Daniel Crespin
Daniel Crespin | |
|---|---|
| Residence | Venezuela |
| Nationality | Venezuelan |
Daniel Crespin is a Venezuelan mathematician, Professor Titular (now retired) of the Escuela de Matemática, Facultad de Ciencias, Universidad Central de Venezuela (UCV) in Caracas. His work spans differential topology and geometry, the mathematical foundations of artificial neural networks, and a critical, deterministic reformulation of quantum wave mechanics.
Biography
Crespin completed his doctorate at Brandeis University under the direction of the topologist Jerome Levine. He was among the first Venezuelan mathematicians of his generation to return home with a doctorate and join the Escuela de Matemática of the Universidad Central de Venezuela, where he spent his academic career and eventually retired as Profesor Titular. Together with Arturo Reyes, Marco Paluzny and Yuli Villaroel he formed the initial core of the topology and geometry research group at the school, an area that grew into one of the established research lines of Venezuelan mathematics.
Work
Neural networks and perceptrons
A substantial part of Crespin's mathematical work is devoted to giving artificial neural networks a rigorous, purely mathematical foundation. In his neural network formalism, networks are defined using only elementary set-theoretic and analytic concepts, without the customary connectionist graphs; the familiar neural diagrams are then derived from the definitions rather than assumed. Within this framework he treats global backpropagation formulas for differentiable neural networks as the minimization of a quadratic error by the gradient method.
His later research on the geometry of perceptron networks seeks theory and computational procedures that calculate the architecture and the weights of a perceptron network directly from the given data vectors, so that the data are recognized exactly and with maximal distance to the decision boundaries, in contrast to statistical training methods. In this line he showed that single-output perceptron networks and the characteristic functions of polyhedrons constitute one and the same class of functions, and that three layers suffice, giving perceptron networks in disjunctive and conjunctive normal form.
Deterministic wave mechanics
Crespin has also argued for a deterministic replacement of standard quantum mechanics. In his analysis, Schrödinger's 1926 wave mechanics was simultaneously a success and a failure: a success in the self-adjoint Schrödinger Hamiltonian operator, but a failure in the time-dependent equation, whose adoption led to the probabilistic Copenhagen interpretation. He proposes that the Schrödinger time-dependent equation be replaced by a deterministic time-dependent equation, and has developed a continuous, causal and fully deterministic — though non-linear — wave mechanics built on the Rayleigh quotient of the Schrödinger operator, yielding a Hamiltonian dynamical system with a well-specified configuration space, state space, energy observable and law of motion. He has pursued this programme under the heading of Hamiltonian projective spectral theorems, aimed at a deterministic theory of the atom.
Publications
Selected works:
- Neural Network Formalism (preprint, Departamento de Matemáticas, Universidad Central de Venezuela, 1995)
- Geometry of Perceptrons (preprint, Departamento de Matemáticas, Universidad Central de Venezuela, 1995)
- A Primer on Perceptrons
- Polyhedrons and Perceptrons Are Functionally Equivalent (arXiv:1311.1090, 2013)
- Quantum Wave Collapse is Unsolvable
- To Quantism and Back
- Deconstruction of Quantum Wave Mechanics
- A Simple Explanation of the Quantum Doctrine