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Introduced by Sadeh et al., the K-star-graph private information retrieval (PIR) problem, so-labeled because the storage graph is a star-graph with K leaf nodes, is comprised of K messages that are stored separately (one-each) at K dedicated servers, and a universal server that stores all K messages, for a total of K + 1 servers. While it is one of the simplest PIR settings to describe, the capacity CK of K-star-graph PIR is open for K ≥ 4. We study the critical K = 4 setting, for which prior work establishes the bounds 2/5 ≤ C4 ≤ 3/7. As our main contribution, we characterize the exact capacity of 4-star-graph PIR as C4 = 5/12, thus improving upon both the prior lower- bound as well as the prior upper-bound. The main technical challenge resides in the new converse bound, whose non-trivial structure is deduced indirectly from the achievable schemes that emerge from the study of a finer tradeoff between the download costs from the dedicated servers versus the universal server. A sharp characterization of this tradeoff is also obtained for K = 4. 

The photophysical properties of naturally occurring chlorophylls depend on the regioisomeric nature of the β-pyrrolic substituents. Such systems are the “gold standard” by which such effects are judged. However, simple extrapolations from what has been learned with chlorophylls may not be appropriate for other partially reduced porphyrinoids. Here we report the synthesis of a series of cis / trans -porphodilactones ( cis / trans -1) and related derivatives ( cis / trans 2–5) designed to probe the effect of regioisomeric substitution in porphyrinoids that incorporate degrees of unsaturation through the β-pyrrolic periphery that exceed those of chlorophyll. These test systems were obtained through β-pyrrolic modifications of the tetrapyrrolic core, which included reduction of β-diazalone to the corresponding dilactol moieties and 1,3-dipolar cycloadditions. In the case of cis - vs. trans -3 bearing two pyrrolidine-fused β-rings we found an unprecedented Δ Q L up to ca. 71 nm (2086 cm −1 ), where Δ Q L ( Q L means the lowest energy transfer band, also the S 0 → S 1 transition band, which is often assigned as Q y (0,0) band) refers to the transition energy difference between the corresponding cis / trans -isomers. The Δ Q L values for these and other systems reported here were found to depend on the differences in the HOMO–LUMO energy gap and to be tied to the degeneracy and energy level splitting of the FMOs, as inferred from a combination of MCD spectral studies and DFT calculations. The aromaticity, estimated from the chemical shifts of the N–H protons and supported by theoretical calculations ( e.g. , AICD plots and NICS(1) values), was found to correlate with the extent of porphyrin periphery saturation resulting from the specific β-modifications. The aromaticity proved inversely proportional to the degree to which the regioisomerism affected the photophysical properties as noted from plots of Δ Q L s in cm −1 vs. the average NICS(1) values for 1–5. Such a finding is not something that can be easily interpolated from prior work and thus reveals how aromaticity may be used to fine-tune photophysical effects in reduced porphyrinoids.