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In a recent paper we used a basic decomposition property of polyanalytic functions of order $2$ in one complex variable to characterize solutions of the classical $\overline{\partial}$-problem for given analytic and polyanalytic data. Our approach suggested the study of a special reproducing kernel Hilbert space that we call the Hörmander-Fock space that will be further investigated in this paper. The main properties of this space are encoded in a specific moment sequence denoted by $η=(η_n)_{n\geq 0}$ leading to a special entire function $\mathsf{E}(z)$ that is used to express the kernel function of the Hörmander-Fock space. We present also an example of a special function belonging to the class ML introduced recently by Alpay et al. and apply a Bochner-Minlos type theorem to this function, thus motivating further connections with the theory of stochastic processes.

We introduce a notion of isolated units, elementary particles or more general physical phenomena that do not significantly affect their surrounding environment, and we build a primitive ontology to describe their evolution and interaction. We further introduce a notion of indistinguishability of distinct spacetime paths of a unit, for which the evolution of the state variables of the unit is the same, and a generalization of the equivalence principle based on indistinguishability. Under a time invertibility condition on the whole set of indistinguishable paths of a unit, we show that the quantization of motion of spinless elementary particles in a general potential field can be derived in this framework, in the limiting case of weak fields and low velocities. Extrapolating this approach to include weak relativistic effects, we explore possible experimental consequences., Comment: 34 pages., minor corrections, added new references and a conclusion

Entire functions in one complex variable are extremely relevant in several areas ranging from the study of convolution equations to special functions. An analog of entire functions in the quaternionic setting can be defined in the slice regular setting, a framework which includes polynomials and power series of the quaternionic variable. In the first chapters of this work we introduce and discuss the algebra and the analysis of slice regular functions. In addition to offering a self-contained introduction to the theory of slice-regular functions, these chapters also contain a few new results (for example we complete the discussion on lower bounds for slice regular functions initiated with the Ehrenpreis-Malgrange, by adding a brand new Cartan-type theorem). The core of the work is Chapter 5, where we study the growth of entire slice regular functions, and we show how such growth is related to the coefficients of the power series expansions that these functions have. It should be noted that the proofs we offer are not simple reconstructions of the holomorphic case. Indeed, the non-commutative setting creates a series of non-trivial problems. Also the counting of the zeros is not trivial because of the presence of spherical zeros which have infinite cardinality. We prove the analog of Jensen and Carath\'eodory theorems in this setting.

Let $F(x)$ be an analytical, real valued function defined on a compact domain $\mathcal {B}\subset\mathbb{R}$. We prove that the problem of establishing the irrationality of $F(x)$ evaluated at $x_0\in \mathcal{B}$ can be stated with respect to the convergence of the phase of a suitable integral $I(h)$, defined on an open, bounded domain, for $h$ that goes to infinity. This is derived as a consequence of a similar equivalence, that establishes the existence of isolated solutions of systems equations of analytical functions on compact real domains in $\mathbb{R}^p$, if and only if the phase of a suitable ``geometric'' complex phase integral $I(h)$ converges for $h\rightarrow \infty$. We finally highlight how the method can be easily adapted to be relevant for the study of the existence of rational or integer points on curves in bounded domains, and we sketch some potential theoretical developments of the method.

With the goal of providing the foundations for a rigorous study of modules of bicomplex holomorphic functions, we develop a general theory of functional analysis with bicomplex scalars. Even though the basic properties of bicomplex number are well known and widely available, our analysis requires some more delicate discussion of the various structures which are hidden in the ring of bicomplex numbers. We study in particular matrices with bicomplex numbers, bicomplex modules, and inner products and norms in bicomplex modules. We consider two kinds of norms on bicomplex modules: a real-valued norm (as one would expect), and a hyperbolic-valued norm. Interestingly enough, while both norms can be used to build the theory of normed bicomplex modules, the hyperbolic-valued norm appears to be much better compatible with the structure of bicomplex modules. We also consider linear functionals on bicomplex numbers. To conclude we describe a bicomplex version of classical Schur analysis

We provide a derivation of the Born Rule in the context of the Everett (Many-Worlds) approach to quantum mechanics. Our argument is based on the idea of self-locating uncertainty: in the period between the wave function branching via decoherence and an observer registering the outcome of the measurement, that observer can know the state of the universe precisely without knowing which branch they are on. We show that there is a uniquely rational way to apportion credence in such cases, which leads directly to the Born Rule. [Editors note: for a video of the talk given by Prof. Carroll at the Aharonov-80 conference in 2012 at Chapman University, see quantum.chapman.edu/talk-14.]

In this paper we further elaborate an extension of Clifford analysis towards super-symmetry, started in our paper [So1]. We discuss the generalized spingroup, the Fischer decomposition and give several examples of canonically defined super-manifolds.

In this paper we survey a series of recent developments in the theory of functions of a hypercomplex variable. The central idea underlying these developments consists in requiring a function to be holomorphic on suitable slices of the space on which the function itself is defined. Specifically, we apply this approach to functions defined on the space H of quaternions, on the space O of octonions, and finally on the Clifford algebra of type (0,3), denoted Cl(0, 3). The properties of these functions resemble those of holomorphic functions, and yet the different nature of the three algebras on which we work introduces new and exciting phenomena.