21,141 research outputs found
Playing with functions of positive type, classical and quantum
A function of positive type can be defined as a positive functional on a
convolution algebra of a locally compact group. In the case where the group is
abelian, by Bochner's theorem a function of positive type is, up to
normalization, the Fourier transform of a probability measure. Therefore,
considering the group of translations on phase space, a suitably normalized
phase-space function of positive type can be regarded as a realization of a
classical state. Thus, it may be called a function of classical positive type.
Replacing the ordinary convolution on phase space with the twisted convolution,
one obtains a noncommutative algebra of functions whose positive functionals we
may call functions of quantum positive type. In fact, by a quantum version of
Bochner's theorem, a continuous function of quantum positive type is, up to
normalization, the (symplectic) Fourier transform of a Wigner quasi-probability
distribution; hence, it can be regarded as a phase-space realization of a
quantum state. Playing with functions of positive type, classical and quantum,
one is led in a natural way to consider a class of semigroups of operators, the
classical-quantum semigroups. The physical meaning of these mathematical
objects is unveiled via quantization, so obtaining a class of quantum dynamical
semigroups that, borrowing terminology from quantum information science, may be
called classical-noise semigroups.Comment: 19 page
Backward orbits and petals of semigroups of holomorphic self-maps of the unit disc
We study the backward invariant set of one-parameter semigroups of
holomorphic self-maps of the unit disc. Such a set is foliated in maximal
invariant curves and its open connected components are petals, which are, in
fact, images of Poggi-Corradini's type pre-models. Hyperbolic petals are in
one-to-one correspondence with repelling fixed points, while only parabolic
semigroups can have parabolic petals. Petals have locally connected boundaries
and, except a very particular case, they are indeed Jordan domains. The
boundary of a petal contains the Denjoy-Wolff point and, except such a fixed
point, the closure of a petal contains either no other boundary fixed point or
a unique repelling fixed point. We also describe petals in terms of geometric
and analytic behavior of K\"onigs functions using divergence rate and
universality of models. Moreover, we construct a semigroup having a repelling
fixed point in such a way that the intertwining map of the pre-model is not
regular.Comment: 35 page
The number of nilpotent semigroups of degree 3
A semigroup is \emph{nilpotent} of degree 3 if it has a zero, every product
of 3 elements equals the zero, and some product of 2 elements is non-zero. It
is part of the folklore of semigroup theory that almost all finite semigroups
are nilpotent of degree 3.
We give formulae for the number of nilpotent semigroups of degree 3 with
elements up to equality, isomorphism, and isomorphism or
anti-isomorphism. Likewise, we give formulae for the number of nilpotent
commutative semigroups with elements up to equality and up to isomorphism
Finite Computational Structures and Implementations
What is computable with limited resources? How can we verify the correctness
of computations? How to measure computational power with precision? Despite the
immense scientific and engineering progress in computing, we still have only
partial answers to these questions. In order to make these problems more
precise, we describe an abstract algebraic definition of classical computation,
generalizing traditional models to semigroups. The mathematical abstraction
also allows the investigation of different computing paradigms (e.g. cellular
automata, reversible computing) in the same framework. Here we summarize the
main questions and recent results of the research of finite computation.Comment: 12 pages, 3 figures, will be presented at CANDAR'16 and final version
published by IEEE Computer Societ
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