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In this paper, we consider sums of generalized polygonal numbers with repeats, generalizing Fermat's polygonal number theorem which was proven by Cauchy. In particular, we obtain the minimal number of generalized m-gonal numbers required to represent every positive integer and we furthermore generalize this result to obtain optimal bounds when many of the generalized m-gonal numbers are repeated r times, where r ∈ N is fixed.

Interfacial reactions between the melts of several borate glasses and titanium have been investigated by analytical scanning electron microscopy (ASEM) and by x-ray photoelectron spectroscopy (XPS). A thin titanium boride interfacial layer is detected by XPS after short (30 minutes) thermal treatments. ASEM analyses after longer thermal treatments (8–120 hours) reveal boron-rich interfacial layers and boride precipitates in the Ti side of the interface.

Among the issues associated with the fabrication of polysilicon thin film transistor (TFT) active matrix displays is the contact resistance between indium tin oxide (ITO) pixel electrodes and the aluminum data lines. We have investigated the use of Co and Ni for use as barrier layers between the Al and ITO in order to provide good ohmic contacts. For comparison, a titanium barrier layer has also been used and measured. A thin, 10 nm Co layer sufficed to provide good ohmic contacts in an ITO-Co-Al stack, while a thicker layer was required in the case of Ni in order to form a similar good ohmic contact. The contacts have been characterized using resistance measurements in conjunction with X-ray photo-electron spectroscopy (XPS) and X-ray diffraction (XRD) in order to correlate the electrical results with the physical metallurgy of the contacts.