In this paper we investigate the abstract hyperbolic model with time dependent stiffness and damping given by 〈ü(t), ψ〉V ∗,V + d(t; u̇(t), ψ) + a(t;u(t), ψ) = 〈f(t), ψ〉V ∗,V where V ⊂ VD ⊂ H ⊂ V ∗ D ⊂ V ∗ are Hilbert spaces with continuous and dense injections, where H is identified with its dual and 〈·, ·〉 denotes the associated duality product. We show under reasonable assumptions on the time-...

In this note, we wish to prove precise theorems for monotone convergence of quadratic forms on a complex Hilbert space, X. For convenience we only consider positive forms although semibounded forms can be treated. In order to describe the existing theorems and to establish some notation, we first review some of the main ideas in the theory [2, 41: a positive quadratic form is a sesquilinear for...

In this paper we survey some work on representations of Bn given by the induced action on a homology module of some space. One of these, called the Lawrence-Krammer representation, recently came to prominence when it was shown to be faithful for all n. We will outline the methods used, applying them to a closely related representation for which the proof is slightly easier. The main tool is the...

1.1. Linear Algebra. Let F = R or F = C and let V be a vector space over F. A finite dimensional vector space over R (C) is isomorphic to Rn (Cn) for some n, where Rn = {x = (x1, . . . , xn)} for xi ∈ R and Cn = {z = (z1, . . . , zn)} for zi ∈ Cn. On Rn, the standard Euclidean inner product is given by 〈x, y〉 = ∑ xiyi and the Euclidean norm is given by ||x|| = 〈x, x〉 2 . On Cn the standard Herm...

These lectures present a relatively recent introduction into the class of discontinuos Galerkin (DG) methods, named Discontinuous Petrov-Galerkin (DPG) methods. DPG methods, in which DG spaces form a critical ingredient, can be thought of as least-square methods in nonstandard norms, or as Petrov-Galerkin methods with special test spaces, or as a nonstandard mixed method. We will pursue all the...

This paper is concerned with the computation of 3D vertex singularities of anisotropic elastic fields with Dirichlet boundary conditions, focusing on the derivation of error estimates for a finite element method on graded meshes. The singularities are described by eigenpairs of a corresponding operator pencil on spherical polygonal domains. The main idea is to introduce a modified quadratic var...

A non-singular sesquilinear form is constructed that is preserved by the Lawrence-Krammer representation. It is shown that if the polynomial variables q and t of the Lawrence-Krammer representation are chosen to be appropriate algebraically independant unit complex numbers, then the form is negativedefinite Hermitian. Since unitary matrices diagonalize, the conjugacy class of a matrix in the un...

Given polar spaces (V, β) and (V, Q) where V is a vector space over a field K, β a reflexive sesquilinear form and Q a quadratic form, we have associated classical isometry groups. Given a subfield F of K and an F -linear function L : K → F we can define new spaces (V, Lβ) and (V, LQ) which are polar spaces over F . The construction so described gives an embedding of the isometry groups of (V, ...

Given an arbitrary sequence of elements ξ = { n } ∈ N $\xi =\lbrace \xi _n\rbrace _{n\in \mathbb {N}}$ a Hilbert space ( H , ⟨ · ⟩ ) $(\mathcal {H},\langle \cdot ,\cdot \rangle )$ the operator T $T_\xi$ is defined as associated to sesquilinear form Ω f g ∑ $\Omega _\xi (f,g)=\sum {N}} \langle _n\rangle _n g\rangle$ for h : | 2 < ∞ $f,g\in \lbrace h\in \mathcal {H}: \sum {N}}|\langle |^2<\infty ...

Let H be a Hilbert space, L(H) the algebra of all bounded linear operators on H and 〈 , 〉A : H×H → C the bounded sesquilinear form induced by a selfadjoint A ∈ L(H), 〈ξ, η〉A = 〈Aξ, η〉, ξ, η ∈ H. Given T ∈ L(H), T is A-selfadjoint if AT = T ∗A. If S ⊆ H is a closed subspace, we study the set of A-selfadjoint projections onto S, P(A,S) = {Q ∈ L(H) : Q = Q , R(Q) = S , AQ = Q∗A} for different choi...