Dimension Reduction by Random Hyperplane Tessellations

Given a subset K of the unit Euclidean sphere, we estimate the minimal number m=m(K) of hyperplanes that generate a uniform tessellation of K, in the sense that the fraction of the hyperplanes separating any pair x,y∈K is nearly proportional to the Euclidean distance between x and y. Random hyperplanes prove to be almost ideal for this problem; they achieve the almost optimal bound m=O(w(K)²) where w(K) is the Gaussian mean width of K. Using the map that sends x∈K to the sign vector with respect to the hyperplanes, we conclude that every bounded subset K of $\mathbb{R}^{n}$ embeds into the Hamming cube {?1,1} ^m with a small distortion in the Gromov–Haussdorff metric. Since for many sets K one has m=m(K)?n, this yields a new discrete mechanism of dimension reduction for sets in Euclidean spaces.     

Mimicking the role of the antenna in photosynthetic photoprotection

One mechanism used by plants to protect against damage from excess sunlight is called nonphotochemical quenching (NPQ). Triggered by low pH in the thylakoid lumen, NPQ leads to conversion of excess excitation energy in the antenna system to heat before it can initiate production of harmful chemical species by photosynthetic reaction centers. Here we report a synthetic hexad molecule that functionally mimics the role of the antenna in NPQ. When the hexad is dissolved in an organic solvent, five zinc porphyrin antenna moieties absorb light, exchange excitation energy, and ultimately decay by normal photophysical processes. Their excited-state lifetimes are long enough to permit harvesting of the excitation energy for photoinduced charge separation or other work. However, when acid is added, a pH-sensitive dye moiety is converted to a form that rapidly quenches the first excited singlet states of all five porphyrins, converting the excitation energy to heat and rendering the porphyrins kinetically incompetent to readily perform useful photochemistry.     

Order-by-disorder in classical oscillator systems

We consider classical nonlinear oscillators on hexagonal lattices. When the coupling between the elements is repulsive, we observe coexisting states, each one with its own basin of attraction. These states differ by their degree of synchronization and by patterns of phase-locked motion. When disorder is introduced into the system by additive or multiplicative Gaussian noise, we observe a non-monotonic dependence of the degree of order in the system as a function of the noise intensity: intervals of noise intensity with low synchronization between the oscillators alternate with intervals where more oscillators are synchronized. In the latter case, noise induces a higher degree of order in the sense of a larger number of nearly coinciding phases. This order-by-disorder effect is reminiscent to the analogous phenomenon known from spin systems. Surprisingly, this non-monotonic evolution of the degree of order is found not only for a single interval of intermediate noise strength, but repeatedly as a function of increasing noise intensity. We observe noise-driven migration of oscillator phases in a rough potential landscape.     

NMR Spectral Patterns in Magneto-Aligned Biaxial Liquid Crystals

Unique NMR spectral patterns can be obtained from biaxially ordered smectic phases where only one of the principal axes of the time averaged nuclear spin interaction is aligned by the magnetic field. It is shown how the asymmetry parameter of the interaction can be obtained directly from edge singularities in these patterns.     

2H-NMR Spectral Patterns in Magneto-Aligned Smectic E and G Phases

Unique ²H-NMR spectral patterns can be obtained from biaxially ordered smectic E and G phases, where only one director of the phase is aligned by the magnetic field. Two effects on the shape of the deuterium spectral pattern are demonstrated: (1) The effect of a motionally induced asymmetry parameter in the time averaged deuterium quadrupole interaction, and (2) the effect when the principal z-axis of the time averaged interaction is not parallel to the aligned director. Spectral patterns for the S_G phase of the compound 4-n-pentoxybenzylidene-4-n'-heptylaniline selectively deuterated on one of the aromatic rings (50.7-d ₄) were obtained for both unaligned and magneto-aligned samples. Both of these spectral patterns were consistent with that expected for an asymmetry parameter, n < 0.2. Spectral patterns in the S_G phase of the compound 50.6 (24.7 wt.%) + 90.4-αd ₂-γd ₂ (75.3 wt.%) were studied showing a different behavior with respect to the spectral patterns obtained in 50.7. We believe that a shift in the orientation of the z principal axis relative to the sample c axis has occurred in this S_G phase. Spectral patterns were also recorded for the S_E phase in magneto-aligned samples of the mixture of compounds 4-cyano-4'-octyloxybiphenyl (80CB) with 50 wt.% 40.8-d ₄. These data also show a small asymmetry parameter but, in addition, a distribution in the orientation of the principal z-axis indicating that the time scale of the motion of the long molecular axis is in a regime where it is of the order of the ²H-NMR measurement. Models for the smectic E and G phases based on these observations are discussed.     

Scaling and universality of the complexity of analog computation

We apply a probabilistic approach to study the computational complexity of analog computers which solve linear programming problems. We numerically analyze various ensembles of linear programming problems and obtain, for each of these ensembles, the probability distribution functions of certain quantities which measure the computational complexity, known as the convergence rate, the barrier and the computation time. We find that in the limit of very large problems these probability distributions are universal scaling functions. In other words, the probability distribution function for each of these three quantities becomes, in the limit of large problem size, a function of a single scaling variable, which is a certain composition of the quantity in question and the size of the system. Moreover, various ensembles studied seem to lead essentially to the same scaling functions, which depend only on the variance of the ensemble. These results extend analytical and numerical results obtained recently for the Gaussian ensemble, and support the conjecture that these scaling functions are universal.     

Using bid data for the management of sequential,multi-unit,online auctions with uniformly distributed bidder valuations

Internet auctions for consumers’ goods are an increasingly popular selling venue. We have observed that many sellers, instead of offering their entire inventory in a single auction, split it into sequential auctions of smaller lots, thereby reducing the negative market impact of larger lots. Information technology also makes it possible to collect and analyze detailed bid data from online auctions. In this paper, we develop and test a new model of sequential online auctions to explore the potential benefits of using real bid data from earlier auctions to improve the management of future auctions. Assuming a typical truth-revealing auction model, we quantify the effect of the lot size on the closing price and derive a closed-form solution for the problem of allocating inventory across multiple auctions when bidder valuation distributions are known. We also develop a decision methodology for allocating inventory across multiple auctions that dynamically incorporates the results of previous auctions as feedback into the management of subsequent auctions, and updating the lot size and number of auctions. We demonstrate how information signals from previous auctions can be used to update the auctioneer’s beliefs about the customers’ valuation distribution, and then to significantly increase the seller’s profit potential. We use several examples to reveal the benefits of using detailed transaction data for the management of sequential, multi-unit, online auctions and we demonstrate how these benefits are influenced by the inventory holding costs, the number of bidders, and the dispersion of consumers’ valuations.     

Semi-primary hereditary rings

A semi-primary hereditary ring Σ, with radicalM and residue ring Γ=Σ/M, is uniquely determined by Γ and a Γ-bimoduleA=M/M ², whenever Σ admits a splitting Σ=Γ+A+M ².