Constructive Approximation in de Branges–Rovnyak Spaces

In most classical holomorphic function spaces on the unit disk in which the polynomials are dense, a function f can be approximated in norm by its dilates \(f_r(z):=f(rz)~(r<1)\). We show that this is not the case for the de Branges–Rovnyak spaces \(\mathcal{H}(b)\). More precisely, we exhibit a space \(\mathcal{H}(b)\) in which the polynomials are dense and a function \(f\in \mathcal{H}(b)\) such that \(\lim _{r\rightarrow 1^-}\Vert f_r\Vert _{\mathcal{H}(b)}=\infty \). On the positive side, we prove the following approximation theorem for Toeplitz operators on general de Branges–Rovnyak spaces \(\mathcal{H}(b)\). If \((h_n)\) is a sequence in \(H^\infty \) such that \(\Vert h_n\Vert _{H^\infty }\le 1\) and \(h_n(0)\rightarrow 1\), then \(\Vert T_{\overline{h}_n}f-f\Vert _{\mathcal{H}(b)}\rightarrow 0\) for all \(f\in \mathcal{H}(b)\). Using this result, we give the first constructive proof that, if b is a nonextreme point of the unit ball of \(H^\infty \), then the polynomials are dense in \(\mathcal{H}(b)\).     

Balanced fuzzy particle swarm optimization

In the present study an extension of particle swarm optimization (PSO) algorithm which is in conformity with actual nature is introduced for solving combinatorial optimization problems. Development of this algorithm is essentially based on balanced fuzzy sets theory. The classical fuzzy sets theory cannot distinguish differences between positive and negative information of membership functions, while in the new method both kinds of information “positive and negative” about membership function are equally important. The balanced fuzzy particle swarm optimization algorithm is used for fundamental optimization problem entitled traveling salesman problem (TSP). For convergence inspecting of new algorithm, method was used for TSP problems. Convergence curves were represented fast convergence in restricted and low iterations for balanced fuzzy particle swarm optimization algorithm (BF-PSO) comparison with fuzzy particle swarm optimization algorithm (F-PSO).     

Predictive control of uncertain nonlinear parabolic PDE systems using a Galerkin/neural-network-based model

In this paper, a model predictive control (MPC) scheme for a class of parabolic partial differential equation (PDE) systems with unknown nonlinearities, arising in the context of transport-reaction processes, is proposed. A spatial operator of a parabolic PDE system is characterized by a spectrum that can be partitioned into a finite slow and an infinite fast complement. In this view, first, Galerkin method is used to derive a set of finite dimensional slow ordinary differential equation (ODE) system that captures the dominant dynamics of the initial PDE system. Then, a Multilayer Neural Network (MNN) is employed to parameterize the unknown nonlinearities in the resulting finite dimensional ODE model. Finally, a Galerkin/neural-network-based ODE model is used to predict future states in the MPC algorithm. The proposed controller is applied to stabilize an unstable steady-state of the temperature profile of a catalytic rod subject to input and state constraints.     

Compact right multipliers on a Banach algebra related to locally compact semigroups

For a locally compact semigroup \({\mathcal{S}}\), let \(L_{0}^{\infty}({\mathcal{S}},M_{a}({\mathcal{S}}))\) be the Banach space of all μ-measurable (\(\mu\in M_{a}({\mathcal{S}})\)) functions vanishing at infinity, where \(M_{a}({\mathcal{S}})\) denotes the algebra of all measures in the measure algebra \(M({\mathcal{S}})\) of \({\mathcal{S}}\) with continuous translations. Here, we study right compact multipliers on the Banach algebra \(L_{0}^{\infty}({\mathcal{S}},M_{a}({\mathcal{S}}))^{*}\) equipped with an Arens product.     

Relative T -Injective Modules and Relative T -Flat Modules

Let T be a Wakamatsu tilting module. A module M is called (n, T)-copure injective (resp. (n, T)-copure flat) if ɛ _T ¹ (N, M) = 0 (resp. Γ₁ ^T (N, M) = 0) for any module N with T-injective dimension at most n (see Definition 2.2). In this paper, it is shown that M is (n, T)-copure injective if and only if M is the kernel of an I _n (T)-precover f: A → B with A ∈ Prod T. Also, some results on Prod T-syzygies are presented. For instance, it is shown that every nth Prod T-syzygy of every module, generated by T, is (n, T)-copure injective.     

Nonlinear vibrations and chaos in electrostatic torsional actuators

Electrostatic torsional micro-mirrors have wide spread use in different industries for diverse purposes. This paper investigates the development of superharmonics and chaotic responses in electrostatic torsional micro-mirrors near the pull-in condition. Appearance of nonlinear phenomena is investigated in models accounting for and disregarding the coupling of torsional and flexural deflections. Analysis of the system response to step and harmonic excitation reveals the appearance of DC and AC symmetry breaking. Increasing the amplitude of harmonic excitation, the response in the form of distinct superharmonics changes to a broad band response, where there is loss of periodicity and the response becomes chaotic. Accounting for flexural deflections in coupled model reduces the voltage thresholds corresponding to symmetry breaking and chaotic responses. It is also shown that damping has a regularizing effect and introduction of damping changes the chaotic undamped response into quasi-periodic one.     

Binomial Sums and Functions of Exponential Type

Let (a_n)_n0 be a sequence of complex numbers, and, for n0, let A number of results are proved relating the growth of the sequences(b_n) and (c_n) to that of (a_n). For example, given p0, if b_n= O(n^p and for all > 0,then a_n=0 for all n > p. Also, given 0 < p < 1, then for all > 0 if and onlyif . It is further shown that, given rß > 1, if b_n,c_n=O(rßⁿ), then a_n=O(ⁿ),where , thereby proving a conjecture of Chalendar, Kellay and Ransford. The principal ingredientsof the proogs are a Phragmén-Lindelöf theorem forentire functions of exponential type zero, and an estimate forthe expected value of e^(X), where X is a Poisson random variable.2000 Mathematics Subject Classification 05A10 (primary), 30D15,46H05, 60E15 (secondary).     

A pointwise convergence for Sobolev space functions

Let 1<p<∞, and k,m be positive integers such that 0(k−2m)pn. Suppose ΩRⁿ is an open set, and Δ is the Laplacian operator. We will show that there is a sequence of positive constants c_j such that for every f in the Sobolev space W^k,p(Ω), for all xΩ except on a set whose Bessel capacity B_k−2m,p is zero.     

A Gleason–Kahane–Żelazko theorem for the Dirichlet space

We show that every linear functional on the Dirichlet space that is non-zero on nowhere-vanishing functions is necessarily a multiple of a point evaluation. Continuity of the functional is not assumed. As an application, we obtain a characterization of weighted composition operators on the Dirichlet space as being exactly those linear maps that send nowhere-vanishing functions to nowhere-vanishing functions.We also investigate possible extensions to weighted Dirichlet spaces with superharmonic weights. As part of our investigation, we are led to determine which of these spaces contain functions that map the unit disk onto the whole complex plane.