1,721,808 research outputs found
From Canvas to Music: Mathematics as a Tool for the Composition of Jackson Time
No full textThe creation of ``Jackson time'' is a project which involves a composer, Davide Amodio, and
a mathematician, Chiara de Fabritiis.
Our common aim was to to ``translate'' a painting by Jackson Pollock, Summertime n. 9,
into a piece of music, making use of different mathematical tools to detect the quantities needed for the
composition. We were inspired by the idea that the painting itself contained some kind of inner--music,
due to the fact that Pollock's moves during the dripping on the canvas had a sort of rhythm, indeed they were often
described by witnesses as a dance.
This paper describes the mathematical background, in particular it illustrates both the analysis
of the painting which was carried out by the two of us and the choice of the mathematical techniques applied to
compute the parameters needed for the composition, which is due to the author. The reader will find a more detailed
report on the composition itself in Davide Amodio's contribution
Transcendental operators acting on slice regular functions
No full textThe aim of this paper is to carry out an analysis of five trascendental operators acting on the space of slice regular functions, namely ∗-exponential, ∗-sine and ∗-cosine and their hyperbolic analogues. The first three of them were introduced by Colombo, Sabadini and Struppa and some features of ∗-exponential were investigated in a previous paper by Altavilla and the author. We show how exp∗(f), sin∗(f), cos∗(f), sinh∗(f) and cosh∗(f) can be written in terms of the real and the vector part of the function f and we examine the relation between cos∗ and cosh∗ when the domain ω is product and when it is slice. In particular we prove that when ω is slice, then cos∗(f) = cosh∗(f ∗ I) holds if and only if f is CI preserving, while in the case ω is product there is a much larger family of slice regular functions for which the above relation holds
s-Regular functions which preserve a complex slice
Full text linkWe study global properties of quaternionic slice regular functions (also called extit{s-regular}) defined on symmetric slice domains.
In particular, thanks to new techniques and points of view, we can characterize the property
of being one-slice preserving in terms of the projectivization of the vectorial part of the function. We also define a ``Hermitian'' product on slice regular functions which gives us the possibility to express the -product of two s-regular functions in terms of the scalar product of suitable functions constructed starting from and .
Afterwards we are able to determine, under different assumptions,
when the sum, the -product and the -conjugation of two slice regular functions preserve a complex slice.
We also study when the -power of a slice regular function has this property or
when it preserves all complex slices.
To obtain these results we prove two factorization theorems: in the first one,
we are able to split a slice regular function into the product of two functions: one keeping track of the zeroes and the other
which is never-vanishing;in the other one we give necessary and sufficient conditions for a slice regular function (which preserves all complex slices) to be the symmetrized of a suitable slice regular one
-exponential of Slice Regular Functions
Full text linkAs in [Entire slice regular functions, Springer, 2016] we define the ∗-exponential of a slice-regular function, which can be seen as a generalization of the complex exponential to quaternions. Explicit formulas for exp ∗ (f) are provided, also in terms of suitable sine and cosine functions. We completely classify under which conditions the ∗-exponential of a function is either slice-preserving or C J -preserving for some J ∈ S and show that exp ∗ (f) is never-vanishing. Sharp necessary and sufficient conditions are given in order that exp ∗ (f + g) = exp ∗ (f) ∗ exp ∗ (g), finding an exceptional and unexpected case in which equality holds even if f and g do not commute. We also discuss the existence of a square root of a slice-preserving regular function, characterizing slice-preserving functions (defined on the circularization of simply connected domains) which admit square roots. Square roots of this kind of function are used to provide a further formula for exp ∗ (f). A number of examples are given throughout the paper
Equivalence of slice semi-regular functions via Sylvester operators
Full text linkThe aim of this paper is to study some features of slice semi-regular
functions on a circular domain contained in the
skew-symmetric algebra of quaternions via the analysis of a family
of linear operators built from left and right -multiplication on
; this class of operators includes the family of
Sylvester-type operators . Our strategy is to give a matrix
interpretation of these operators as we show that can be
seen as a -dimensional vector space on the field
. We then study the rank of
and describe its kernel and image when it is not
invertible. By using these results, we are able to characterize when the
functions and are either equivalent under -conjugation or
intertwined by means of a zero divisor, thus proving a number of statements on
the behaviour of slice semi-regular functions. We also provide a complete
classification of idempotents and zero divisors on product domains of
Linear Operators on generalized Bergman Spaces
No full textIn this paper we discuss several features of Hilbert spaces of holomorphic functions on domains of Cn. We study composition and multiplication operators on generalized Bergman spaces and give results on the dynamical behaviour (i.e. cyclicity, hypercyclicity, compactness) of the first ones and on the algebraic properties of the space that the second one interprets. In particular we underline the analogies and differences between the case of bounded and unbounded domains in C and Cn
Applications of the Sylvester operator in the space of slice semi-regular functions
No full textIn this paper we apply the results obtained in [3] to establish some outcomes of the study of the behaviour of a class of linear operators, which include the Sylvester ones, acting on slice semi-regular functions. We first present a detailed study of the kernel of the linear operator ℒf,g (when not trivial), showing that it has dimension 2 if exactly one between f and g is a zero divisor, and it has dimension 3 if both f and g are zero divisors. Afterwards, we deepen the analysis of the behaviour of the -product, giving a complete classification of the cases when the functions fv, gv and fvgv are linearly dependent and obtaining, as a by-product, a necessary and sufficient condition on the functions f and g in order their *-product is slice-preserving. At last, we give an Embry-type result which classifies the functions f and g such that for any function h commuting with f + g and f * g, we have that h commutes with f and g, too
One-parameter groups of volume-preserving automorphisms of C2
No full textSi esaminano i gruppi ad un parametro nel gruppo degli automorfismi
polinomiali di C e nel gruppo degli shears, provando che sono
coniugati a gruppi a un parametro nel gruppo degli automorfismi affini
di C o nel gruppo degli automorfismi elementari; da ciò si
deducono risultati sul comportamento asintotico del gruppo ad un parametro,
sui suoi punti periodici e sui suoi punti fissi.In this work we study the one-parameter groups in the group of all
polynomial automorphisms of C and in the group of all shears.
We prove that any such one-parameter group is conjugated to a one-parameter
group contained either in the group of all affine automorphisms of
C or in the group of elementary automorphisms. This implies
some results on the asymptotic behaviour of the one-parameter group,
on its periodic points and on its fixed points
- …
