HR: A System for Machine Discovery in Finite Algebras

We describe the HR concept formation program which invents mathematical definitions and conjectures in finite algebras such as group theory and ring theory. We give the methods behind and the reasons for the concept formation in HR, an evaluation of its performance in its training domain, group theory, and a look at HR in domains other than group theory

The Use of Classification in Automated Mathematical Concept Formation

Concept formation programs aim to produce a high yield of concepts which are considered interesting. One intelligent way to do this is to base a new concept on one or more concepts which are already known to be interesting. This requires a concrete notion of the `interestingness' of a particular concept. Restricting the concepts formed to mathematical definitions in finite group theory, we derive three measures of the importance of a concept. These measures are based on how much the concept improves a classification of finite groups. Introduction One approach to automatic mathematical concept formation is to perform a heuristic search through a space of sentences which define mathematical concepts. In the space, there will be some sentences which are rubbish, some which are plausible but not very exciting, and some which are important. In order to be able to do an effective search, reducing the number of rubbish sentences, and increasing the yield of important concepts, it is..

"Oxide-free" tip for scanning tunneling microscopy

We report a new tip for scanning tunneling microscopy and a tip repair procedure that allows one to reproducibly obtain atomic images of highly oriented pyrolytic graphite with previously inoperable tips. The tips are shown to be relatively oxide-free and highly resistant to oxidation. The tips are fabricated with graphite by two distinct methods

Active Exterior Cloaking

A new method of cloaking is presented. For two-dimensional quasistatics it is proven how a single active exterior cloaking device can be used to shield an object from surrounding fields, yet produce very small scattered fields. The problem is reduced to finding a polynomial which is approximately one within one disk and zero within a second disk, and such a polynomial is constructed. For the two-dimensional Helmholtz equation, it is numerically shown that three active exterior devices placed around the object suffice to produce very good cloaking.Comment: 4 pages, 3 figures, submitted to Physical Review Letter

Proof-Pattern Recognition and Lemma Discovery in ACL2

We present a novel technique for combining statistical machine learning for proof-pattern recognition with symbolic methods for lemma discovery. The resulting tool, ACL2(ml), gathers proof statistics and uses statistical pattern-recognition to pre-processes data from libraries, and then suggests auxiliary lemmas in new proofs by analogy with already seen examples. This paper presents the implementation of ACL2(ml) alongside theoretical descriptions of the proof-pattern recognition and lemma discovery methods involved in it

Automatic Invention of Integer Sequences

We report on the application of the HR program (Colton, Bundy, & Walsh 1999) to the problem of automatically inventing integer sequences. Seventeen sequences invented by HR are interesting enough to have been accepted into the Encyclopedia of Integer Sequences (Sloane 2000) and all were supplied with interesting conjectures about their nature, also discovered by HR. By extending HR, we have enabled it to perform a two stage process of invention and investigation. This involves generating both the definition and terms of a new sequence, relating it to sequences already in the Encyclopedia and pruning the output to help identify the most surprising and interesting results

Thermal Radiation From Carbon Nanotube in Terahertz Range

The thermal radiation from an isolated finite-length carbon nanotube (CNT) is theoretically investigated both in near- and far-field zones. The formation of the discrete spectrum in metallic CNTs in the terahertz range is demonstrated due to the reflection of strongly slowed-down surface-plasmon modes from CNT ends. The effect does not appear in semiconductor CNTs. The concept of CNT as a thermal nanoantenna is proposed.Comment: 5 pages, 3 figure

Graphs Obtained From Collections of Blocks

Given a collection of $d$-dimensional rectangular solids called blocks, no two of which sharing interior points, construct a block graph by adding a vertex for each block and an edge if the faces of the two corresponding blocks intersect nontrivially. It is known that if $d \geq 3$, such block graphs can have arbitrarily large chromatic number. We prove that the chromatic number can be bounded with only a mild restriction on the sizes of the blocks. We also show that block graphs of block configurations arising from partitions of $d$-dimensional hypercubes into sub-hypercubes are at least $d$-connected. Bounds on the diameter and the hamiltonicity of such block graphs are also discussed

Determining the shape of defects in non-absorbing inhomogeneous media from far-field measurements

International audienceWe consider non-absorbing inhomogeneous media represented by some refraction index. We have developed a method to reconstruct, from far-field measurements, the shape of the areas where the actual index differs from a reference index. Following the principle of the Factorization Method, we present a fast reconstruction algorithm relying on far field measurements and near field values, easily computed from the reference index. Our reconstruction result is illustrated by several numerical test cases