46 research outputs found
Kirchhoff's Rule for Quantum Wires
In this article we formulate and discuss one particle quantum scattering
theory on an arbitrary finite graph with open ends and where we define the
Hamiltonian to be (minus) the Laplace operator with general boundary conditions
at the vertices. This results in a scattering theory with channels. The
corresponding on-shell S-matrix formed by the reflection and transmission
amplitudes for incoming plane waves of energy is explicitly given in
terms of the boundary conditions and the lengths of the internal lines. It is
shown to be unitary, which may be viewed as the quantum version of Kirchhoff's
law. We exhibit covariance and symmetry properties. It is symmetric if the
boundary conditions are real. Also there is a duality transformation on the set
of boundary conditions and the lengths of the internal lines such that the low
energy behaviour of one theory gives the high energy behaviour of the
transformed theory. Finally we provide a composition rule by which the on-shell
S-matrix of a graph is factorizable in terms of the S-matrices of its
subgraphs. All proofs only use known facts from the theory of self-adjoint
extensions, standard linear algebra, complex function theory and elementary
arguments from the theory of Hermitean symplectic forms.Comment: 40 page
Dirac Operators and the Calculation of the Connes Metric on arbitrary (Infinite) Graphs
As an outgrowth of our investigation of non-regular spaces within the context
of quantum gravity and non-commutative geometry, we develop a graph Hilbert
space framework on arbitrary (infinite) graphs and use it to study spectral
properties of graph-Laplacians and graph-Dirac-operators. We define a spectral
triplet sharing most of the properties of what Connes calls a spectral triple.
With the help of this scheme we derive an explicit expression for the
Connes-distance function on general directed or undirected graphs. We derive a
series of apriori estimates and calculate it for a variety of examples of
graphs. As a possibly interesting aside, we show that the natural setting of
approaching such problems may be the framework of (non-)linear programming or
optimization. We compare our results (arrived at within our particular
framework) with the results of other authors and show that the seeming
differences depend on the use of different graph-geometries and/or Dirac
operators.Comment: 27 pages, Latex, comlementary to an earlier paper, general treatment
of directed and undirected graphs, in section 4 a series of general results
and estimates concerning the Connes Distance on graphs together with examples
and numerical estimate
Boundary relations and generalized resolvents of symmetric operators
The Kre\u{\i}n-Naimark formula provides a parametrization of all selfadjoint
exit space extensions of a, not necessarily densely defined, symmetric
operator, in terms of maximal dissipative (in \dC_+) holomorphic linear
relations on the parameter space (the so-called Nevanlinna families). The new
notion of a boundary relation makes it possible to interpret these parameter
families as Weyl families of boundary relations and to establish a simple
coupling method to construct the generalized resolvents from the given
parameter family. The general version of the coupling method is introduced and
the role of boundary relations and their Weyl families for the
Kre\u{\i}n-Naimark formula is investigated and explained.Comment: 47 page
Generalized boundary triples, I. Some classes of isometric and unitary boundary pairs and realization problems for subclasses of Nevanlinna functions
© 2020 The Authors. Mathematische Nachrichten published by Wiley‐VCH Verlag GmbH & Co. KGaA. This is an open access article under the terms of the Creative Commons Attribution License, which permits use, distribution and reproduction in any medium, provided the original work is properly cited.fi=vertaisarvioitu|en=peerReviewed
Towards Comprehensive Foundations of Computational Intelligence
Abstract. Although computational intelligence (CI) covers a vast variety of different methods it still lacks an integrative theory. Several proposals for CI foundations are discussed: computing and cognition as compression, meta-learning as search in the space of data models, (dis)similarity based methods providing a framework for such meta-learning, and a more general approach based on chains of transformations. Many useful transformations that extract information from features are discussed. Heterogeneous adaptive systems are presented as particular example of transformation-based systems, and the goal of learning is redefined to facilitate creation of simpler data models. The need to understand data structures leads to techniques for logical and prototype-based rule extraction, and to generation of multiple alternative models, while the need to increase predictive power of adaptive models leads to committees of competent models. Learning from partial observations is a natural extension towards reasoning based on perceptions, and an approach to intuitive solving of such problems is presented. Throughout the paper neurocognitive inspirations are frequently used and are especially important in modeling of the higher cognitive functions. Promising directions such as liquid and laminar computing are identified and many open problems presented.