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We consider the relationship between linear continuous operators acting on the space of entire functions of one variable of a given order and linear differential operators of infinite order satisfying certain growth conditions for the coefficients. We found that these two classes of operators are equivalent.

By integrating the simplicial model of social aggregation with existing research on opinion leadership and diffusion networks, this article introduces the constructs of simplicial diffusers (mathematically defined as nodes embedded in simplexes; a simplex is a socially bonded cluster) and simplicial diffusing sets (mathematically defined as minimal covers of a simplicial complex; a simplicial complex is a social aggregation in which socially bonded clusters are embedded) to propose a strategic approach for information diffusion of cancer screenings as a health intervention on Facebook for community cancer prevention and control. This approach is novel in its incorporation of interpersonally bonded clusters, culturally distinct subgroups, and different united social entities that co-exist within a larger community into a computational simulation to select sets of simplicial diffusers with the highest degree of information diffusion for health intervention dissemination. The unique contributions of the article also include seven propositions and five algorithmic steps for computationally modeling the simplicial model with Facebook data.

We prove that the existence of isolated solutions of systems of equations of analytical functions on compact real domains in Rp, is equivalent to the convergence of the phase of a suitable complex valued integral I(h) for h→∞. As an application, we then use this result to prove that the problem of establishing the irrationality of the value of an analytic function F(x) at a point x0 can be rephrased in terms of a similar phase convergence.

The advent of computers, and especially high-performance computers, has had a dramatic impact on the way in which mathematics is done and even more on how mathematics is applied, as demonstrated by the growth of computational mathematics as well as what goes under the name of “experimental mathematics”, to which a journal is now devoted. More importantly, computers are now used to perform highly complex computations in order to apply mathematical models to a variety of empirical problems that could never be attacked otherwise.

In the classical theory of several complex variables, holomorphic mappings are just n-tuples of holomorphic functions in m variables, with arbitrary n and m, and no relations between these functions are assumed. Some 30 years ago John Ryan introduced complex, or complexified, Clifford analysis which is, in a sense, the study of certain classes of holomorphic mappings where the components are not independent, and instead obey the relations generated by the Cauchy– Riemann and Dirac-type operators. In this paper, we take a closer look at this theory emphasizing some additional properties that holomorphic mappings satisfy in this context. Our attention is mostly restricted to the case of low dimensions where it is possible to identify new and interesting properties and to single out the special role played by bicomplex analysis.

Weak measurement and weak values have a very deep meaning in quantum mechanics, and new phenomena associated to them have recently been observed experimentally. These measurements give rise to the notion of superoscillating sequences of functions. In the recent years the authors have started an intensive study of this topic from the mathematical point of view. In this paper we use a generalization of the Schrodinger equation in which the spatial derivative is replaced by a suitable convolution operator to prove the existence of a large class of superoscillating sequences. The method we use also allows us to construct such sequences explicitly.

In this paper we introduce and study a functional calculus for bicomplex linear bounded operators. The study is based on the decomposition of bicomplex numbers and of linear operators using the two nonreal idempotents. We show that, due to the presence of zero divisors in the bicomplex numbers, the spectrum of a bounded operator is unbounded. We therefore introduce a different spectrum (called reduced spectrum) which is bounded and turns out to be the right tool to construct the bicomplex holomorphic functional calculus. Finally we provide some properties of the calculus.

In this paper, we study the evolution of superoscillating initial data for the quantum driven harmonic oscillator. Our main result shows that superoscillations are amplified by the harmonic potential and that the analytic solution develops a singularity in finite time. We also show that for a large class of solutions of the Schrodinger equation, superoscillating behavior at any given time implies superoscillating behavior at any other time.