Bound on the projective dimension of three cubics

We show that given any polynomial ring R over a field, and any ideal J in R which is generated by three cubic forms, the projective dimension of R/J is at most 36. We also settle the question whether ideals generated by three cubic forms can have projective dimension greater than 4, by constructing one with projective dimension equal to 5.Comment: to appear in Journal of Symbolic Computatio

'Growing Evanescent Envelopes and Anomalous Tunneling' in Cascaded Sets of Frequency-Selective Surfaces in Their Stop Bands

The presence of wave tunneling and the 'growing evanescent envelope' for field distributions in suitably designed, periodically layered stacks of frequency selective surfaces (FSS) is discussed in this paper. Here it is shown that a setup completely different completely different from the Pendry's lens allows an analogous buildup of evanescently modulated waves. In particular, it is shown how an interface resonance phenomenon similar to the one present at the interface between metamaterials with oppositely signed constitutive parameters may be induced by a proper choice of the periodicities of the FSS stacks and the geometrical properties of these surfaces. The analysis is performed through an equivalent transmission-line approach, and some physical insights into this phenomenon are presented. Salient features, such as the complete wave tunneling through the pair of cascaded FSS, each operating at its bandgap, are presented and discussed.Comment: 5 pages, 2 figures, submitted to Physical review

Physical Insight into the 'Growing' Evanescent Fields of Double-Negative Metamaterial Lenses Using their Circuit Equivalence

Pendry in his paper [Phys. Rev. Lett., 85, 3966 (2000)] put forward an idea for a lens made of a lossless metamaterial slab with n = -1, that may provide focusing with resolution beyond the conventional limit. In his analysis, the evanescent wave inside such a lossless double-negative (DNG) slab is 'growing', and thus it 'compensates' the decaying exponential outside of it, providing the sub-wavelength lensing properties of this system. Here, we examine this debated issue of 'growing exponential' from an equivalent circuit viewpoint by analyzing a set of distributed-circuit elements representing evanescent wave interaction with a lossless slab of DNG medium. Our analysis shows that, under certain conditions, the current in series elements and the voltage at the element nodes may attain the dominant increasing due to the suitable resonance of the lossless circuit, providing an alternative physical explanation for 'growing exponential' in Pendry's lens and similar sub-wavelength imaging systems.Comment: 7 pages, 2 figures, 1 table, submitted to IEEE Transactions on Antennas and Propagatio

Optical Nanotransmission Lines: Synthesis of Planar Left-Handed Metamaterials in the Infrared and Visible Regimes

Following our recent theoretical development of the concept of nano-inductors, nano-capacitors and nano-resistors at optical frequencies and the possibility of synthesizing more complex nano-scale circuits, here we theoretically investigate in detail the problem of optical nano-transmission-lines (NTL) that can be envisioned by properly joining together arrays of these basic nano-scale circuit elements. We show how, in the limit in which these basic circuit elements are closely packed together, the NTLs can be regarded as stacks of plasmonic and non-plasmonic planar slabs, which may be designed to effectively exhibit the properties of planar metamaterials with forward (right-handed) or backward (left-handed) operation. With the proper design, negative refraction and left-handed propagation are shown to be possible in these planar plasmonic guided-wave structures, providing possibilities for sub-wavelength focusing and imaging in planar optics, and laterally-confined waveguiding at IR and visible frequencies. The effective material parameters for such NTLs are derived, and the connection and analogy between these optical NTLs and the double-negative and double-positive metamaterials are also explored. Physical insights and justification for the results are also presented.Comment: 26 pages, 12 figures, accepted for publication in JOSA B, scheduled to appear March 200