Hanna E. Makaruk

Deeper Sparsely Nets can be Optimal

The starting points of this paper are two size-optimal solutions: (i) one for implementing arbitrary Boolean functions [1]; and (ii) another one for implementing certain sub-classes of Boolean functions [2]. Because VLSI implementations do not cope well with highly interconnected nets – the area of a chip grows with the cube of the fan-in [3] – this paper will analyse the influence of limited fan-in on the size optimality for the two solutions mentioned. First, we will extend a result from Horne & Hush [1] valid for fan-ins Δ = 2 to arbitrary fan-in. Second, we will prove that size-optimal solutions are obtained for small constant fan-in for both constructions, while relative minimum size solutions can be obtained for fan-ins strictly lower than linear. These results are in agreement with similar ones proving that for small constant fan-ins (Δ = 6 ... 9), there exist VLSI-optimal (i.e., minimising AT2) solutions [4], while there are similar small constants relating to our capacity of processing information [5].

SHOCK PHYSICS DATA RECONSTRUCTION USING SUPPORT VECTOR REGRESSION

This paper considers a set of shock physics experiments that investigate how materials respond to the extremes of deformation, pressure, and temperature when exposed to shock waves. Due to the complexity and the cost of these tests, the available experimental data set is often very sparse. A support vector machine (SVM) technique for regression is used for data estimation of velocity measurements from the underlying experiments. Because of good generalization performance, the SVM method successfully interpolates the experimental data. The analysis of the resulting velocity surface provides more information on the physical phenomena of the experiment. Additionally, the estimated data can be used to identify outlier data sets, as well as to increase the understanding of the other data from the experiment.

On limited fan-in optimal neural networks

This paper analyses the influence of limited fan-in on the size and VLSI optimality of highly interconnected nets. Two different approaches show that VLSI- and size-optimal discrete neural networks can be obtained for small fan-in values. They have applications to hardware implementations of neural networks. The first approach is based on implementing a certain sub-class of Boolean functions, F/sub n,m/ functions. We show that this class of functions can be implemented in VLSI-optimal (i.e., minimising AT/sup 2/) neural networks of small constant fan-ins. The second approach is based on implementing Boolean functions for which the classical Shannons decomposition can be used. Such a solution has already been used by Alon-Bruck (1991) to prove bounds on neural networks with fan-ins limited to 2. We generalise the result presented there to arbitrary fan-in, and prove that the size is minimised by small fan-in values, while relative minimum size solutions can be obtained for fan-ins strictly lower than linear. Finally, a size-optimal neural network having small constant fan-ins is suggested for F/sub n,m/ functions.

Generalized noiseless quantum codes utilizing quantum enveloping algebras

A generalization of the results of Rasetti and Zanardi concerning avoiding errors in quantum computers by using states preserved by evolution is presented. The concept of dynamical symmetry is generalized from the level of classical Lie algebras and groups to the level of dynamical symmetry based on quantum Lie algebras and quantum groups (in the sense of Woronowicz). A natural connection is proved between states preserved by representations of a quantum group and states preserved by evolution with dynamical symmetry of the appropriate universal enveloping algebra. Illustrative examples are discussed.

Small fan-in is beautiful

The starting points of this paper are two size-optimal solutions: (i) one for implementing arbitrary Boolean functions; and (ii) another one for implementing certain sub-classes of Boolean functions. Because VLSI implementations do not cope well with highly interconnected nets-the area of a chip grows with the cube of the fan-in-this paper analyses the influence of limited fan-in on the size optimality for the two solutions mentioned. First, we extend one result from Horne and Hush (1994) valid for fan-in /spl Delta/=2 to arbitrary fan-in. Second, we prove that size-optimal solutions are obtained for small constant fan-ins for both constructions, while relative minimum size solutions can be obtained for fan-ins strictly lower than linear. These results are in agreement with similar ones proving that for small constant fan-ins (/spl Delta/=6...9) there exist VLSI-optimal (i.e., minimising AT/sup 2/) solutions, while there are similar small constants relating to our capacity of processing information.

Quantum computing and second quantization

Quantum computers by their nature are many particle quantum systems. Both the many-particle arrangement and being quantum are necessary for the existence of the entangled states, which are responsible for the parallelism of the quantum computers. Second quantization is a very important approximate method of describing such systems. This lecture will present the general idea of the second quantization, and discuss shortly some of the most important formulations of second quantization.

PREDICTIONS AND DIAGNOSTICS IN EXPERIMENTAL DATA USING SUPPORT VECTOR REGRESSION

In this paper we present a novel support vector machine (SVM) based framework for prognosis and diagnosis. We apply the framework to sparse physics data sets, although the method can easily be extended to other domains. Experiments in applied fields, such as experimental physics, are often complicated and expensive. As a result, experimentalists are unable to conduct as many experiments as they would like, leading to very unbalanced data sets that can be dense in one dimension and very sparse in others. Our method predicts the data values along the sparse dimension providing more information to researchers. Often experiments deviate from expectations due to small misalignments in initial parameters. Our method detects these outlier experiments.

Symbolic Package for Quantum Groups, Noncommutative Algebras, and Logics

This paper describes packages for symbolic calculations in quantum groups, noncommutative differential geometry, and multivalued logic. The package for quantum groups and the program for logic are written in Mathematica 3.0 and/or 4.0. As an example, some results in the logic obtained using these packages are presented.