Growth Rates and Explosions in Sandpiles

We study the abelian sandpile growth model, where n particles are added at the origin on a stable background configuration in ? ^d . Any site with at least 2d particles then topples by sending one particle to each neighbor. We find that with constant background height h≤2d?2, the diameter of the set of sites that topple has order n ^1/d . This was previously known only for h<d. Our proof uses a strong form of the least action principle for sandpiles, and a novel method of background modification. We can extend this diameter bound to certain backgrounds in which an arbitrarily high fraction of sites have height 2d?1. On the other hand, we show that if the background height 2d?2 is augmented by 1 at an arbitrarily small fraction of sites chosen independently at random, then adding finitely many particles creates an explosion (a sandpile that never stabilizes).     

A Macrocyclic Furan with Accessible Oxidation States: Switching Between Aromatic and Antiaromatic Global Ring Currents

Macrocyclic furans are predicted to switch between global aromaticity and antiaromaticity, depending on their oxidation states. However, the macrocyclic furans reported to date are stabilized by electron withdrawing groups, which result in inaccessible oxidation states. To circumvent this problem, a post-macrocyclization approach was applied to introduce methylene-substituted macrocyclic furans, which display an extremely low oxidation potential of −0.23 vs. Fc/Fc⁺, and are partially oxidized in ambient conditions. Additional oxidation to the dication results in aromaticity switching to a global 30πe⁻ aromatic state, as indicated by the formation of a strong diatropic current observed in the ¹H NMR spectrum. NICS and ACID calculations support this trend and provide evidence for a different pathway for the global current in the neutral and dicationic states. According to these findings, macrocyclic furans can be rendered as promising p-type materials with stable oxidation states.     

Stability Testing of Two-Dimensional Discrete-Time Systems by a Scattering-Type Stability Table and Its Telepolation

Stability testing of two-dimensional (2-D) discrete-time systems requires decision on whether a 2-D (bivariate) polynomial does not vanish in the closed exterior of the unit bi-circle. The paper reformulates a tabular test advanced by Jury to solve this problem. The 2-D tabular test builds for a real 2-D polynomial of degree (n ₁, n ₂) a sequence of n ₂ matrices or 2-D polynomials (the 2-D table). It then examines its last polynomial - a 1-D polynomial of degree 2n ₁ n ₂ - for no zeros on the unit circle. A count of arithmetic operations for the tabular test is performed. It shows that the test has O(n ⁶) complexity (assuming n ₁ = n ₂ = n)- a significant improvement compared to previous tabular tests that used to be of exponential complexity. The analysis also reveals that, even though the testing of the condition on the last polynomial requires O(n ⁴) operations, the count of operations required for the table's construction makes the overall complexity O(n ⁶). Next it is shown that it is possible to telescope the last polynomial of the table by interpolation and circumvent the construction of the 2-D table. The telepolation of the tabular test replaces the table by n ₁ n ₂ + 1 stability tests of 1-D polynomials of degree n ₁ or n ₂ of certain form. The resulting new 2-D stability testing procedure requires a very low O(n ⁴) count of operations. The paper also brings extension for the tabular test and its simplification by telepolation to testing 2-D polynomials with complex valued coefficients.     

Geometric nature of the environment-induced Berry phase and geometric dephasing

We investigate the geometric phase or Berry phase acquired by a spin half which is both subject to a slowly varying magnetic field and weakly coupled to a dissipative environment (either quantum or classical). We study how this phase is modified by the environment and find that the modification is of a geometric nature. While the original Berry phase (for an isolated system) is the flux of a monopole field through the loop traversed by the magnetic field, the environment-induced modification of the phase is the flux of a quadrupolelike field. We find that the environment-induced phase is complex, and its imaginary part is a geometric contribution to dephasing. Its sign depends on the direction of the loop. Unlike the Berry phase, this geometric dephasing is gauge invariant for open paths of the magnetic field.     

The threshold for random -SAT is

Let be a random -CNF formula formed by selecting uniformly and independently out of all possible -clauses on variables. It is well known that if , then is unsatisfiable with probability that tends to 1 as . We prove that if , where , then is satisfiable with probability that tends to 1 as .

Our technique, in fact, yields an explicit lower bound for the random -SAT threshold for every . For our bounds improve all previously known such bounds.

     

Nonproteolytic cleavage of aspartyl proline bonds in the cellulosomal scaffoldin subunit from Clostridium thermocellum

Previous work from our group [Morag (Morgenstern), E., Bayer, E. A., and Lamed, R. (1991), Appl. Biochem. Biotechnol. 30, 129–136] has demonstrated an anomalous electrophoretic mobility pattern for scaffoldin, the 210-kDa cellulosome-integrating subunit of Clostridium thermocellum. Subsequent evidence [Morag, E., Bayer, E. A., and Lamed, R. (1992), Appl. Biochem. Biotechnol. 33, 205–217] indicated that the effect could be attributed to a nonproteolytic fragmentation of the subunit into a defined series of lowermolecular-weight bands. In the present work, a recombinant segment of the scaffoldin subunit was employed to determine the site(s) of bond breakage. An Asp-Pro sequence within the cohesin domain was identified to be the sensitive peptide bond. This sequence appears quite frequently in the large cellulosomal proteins, and the labile bond may be related to an as yet undescribed physiological role in the hydrolysis of cellulose by cellulosomes.     

Motivic torsors

The torsorP _δ=Hom^⊗ (H _DR,H _σ) under the motivic Galois groupG _σ=Aut^⊗ H _δ of the Tannakian category generated by one-motives related by absolute Hodge cycles over a field κ with an embedding σ:k↪ℂ is shown to be determined by its projectionP _σ→P _σ/G _σ ⁰ to a Gal( )-torsor, and by its localizationsP _σ⊕_k k _ξ at a dense subset of orderings ϕ of the field κ, provided κ has virtual cohomological dimension (vcd) one. This result is an application of a recent local-global principle for connected linear algebraic groups over a field κ of vcd ≤1.     

Equivalence of positive Hausdorff measure and the open set condition for self-conformal sets

A compact set is self-conformal if it is a finite union of its images by conformal contractions. It is well known that if the conformal contractions satisfy the ``open set condition' (OSC), then has positive -dimensional Hausdorff measure, where is the solution of Bowen's pressure equation. We prove that the OSC, the strong OSC, and positivity of the -dimensional Hausdorff measure are equivalent for conformal contractions; this answers a question of R. D. Mauldin. In the self-similar case, when the contractions are linear, this equivalence was proved by Schief (1994), who used a result of Bandt and Graf (1992), but the proofs in these papers do not extend to the nonlinear setting.

     

Thin points for Brownian motion

Let Θ(x,r) denote the occupation measure of the ball of radius r centered at x for Brownian motion {W_t}_0≤t≤1 in . We prove that for any analytic set E in [0,1], we have

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, where dim_P(E) is the packing dimension of E. We deduce that for any a≥1, the Hausdorff dimension of the set of “thin points” x for which

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, is almost surely 2−2/a; this is the correct scaling to obtain a nondegenerate “multifractal spectrum” for the “thin” part of Brownian occupation measure. The methods of this paper differ considerably from those of our work on Brownian thick points, due to the high degree of correlation in the present case. To prove our results, we establish general criteria for determining which deterministic sets are hit by random fractals of ‘limsup type' in the presence of long-range correlations. The hitting criteria then yield lower bounds on Hausdorff dimension. This refines previous work of Khoshnevisan, Xiao and the second author, that required decay of correlations.     

Work allocation to stations with varying learning slopes and without buffers

In this paper, we address the problem of allocating the work elements, belonging to the products of a lot, to the stations of an assembly line so as to minimize the makespan. The lots that are processed on the assembly line are characterized by a low overall demand for each product. There is no buffer permitted in between the stations, and the line operates under learning. In particular, the stations’ learning slopes are assumed to be different. We present a procedure to determine the optimal assignments of the workload to the stations under learning variability and show that it considerably affects these assignments.