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Finite Element Method
Published in Andrei V. Lavrinenko, Jesper Lægsgaard, Niels Gregersen, Frank Schmidt, Thomas Søndergaard, Numerical Methods in Photonics, 2018
Andrei V. Lavrinenko, Jesper Lægsgaard, Niels Gregersen, Frank Schmidt, Thomas Søndergaard
The integral in (8.9) is the core of the weak form. The term is called a sesquilinear form because it is anti-linear in its first argument (the v, due to complex conjugation) and linear in its second argument (the u). Hence, a(v, u) maps two complex functions via integration to a complex number. In addition to the integration by parts, we would like to get rid of the use of the boundary conditions (8.6) and (8.7) in terms of explicit constraints. This is easily accomplished for the DtN condition (8.7). We just set v∗(x)∂∗u(x)|L=v∗(L)∂xu(L)=v∗(L)iku(L).
Matrix Groups
Published in Leslie Hogben, Richard Brualdi, Anne Greenbaum, Roy Mathias, Handbook of Linear Algebra, 2006
A φ-sesquilinear form B is φ-Hermitian if B(v, w) = φ(B(w, v)) for all v, w ∈ V. In the case where F = ℂ and φ is conjugation, a φ-Hermitian form is called a Hermitian form.
A generalization of van der Corput's difference theorem with applications to recurrence and multiple ergodic averages
Published in Dynamical Systems, 2023
Let and denote the norm and inner product on and let and denote the norm and inner product on . We denote the collection of square averageable sequences by Let and observe that It follows that we may use diagonalization to construct an increasing sequence of positive integers for which exists whenever and . We now construct a new Hilbert space from and as follows. For all and , we define so is a sesquilinear form on with scalar multiplication and addition occuring pointwise. Letting and we see that is a pre-Hilbert space. We will soon see that is sequentially closed under the topology induced by , so we define . We call the Hilbert space induced by , and we may write in place of if is understood from the context.