901 research outputs found
S-regular functions which preserve a complex slice
We study global properties of quaternionic slice regular functions (also
called 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.Comment: 23 pages, to appear in Annali di Matematica Pura e Applicat
-exponential of slice-regular functions
According to [5] 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 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
-preserving for some and show that
is never-vanishing. Sharp necessary and sufficient conditions are given in
order that , finding an exceptional and
unexpected case in which equality holds even if and 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 functions are used to provide a further formula for
. A number of examples is given throughout the paper.Comment: 15 pages; to appear in Proceedings of the American Mathematical
Societ
Performance of the Cell processor for biomolecular simulations
The new Cell processor represents a turning point for computing intensive
applications. Here, I show that for molecular dynamics it is possible to reach
an impressive sustained performance in excess of 30 Gflops with a peak of 45
Gflops for the non-bonded force calculations, over one order of magnitude
faster than a single core standard processor
On Size and Growth of Business Firms
We study size and growth distributions of products and business firms in the context of a given industry. Firm size growth is analyzed in terms of two basic mechanisms, i.e. the increase of the number of new elementary business units and their size growth. We find a power-law relationship between size and the variance of growth rates for both firms and products, with an exponent between -0.17 and -0.15, with a remarkable stability upon aggregation. We then introduce a simple and general model of proportional growth for both the number of firm independent constituent units and their size, which conveys a good representation of the empirical evidences. This general and plausible generative process can account for the observed scaling in a wide variety of economic and industrial systems. Our findings contribute to shed light on the mechanisms that sustain economic growth in terms of the relationships between the size of economic entities and the number and size distribution of their elementary components.Firm Growth; Power Laws, Gibrat's Law; Economic Growth; Pharmaceutical Industry
Multiscale modelling of liquids with molecular specificity
The separation between molecular and mesoscopic length and time scales poses
a severe limit to molecular simulations of mesoscale phenomena. We describe a
hybrid multiscale computational technique which address this problem by keeping
the full molecular nature of the system where it is of interest and
coarse-graining it elsewhere. This is made possible by coupling molecular
dynamics with a mesoscopic description of realistic liquids based on Landau's
fluctuating hydrodynamics. We show that our scheme correctly couples
hydrodynamics and that fluctuations, at both the molecular and continuum
levels, are thermodynamically consistent. Hybrid simulations of sound waves in
bulk water and reflected by a lipid monolayer are presented as illustrations of
the scheme
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