Bounding Embeddings of VC Classes into Maximum Classes

One of the earliest conjectures in computational learning theory-the Sample Compression conjecture-asserts that concept classes (equivalently set systems) admit compression schemes of size linear in their VC dimension. To-date this statement is known to be true for maximum classes---those that possess maximum cardinality for their VC dimension. The most promising approach to positively resolving the conjecture is by embedding general VC classes into maximum classes without super-linear increase to their VC dimensions, as such embeddings would extend the known compression schemes to all VC classes. We show that maximum classes can be characterised by a local-connectivity property of the graph obtained by viewing the class as a cubical complex. This geometric characterisation of maximum VC classes is applied to prove a negative embedding result which demonstrates VC-d classes that cannot be embedded in any maximum class of VC dimension lower than 2d. On the other hand, we show that every VC-d class C embeds in a VC-(d+D) maximum class where D is the deficiency of C, i.e., the difference between the cardinalities of a maximum VC-d class and of C. For VC-2 classes in binary n-cubes for 4 <= n <= 6, we give best possible results on embedding into maximum classes. For some special classes of Boolean functions, relationships with maximum classes are investigated. Finally we give a general recursive procedure for embedding VC-d classes into VC-(d+k) maximum classes for smallest k.Comment: 22 pages, 2 figure

Microgravity acoustic mixing for particle cloud combustors

Experimental and theoretical investigations of acoustic mixing procedures designed to uniformly distribute fuel particles in a combustion tube for application in the proposed Particle Cloud Combustion Experiment (PCCE) are described. Two acoustic mixing methods are investigated: mixing in a cylindrical tube using high frequency spinning modes generated by suitably phased, or quadrature speakers, and acoustic premixing in a sphere. Quadrature mixing leads to rapid circumferential circulation of the powder around the tube. Good mixing is observed in the circulating regions. However, because axial inhomogeneities are necessarily present in the acoustic field, this circulation does not extend throughout the tube. Simultaneous operation of the quadrature-speaker set and the axial-speaker was observed to produce considerably enhanced mixing compared to operation of the quadrature-speaker set alone. Mixing experiments using both types of speakers were free of the longitudinal powder drift observed using axial-speakers alone. Vigorous powder mixing was obtained in the sphere for many normal modes: however, in no case was the powder observed to fill the sphere entirely. Theoretical analysis indicated that mixing under steady conditions cannot fill more than a hemisphere except under very unusual conditions. Premixing in a hemisphere may be satisfactory; otherwise, complete mixing in microgravity might be possible by operating the speaker in short bursts. A general conclusion is that acoustic transients are more likely to produce good mixing than steady state conditions. The reason is that in steady conditions, flow structures like nodal planes are possible and often even unavoidable. These tend to separate the mixing region into cells across which powder cannot be transferred. In contrast, transients not only are free of such structures, they also have the characteristics, desirable for mixing, of randomness and disorder. This conclusion is corroborated by mixing experiments using axial waves

Structural tailoring of select fiber composite structures

A multidisciplinary design process for aerospace propulsion composite structures was formalized and embedded into computer codes. These computer codes are streamlined to obtain tailored designs for select composite structures. The codes available are briefly described with sample cases to illustrate their applications. The sample cases include aircraft engine blades, propfans (turboprops), flat, and cylindrical panels. Typical results illustrate that the use of these codes enable the designer to obtain designs which meet all the design requirements with maximum benefits in efficiency, noise, weight or thermal distortions