Generalized Bernoulli Wick differential equation

Based on the distributions space on [Formula: see text] (denoted by [Formula: see text]) which is the topological dual space of the space of entire functions with exponential growth of order [Formula: see text] and of minimal type, we introduce a new type of differential equations using the Wick derivation operator and the Wick product of elements in [Formula: see text]. These equations are called generalized Bernoulli Wick differential equations which are the analogue of the classical Bernoulli differential equations. We solve these generalized Wick differential equations. The present method is exemplified by several examples.

Temporal derivation operator applied on the historic and school case of slab waveguides families eigenvalue equations: another method for computation of variational expressions

Starting from the well-known and historic eigenvalue equations describing the behavior of 3-layer and 4-layer slab waveguides, this paper presents another specific analytical framework providing time-laws of evolution of the effective propagation constant associated to such structures, in case of temporal variation of its various geometrical features. So as to develop such kind of time-propagator formulation and related principles, a temporal derivation operator is applied on the studied school case equations, considering then time varying values of all the geometrical characteristics together with the effective propagation constant. Relevant calculations are performed on three different cases. For example, we first investigate the variation of the height of the guiding layer for the family of 3-layer slab waveguides: then, considering the 4-layer slab waveguide's family, we successively address the variation of its guiding layer and of its first upper cladding. As regards the family of 4-layer waveguides, calculations are performed for two different families of guided modes and light cones. Such another approach yields rigorous new generic analytical relations, easily implementable and highly valuable to obtain and trace all the family of dispersion curves by one single time-integration and one way.

Model Predictive Control of Fractional Order Systems

This paper provides the model predictive control (MPC) of fractional order systems. The direct method will be used as internal model to predict the future dynamic behavior of the process, which is used to achieve the control law. This method is based on the Grünwald–Letnikov's definition that consists of replacing the noninteger derivation operator of the adopted system representation by a discrete approximation. The performances and the efficiency of this approach are illustrated with practical results on a thermal system and compared to the MPC based on the integer ARX model.

Numerical Range of the Derivation of an Induced Operator

Let V be an n–dimensional inner product space over , let H be a subgroup of the symmetric group on {l,…, m}, and let x: H → be an irreducible character. Denote by (H) the symmetry class of tensors over V associated with H and x. Let K (T) ∈ End((H)) be the operator induced by T ∈ End(V), and let DK(T) be the derivation operator of T. The decomposable numerical range W*(DK(T)) of DK(T) is a subset of the classical numerical range W(DK(T)) of DK(T). It is shown that there is a closed star-shaped subset of complex numbers such that⊆ W*(DK(T)) ⊆ W(DK(T)) = con where conv denotes the convex hull of . In many cases, the set is convex, and thus the set inclusions are actually equalities. Some consequences of the results and related topics are discussed.

Universal approximation theorem for Dirichlet series

The paper deals with an extension theorem by Costakis and Vlachou on simultaneous approximation for holomorphic function to the setting of Dirichlet series, which are absolutely convergent in the right half of the complex plane. The derivation operator used in the analytic case is substituted by a weighted backward shift operator in the Dirichlet case. We show the similarities and extensions in comparing both results. Several density results are proved that finally lead to the main theorem on simultaneous approximation.

Two Results About H∞ Functional Calculus on Analytic umd Banach Spaces

Let X be a Banach space with the analytic UMD property, and let A and B be two commuting sectorial operators on X which admit bounded H∞ functional calculi with respect to angles θ1 and θ2 satisfying θ1 + θ2 > π. It was proved by Kalton and Weis that in this case, A + B is closed. The first result of this paper is that under the same conditions, A + B actually admits a bounded H∞ functional calculus. Our second result is that given a Banach space X and a number 1 ≦ p < ∞, the derivation operator on the vector valued Hardy space Hp (R; X) admits a bounded H∞ functional calculus if and only if X has the analytic UMD property. This is an ‘analytic’ version of the well-known characterization of UMD by the boundedness of the H∞ functional calculus of the derivation operator on vector valued Lp-spaces Lp (R; X) for 1 < p < ∞ (Dore-Venni, Hieber-Prüss, Prüss).

MALLIAVIN CALCULUS AND SKOROHOD INTEGRATION FOR QUANTUM STOCHASTIC PROCESSES

A derivation operator and a divergence operator are defined on the algebra of bounded operators on the symmetric Fock space over the complexification of a real Hilbert space [Formula: see text] and it is shown that they satisfy similar properties as the derivation and divergence operator on the Wiener space over [Formula: see text]. The derivation operator is then used to give sufficient conditions for the existence of smooth Wigner densities for pairs of operators satisfying the canonical commutation relations. For [Formula: see text], the divergence operator is shown to coincide with the Hudson–Parthasarathy quantum stochastic integral for adapted integrable processes and with the noncausal quantum stochastic integrals defined by Lindsay and Belavkin for integrable processes.

A GENERALIZATION OF GAUGE COUPLING CONSTANTS TO GAUGE COUPLING FIELDS

A natural generalization of the notion of the gauge coupling constant appearing in the covariant derivation operator is obtained by replacing it with a field Γ which takes values in the linear hermitian invertible mappings [Formula: see text] ([Formula: see text] is the Lie algebra of the gauge group G equipped with a G-invariant inner product). In this case the eigenvalues of Γ(x) for each point x from the space-time take the role of the usual single gauge coupling constant.