Edges within a Specified Subset Preserving the Edge-Connectivity of the Graph

Splitting off a pair su, sv of edges in a graph G means the operation that deletes su and sv and adds a new edge uv. Given a graph G = (V + s, E) which is k-edge-connected (k ≥ 2) between vertices of V and a specified subset R  V, first we consider the problem of finding a longest possible sequence of disjoint pairs of edges sx, sy, (x ,y  R) which can be split off preserving k-edge-connectivity in V. If R = V and d(s) is even then a well-known theorem of Lovász asserts that a complete R-splitting exists, that is, all the edges connecting s to R can be split off in pairs. This is not the case in general. We characterize the graphs possessing a complete R-splitting and give a formula for the length of a longest R-splitting sequence. Motivated by the connection between splitting off results and connectivity augmentation problems we also investigate the following problem that we call the split completion problem: given G and R as above, find a smallest set F of new edges incident to s such that G′ = (V + s, E + F) has a complete R-splitting. We give a min-max formula for F as well as a polynomial algorithm to find a smallest F. As a corollary we show a polynomial algorithm which finds a solution of size at most k/2 + 1 more than the optimum for the following augmentation problem, raised in [[2]]: given a graph H = (V, E), an integer k ≥ 2, and a set R  V, find a smallest set F′ of new edges for which H′ = (V, E + F′) is k-edge-connected and no edge of F′ crosses R.     

Bicommutants and ranges of derivations

Let V be a vector space, V* its dual space and L(V) the algebra of all linear operators on V. For an operator a?∈?L(V) let a* be its adjoint acting on V*, and for a subset R of L(V) let R″ be its bicommutant. If R is a left noetherian subalgebra of L(V), then {a*: a?∈?R}″?=?{a*: a?∈?R″}. When R is singly generated R″ is described precisely. Further, for any two operators a, b?∈?L(V), b?∈?(a)″ if and only if the derivations d _a and d _b satisfy d _b (F(V))???d _a (F(V)), where F(V) is the set of all finite rank operators on V. In this case the inclusion d _b (L(V))???d _a (L(V)) also holds.     

On V-orthogonal projectors associated with a semi-norm

For any n ×  p matrix X and n ×  n nonnegative definite matrix V, the matrix X(X′V X)⁺ X′V is called a V-orthogonal projector with respect to the semi-norm , where (·)⁺ denotes the Moore-Penrose inverse of a matrix. Various new properties of the V-orthogonal projector were derived under the condition that rank(V X) =  rank(X), including its rank, complement, equivalent expressions, conditions for additive decomposability, equivalence conditions between two (V-)orthogonal projectors, etc.     

Unique addition modules

LetV be a module over a ring R. We define V to be a unique addition module (UAM) if it is not possible to change the addition of V without changing the action of R on V. If R is a domain such that 1 is not the only unit in R and V is a torsion-free i2-module, then we show that V is a UAM if and only if V has rank 1 (or if V={0}). We also classify finitely generated unique addition modules over commutative Artinian rings.     

Exponential decay of solutions to nonlinear elliptic equations with potentials

Exponential decay estimates are obtained for complex-valued solutions to nonlinear elliptic equations in \mathbbRⁿ ,\mathbb{R}^{n} , where the linear term is given by Schr?dinger operators H =  − Δ  +  V with nonnegative potentials V and the nonlinear term is given by a single power with subcritical Sobolev exponent in the attractive case. We describe specific rates of decay in terms of V, some of which are shown to be optimal. Moreover, our estimates provide a unified understanding of two distinct cases in the available literature, namely, the vanishing potential case V = 0 and the harmonic potential case V(x) = |x|².     

Lax Operator for the Quantised Orthosymplectic Superalgebra Uq[osp(m|n)]

Representations of quantum superalgebras provide a natural framework in which to model supersymmetric quantum systems. Each quantum superalgebra, belonging to the class of quasi-triangular Hopf superalgebras, contains a universal R-matrix which automatically satisfies the Yang–Baxter equation. Applying the vector representation π, which acts on the vector module V, to the left-hand side of a universal R-matrix gives a Lax operator. In this article a Lax operator is constructed for the quantised orthosymplectic superalgebras U _q [osp(m|n)] for all m > 2, n ≥ 0 where n is even. This can then be used to find a solution to the Yang–Baxter equation acting on V ⊗ V ⊗ W, where W is an arbitrary U _q [osp(m|n)] module. The case W = V is studied as an example. Presented by A. Verschoren.     

The Discrete Spectrum in the Spectral Gaps of Semibounded Operators with Non-sign-definite Perturbations

Given two self-adjoint operators A and V = V₊  − V₋ , we study the motion of the eigenvalues of the operator A(t) = A − tV as t increases. Let α > 0 and let λ be a regular point for A. We consider the quantities N₊ (V; λ, α), N₋ (V; λ, α), and N₀(V; λ, α) defined as the number of eigenvalues of the operator A(t) that pass point λ from the right to the left, from the left to the right, or change the direction of their motion exactly at point λ, respectively, as t increases from 0 to α > 0. We study asymptotic characteristics of these quantities as α → ∞. In the present paper, the results obtained previously [O. L. Safronov, Comm. Math. Phys.193 (1998), 233–243] are extended and given new applications to differential operators.     

Norm Attaining Operators and Pseudospectrum

It is shown that if 1 < p < ∞ and X is a subspace or a quotient of an ℓ_p-direct sum of finite dimensional Banach spaces, then for any compact operator T on X such that ∥I + T∥ > 1, the operator I + T attains its norm. A reflexive Banach space X and a bounded rank one operator T on X are constructed such that ∥I + T∥  > 1 and I + T does not attain its norm. The author would like to thank E. Shargorodsky for his interest and comments.     

Lower bounds on the eigenvalue sums of the Schrödinger operator and the spectral conservation law

We consider the Schr?dinger operator H = −Δ − V(x), V > 0, acting in the space L² (\mathbbR^d) L^2 (\mathbb{R}^d) and study relations between the behavior of V at infinity and properties of the negative spectrum of H. Bibliography: 34 titles.     

Note: Combinatorial Alexander Duality—A Short and Elementary Proof

Let X be a simplicial complex with ground set V. Define its Alexander dual as the simplicial complex X ^*={σ⊆V∣V∖σ ∉ X}. The combinatorial Alexander duality states that the ith reduced homology group of X is isomorphic to the (|V|−i−3)th reduced cohomology group of X ^* (over a given commutative ring R). We give a self-contained proof from first principles accessible to a nonexpert.     

Lifting chains of prime ideals to paravaluation rings

Résumé  SoitR ⊂T une extension des anneaux commutatifs et soit {P _α :α ∈I} une cha?ne croissante des idéaux premiers deR (I étant un ensemble totalement ordonné, peut-être infini). Alors il existe un anneau de paravaluationV deT et une cha?ne {Q _α} des idéaux premiers deV de sorte queR ⊂V etQ _α ∩R =P _α pour toutα ∈I. Tout d’abord, on établit le cas spécial dans lequelT est un corps; dans ce cas, on trouve en effet un tel anneau de valuationV deT. Ensuite, l’assertion ci-dessus pour le cas général découle comme conséquence. Dans le cas général, on peut aussi remplacer le mot “paravaluation” avec le mot “valuation” siR est un anneau de Marot etT est son anneau total de fractions.      

Numerical verification of condition for approximately midconvex functions

Let X be a normed space and V be a convex subset of X. Let a\colon \mathbbR₊ ? \mathbbR₊{\alpha \colon \mathbb{R}_+ \to \mathbb{R}_+}. A function f \colon V ? \mathbbR{f \colon V \to \mathbb{R}} is called α-midconvex if

On the Roman Bondage Number of Planar Graphs

A Roman dominating function on a graph G is a function f : V(G) → {0, 1, 2} satisfying the condition that every vertex u for which f (u) = 0 is adjacent to at least one vertex v for which f (v) = 2. The weight of a Roman dominating function is the value f (V(G)) = ?_{u ? V(G)} f (u){f (V(G)) = \sum_{u\in V(G)} f (u)}. The Roman domination number, γ _R (G), of G is the minimum weight of a Roman dominating function on G. The Roman bondage number b _R (G) of a graph G with maximum degree at least two is the minimum cardinality of all sets E^￠ í E(G){E^{\prime} \subseteq E(G)} for which γ _R (G − E′) > γ _R (G). In this paper we present different bounds on the Roman bondage number of planar graphs.     

On spectral properties of periodic polyharmonic matrix operators

We consider a matrix operatorH = (-Δ)^l +V inR ⁿ, wheren ≥ 2,l ≥ 1, 4l > n + 1, andV is the operator of multiplication by a periodic inx matrixV(x). We study spectral properties ofH in the high energy region. Asymptotic formulae for Bloch eigenvalues and the corresponding spectral projections are constructed. The Bethe-Sommerfeld conjecture, stating that the spectrum ofH can have only a finite number of gaps, is proved.     

On m-Accretivity of Perturbed Bochner Laplacian in L p Spaces on Riemannian Manifolds

We consider a differential expression ${H=\nabla^*\nabla+V}We consider a differential expression H=?^*?+V{H=\nabla^*\nabla+V}, where ?{\nabla} is a Hermitian connection on a Hermitian vector bundle E over a manifold of bounded geometry (M, g) with metric g, and V is a locally integrable section of the bundle of endomorphisms of E. We give a sufficient condition for H to have an m-accretive realization in the space L ^p (E), where 1 < p <  +∞. We study the same problem for the operator Δ _M  + V in L ^p (M), where 1 < p < ∞, Δ _M is the scalar Laplacian on a complete Riemannian manifold M, and V is a locally integrable function on M.     

Operator Machines on Directed Graphs

We show that if an infinite-dimensional Banach space X has a symmetric basis then there exists a bounded, linear operator ${R\,:\,X\longrightarrow\, X}We show that if an infinite-dimensional Banach space X has a symmetric basis then there exists a bounded, linear operator R : X? X{R\,:\,X\longrightarrow\, X} such that the set

A = {x ? X : ||Rn x||? ￥}A = \{x \in X\,:\,{\left|\left|{R^n x}\right|\right|}\rightarrow \infty\}