We show that the Desch–Schappacher perturbation and the Miyadera–Voigt perturbation of an immediately norm continuous semigroup are immediately norm continuous. We also show that a perturbation theorem of C. Batty, C. Kaiser and L. Weis based on a generation theorem of A.M. Gomilko, D.-X. Feng and D.-H. Shi also preserves the immediate norm continuity of semigroups. The novelty of these results...

For a 3-manifold M , McMullen derived from the Alexander polynomial of M a norm on H(M,R) called the Alexander norm. He showed that the Thurston norm is an upper bound for the Alexander norm. He asked if these two norms were the same when M fibers over the circle. Here, I give examples that show this is not the case. This question relates to the faithfulness of the Gassner representations of th...

Abstract. We analyse dissipative boundary conditions for nonlinear hyperbolic systems in one space dimension. We show that a known sufficient condition for exponential stability with respect to the H-norm is not sufficient for the exponential stability with respect to the C-norm. Hence, due to the nonlinearity, even in the case of classical solutions, the exponential stability depends strongly ...

In this paper we show how to compute the Λα norm , α ≥ 0, using the dyadic grid. This result is a consequence of the description of the Hardy spaces H(R ) in terms of dyadic and special atoms. Recently, several novel methods for computing the BMO norm of a function f in two dimensions were discussed in [9]. Given its importance, it is also of interest to explore the possibility of computing the...

and Applied Analysis 3 Remark 5. Assumption (13) holds if the system has a unique solution. For brevity, we rewrite system (2)∼(5) in the vector form by introducing W(x, t) = (u(x, t), n(x, t)) and Y(x, t) = (a(x, t), b(x, t)). We denote by E 0 = H 2 ⋂H 1 0 (Ω) × H 1 0 (Ω) the space of vector functionsW(x, t) = (u(x, t), n(x, t)) with norm E0 = {‖u‖ 2 H 2 + ‖n‖ 2 H 1} 1/2 . (14) Similarly, we d...

Masterclass “Topological quantum field theories, quantum groups and 3-manifold invariants”) This is a follow-up notes for the introductory talks (45min × 2) on the hyperfinite II1 factor and the standard form prepared for Prof. Ryszard Nest’s talks Oct.9-10. There may be typos or errors. 1 Talk 1 (Oct. 6 2014, 10:30-11:15) We define the notion of II1 factors and see one example, namely the hype...

If $f : S' \to S$ is a finite locally free morphism of schemes, we construct symmetric monoidal "norm" functor $f_\otimes \mathcal{H}_{\bullet}(S')\to \mathcal{H}_{\bullet}(S)$, where $\mathcal{H}_\bullet(S)$ the pointed unstable motivic homotopy category over $S$. $f$ étale, show that it stabilizes to \mathcal{S}\mathcal{H}(S') \mathcal{S}\mathcal{H}(S)$, $\mathcal{S}\mathcal{H}(S)$ $\mathbb{P...

Abstract. To deal with the divergence-free constraint in a double curl problem: curlμ−1curlu = f and div εu = 0 in Ω, where μ and ε represent the physical properties of the materials occupying Ω, we develop a δ-regularization method: curlμcurluδ + δεuδ = f to completely ignore the divergence-free constraint div εu = 0. It is shown that uδ converges to u in H(curl ; Ω) norm as δ → 0. The edge fi...

and Applied Analysis 3 A finite subset of j consecutive positive integers starting with 1 is denoted by j : {1, 2, . . . , j}. The set R−h : −h, 0 ∪ R0 will be used to define the solution of functional differential equations on R0 including its initial condition in −h, 0 . C i R0 ,Rm0 is the set of m-vector real functions of class C i and definition domain R0 and PC i R0 ,Rm0 is the set of m-ve...

In this paper we address the H∞ control analysis, the output feedback stabilization, and the output feedback H∞ control synthesis problems for state-space symmetric systems. Using a particular solution of the Bounded Real Lemma for an open-loop symmetric system we obtain an explicit expression to compute the H∞ norm of the system. For the output feedback stabilization problem we obtain an expli...