Entire slice regular functions

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

Extension results for slice regular functions of a quaternionic variable

In this paper we prove a new representation formula for slice regular functions, which shows that the value of a slice regular function $f$ at a point $q=x+yI$ can be recovered by the values of $f$ at the points $q+yJ$ and $q+yK$ for any choice of imaginary units $I, J, K.$ This result allows us to extend the known properties of slice regular functions defined on balls centered on the real axis to a much larger class of domains, called axially symmetric domains. We show, in particular, that axially symmetric domains play, for slice regular functions, the role played by domains of holomorphy for holomorphic functions

Laser dimpling and remote welding of zinc-coated steels for automotive applications

"The flexibility in terms of beam shapebility and amplitude in the range of process parameters offered by the emerging high power and quality active fiber lasers can be advantageously used in overcoming the well-known problems in remote laser welding of zinc-coated steel components in the car mass production. The paper explores the potentiality offered by the combination of the scanner technology with high brilliance fiber lasers when laser dimpling and remote welding are executed in a successive order: the first one to realize gap spacers and the second one to weld together clamped sheets in zinc-coated steels. In particular, the substitution of the traditional mechanical dimpling with the laser dimpling is investigated in order to highlight the potentiality of a solution that is flexible, unaffected by tool wear and highly productive.

A new functional calculus for non-commuting operators

In this paper we use the notion of slice monogenic functions \cite{slicecss} to define a new functional calculus for an $n$-tuple $T$ of not necessarily commuting operators. This calculus is different from the one discussed in \cite{jefferies} and it allows the explicit construction of the eigenvalue equation for the $n$-tuple $T$ based on a new notion of spectrum for $T$. Our functional calculus is consistent with the Riesz-Dunford calculus in the case of a single operator.Comment: to appear in Journal of Functional Analysi

Analisi della migrazione cellulare collettiva in presenza e in assenza di un campo elettrico

L'elaborato di tesi analizza la migrazione collettiva cellulare in presenza e assenza di un campo elettrico esterno. Essa è un processo fondamentale per la fisiologia e lo sviluppo degli animali, per esempio mantiene intatto e continuo un tessuto o una struttura rimodellandola. Inoltre è stato scoperto che le cellule si muovono in risposta a deboli campi elettrici in un processo chiamato galvanotassi. Ci sono vari modelli che provano a spiegare questi fenomeni. Per analizzarli sono state osservate delle colture cellulari di glioblastoma multiforme mentre rigeneravano un taglio. Una coltura in presenza e una in assenza di un campo elettrico. Con l'ausilio del programma “Imagej” e di python sono stati calcolati i valori del MSD (Mean Square Displacement), della VACF (Velocity Autocorellation Function) e della curvatura dei fronti cellulari. Per i primi due valori è stato fatto un fit rispetto alla funzione F=Dt^b, per vedere la loro dipendenza nel tempo. I parametri di b per il MSD per i diversi fronti sono risultati compresi tra 1.1/1.6, evidenziando un andamento super-diffusivo. I parametri di b per la VACF sono compresi tra -1.2/-0.2 dimostrando la presenza di forze deboli che agiscono sulla migrazione rendendo la velocità non autocorrelata. Dallo studio della curvatura si è visto che le zone con questa positiva (in assenza di sporgenze) tendono a non formare protrusioni mentre le zone in presenza di esse tendono a rimanere in questa posizione. Inoltre vi è una relazione tra la curvatura e la diffusione, cioè a zone più negative di curvatura corrispondono zone con diffusività maggiore. Non sono state rivelate differenze significative tra l'acquisizione con e senza campo elettrico. Si è visto nel primo caso una diversa diffusività tra i due fronti, destro e sinistro, maggiore nel fronte concorde al campo elettrico. Questa incongruenza si è però osservata anche nell'altro caso per cui non si è potuto concludere che il campo elettrico abbia influito alla migrazione

A Panorama on Superoscillations

Purpose of this note is to give an overview on superoscillating sequences and some of their properties. We discuss their persistence in time under Schrödinger equation, we propose various classes of superoscillating functions and we also briefly mention how they can be used to approximate some generalized functions

Superoscillating Sequences Towards Approximation in S or S′ -Type Spaces and Extrapolation

Aharonov–Berry superoscillations are band-limited sequences of functions that happen to oscillate asymptotically faster than their fastest Fourier component. In this paper we analyze in what sense functions in the Schwartz space S(R,C) role= presentation \u3eS(R,C) or in some of its subspaces, tempered distributions or also ultra-distributions, could be approximated over compact sets or relatively compact open sets (depending on the context) by such superoscillating sequences. We also show how one can profit of the existence of such sequences in order to extrapolate band-limited signals with finite energy from a given segment of the real line

The Pompeiu Formula for Slice Hyperholomorphic Functions

The fundamental result that makes complex analysis into a new discipline, independent from the theory of real variables, is the Cauchy formula, which allows the representation of any holomorphic function through a reproducing holomorphic kernel. This result is in fact an almost immediate application of the Stokes formula, which, in the more general case, offers an integral representation formula for c1 functions. This general representation formula is often known as the Pompeiu formula and can be stated as follows

Bicomplex Holomorphic Functional Calculus

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

Continuity Theorems for a Class of Convolution Operators and Applications to Superoscillations

We prove a new theorem on the continuity of convolution operators with variable coefficients, and we use it to deduce that the limit of a superoscillating sequence maintains the superoscillatory behaviour for all values of time, when evolved according to Schrödinger equations with time-dependent potentials