108 research outputs found
Łukasiewicz-Moisil Many-Valued Logic Algebra of Highly-Complex Systems
A novel approach to self-organizing, highly-complex systems (HCS), such as living organisms and artificial intelligent systems (AIs), is presented which is relevant to Cognition, Medical Bioinformatics and Computational Neuroscience. Quantum Automata (QAs) were defined in our previous work as generalized, probabilistic automata with quantum state spaces (Baianu, 1971). Their next-state functions operate through transitions between quantum states defined by the quantum equations of motion in the Schroedinger representation, with both initial and boundary conditions in space-time. Such quantum automata operate with a quantum logic, or Q-logic, significantly different from either Boolean or Łukasiewicz many-valued logic. A new theorem is proposed which states that the category of quantum automata and automata--homomorphisms has both limits and colimits. Therefore, both categories of quantum automata and classical automata (sequential machines) are bicomplete. A second new theorem establishes that the standard automata category is a subcategory of the quantum automata category. The quantum automata category has a faithful representation in the category of Generalized (M,R)--Systems which are open, dynamic biosystem networks with defined biological relations that represent physiological functions of primordial organisms, single cells and higher organisms
Orthogonal Appell bases for Hodge-de Rham systems in Euclidean spaces
Recently the Gelfand-Tsetlin construction of orthogonal bases has been
explicitly described for the spaces of k-homogeneous polynomial solutions of
the Hodge-de Rham system in the Euclidean space R^m which take values in the
space of s-vectors. In this paper, we give another construction of these bases
and, mainly, we show that the bases even form complete orthogonal Appell
systems. Moreover, we study the corresponding Taylor series expansions. As an
application, we construct quite explicitly orthogonal bases for homogeneous
solutions of an arbitrary generalized Moisil-Theodoresco system.Comment: submitte
The Gelfand-Tsetlin bases for Hodge-de Rham systems in Euclidean spaces
The main aim of this paper is to construct explicitly orthogonal bases for
the spaces of k-homogeneous polynomial solutions of the Hodge-de Rham system in
the Euclidean space R^m which take values in the space of s-vectors. Actually,
we describe even the so-called Gelfand-Tsetlin bases for such spaces in terms
of Gegenbauer polynomials. As an application, we obtain an algorithm how to
compute an orthogonal basis of the space of homogeneous solutions of a
generalized Moisil-Theodoresco system in R^m.Comment: submitte
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