By Joji Kajiwara, Zhong Li, Kwang Ho Shon

This quantity offers the court cases of the 7th foreign Colloquium on Finite or countless Dimensional complicated research held in Fukuoka, Japan. The contributions supply a number of views and various study examples on complicated variables, Clifford algebra variables, hyperfunctions and numerical research.

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For the proof of Theorem 1 we need the following auxiliary lemmas. Lemma 3. The lowest positive and the largest negative eigenvalue satisfy the following variational principles. (i) If σ < σ0 (D) < 0 then 1 = sup a(v, v), K0 := {v ∈ W 1,2 (D), v, v = 1}. λ1 (D) K0 (ii) If 0 > σ > σ0 (D) then 1 = inf a(v, v). K0 λ−1 (D) Proof. Let v ∈ K0 be ﬁxed and let v0 be any real number. Clearly v + v0 is also an element of K0 and a(v + v0 , v + v0 ) = a(v, v) + a(1, 1)v02 + 2a(v, 1)v0 . Since by assumption a(1, 1) < 0 the function f (v0 ) = a(1, 1)v02 + 2a(v, 1)v0 takes its maximum for v0∗ = − a(v,1) a(1,1) .

Ch International Series of Numerical Mathematics, Vol. 157, 13–22 c 2008 Birkh¨ auser Verlag Basel/Switzerland Lower and Upper Bounds for Sloshing Frequencies Henning Behnke Abstract. The calculation of the frequencies ω for small oscillations of an ideal liquid in a container results in a Steckloﬀ eigenvalue problem. A procedure for calculating lower and upper bounds to the smallest eigenvalues is proposed. For the lower bound computation Goerisch’s generalization of Lehmann’s method is applied, trial functions are constructed with ﬁnite elements.

Setting for the problem Let Ha and Hb be two separable, complex Hilbert spaces with inner products a( . , . ) and b( . , . ), respectively. Suppose Ha is a dense subspace of Hb continuously embedded in Hb such that for κ > 0 κ b(u, u) ≤ a(u, u) for all u ∈ Ha holds true. The following variationally posed eigenvalue problem is considered: Find eigenpairs (λ, u) ∈ R × Ha , u = 0 , such that a(u, v) = λb(u, v) holds for all v ∈ Ha . (2) Denote by B ∈ L(Ha ) the bounded self-adjoint operator that satisﬁes a(Bu, v) = b(u, v) for all u, v ∈ Ha .