We consider summation of consecutive values φ(v), φ(v+1), . . . , φ(w) of a meromorphic function φ(z) where v, w ∈ ZZ. We assume that φ(z) satisfies a linear difference equation L(y) = 0 with polynomial coefficients, and that a summing operator for L exists (such an operator can be found – if it exists – by the Accurate Summation algorithm, or alternatively, by Gosper’s algorithm when ordL = 1)...

The paper contains a new prof for the sufficiency in Joel H. Shapiro’s recent characterization of inner functions saying that an analytic selfmap φ of the open unit disk is an inner function if and only if the essential norm of the composition operator of symbol φ is equal to √ 1+|φ(0)| 1−|φ(0)| . The main ingredient in the proof is a formula for the essential norm of a composition operator in ...

We say that the quasi-norm Φ of a symmetric sequence space E has property P if, for all x, y ∈ E with 0 ≤ xn ≤ yn for all n, and xN < yN for some N , we have Φ (xn) < Φ (yn) . Simon [6] proved that if the Banach norm Φ has property P , then whenever An → A in the weak operator topology on the unitary matrix space CE and Φ(An)→ Φ(A), then An → A in the strong operator topology on CE . In this no...

Defining the observable φ canonically conjugate to the number observable N has long been an open problem in quantum theory. The problem stems from the fact that N is bounded from below. Here we show how to define the absolute phase observable Φ ≡ |φ| by suitably restricting the Hilbert space of x and p like variables. This Φ is actually the absolute value of the phase and has the correct classi...

The purpose of this paper is to explore structures of functions in optimization. We will assume that the functions are composed of user-defined functions and are given as computer programs. Factorable functions and factorable programming problems were developed from 1967 through 1990 and are early examples of structure in nonlinear optimization. We explore the relationship between source code t...

We present a modal extension of classical propositional logic in which intuitionistic propositional logic is mirrored by means of the modal operator. In this sense, the modal extension combines classical and intuitionistic propositional logic avoiding the collapsing problem. Adopting ideas from a recent paper (S. Lewitzka, A denotational semantics for a Lewis-style modal system close to S1, 201...

Let Dn be the unit polydisc of Cn. The class of all holomorphic functions with domain Dn will be denoted by H(Dn). Let φ be a holomorphic self-map of Dn, the composition operator Cφ induced by φ is defined by (Cφ f )(z) = f (φ(z)) for z ∈Dn and f ∈H(Dn). If, in addition, ψ is a holomorphic function defined on Dn, the weighted composition operator ψCφ induced by ψ and φ is defined by ψCφ(z) = ψ(...

Let φ(z) = (φ1(z), · · · , φn(z)) be a holomorphic self-map of Bn and ψ(z) a holomorphic function on Bn, and H(Bn) the class of all holomorphic functions on Bn, where Bn is the unit ball of C , the weight composition operator Wψ,φ is defined by Wψ,φ = ψf(φ) for f ∈ H(Bn). In this paper we estimate the essential norm for the weighted composition operator Wψ,φ acting from the Hardy space H to H (...

Certain functions of the covariance operator such as the square root of a regularized inverse are important components of many statistics employed for functional data analysis. If Σ is a covariances operator on a Hilbert space, ̂ Σ a sample analogue of this operator, and φ a function on the complex plane, which is analytic on a domain containing a contour around the spectrum of Σ, a tool of gene...

In this paper we present a theory of qualitative probability. Work in the area goes back at least to de Finetti. The usual approach is to specify a binary operator with φ ψ having the intended interpretation that φ is not more probable than ψ. We generalise these approaches by extending the domain of the operator from the set of events to the set of finite sequences of events. If Φ and Ψ are fi...