Euclidean quotients of finite metric spaces

This paper is devoted to the study of quotients of finite metric spaces. The basic type of question we ask is: Given a finite metric space M and α?1, what is the largest quotient of (a subset of) M which well embeds into Hilbert space. We obtain asymptotically tight bounds for these questions, and prove that they exhibit phase transitions. We also study the analogous problem for embeddings into ?_p, and the particular case of the hypercube.     

Transition from zero-dimensional to one-dimensional behavior in InAs and CdSe nanorods

Tunneling and optical spectroscopy performed on InAs nanorods 7–25 nm long, reveal a clear dependence of the band gap on length. This (zero-dimension like) behavior is different from that of CdSe rods, where the band gap is nearly independent of length, a signature of quasi-one-dimensionality. The transition between these two regimes is governed by the ratio between the Bohr radius and the nanorods length. The gaps measured by tunneling spectroscopy are larger than the optical gaps by a factor that depends on the tunneling configuration. This is attributed to a combination of the Coulomb interaction and the voltage division between the two tunnel junctions in the STM experiment. However, the tunneling gaps were found to reduce in dense aggregates of rods.     

Synthesis of Biaryl-Bridged Cyclic Peptides via Catalytic Oxidative Cross-Coupling Reactions

Biaryl-bridged cyclic peptides comprise an intriguing class of structurally diverse natural products with significant biological activity. Especially noteworthy are the antibiotics arylomycin and its synthetic analogue G0775, which exhibits potent activity against Gram-negative bacteria. Herein, we present a simple, flexible, and reliable strategy based on activating-group-assisted catalytic oxidative coupling for assembling biaryl-bridged cyclic peptides from natural amino acids. The synthetic approach was utilized for preparing a number of natural and unnatural biaryl-bridged cyclic peptides, including arylomycin/G0775 and RP 66453 cyclic cores.     

A Pieri rule for skew shapes

The Pieri rule expresses the product of a Schur function and a single row Schur function in terms of Schur functions. We extend the classical Pieri rule by expressing the product of a skew Schur function and a single row Schur function in terms of skew Schur functions. Like the classical rule, our rule involves simple additions of boxes to the original skew shape. Our proof is purely combinatorial and extends the combinatorial proof of the classical case.     

Wreath products and representations of p-local finite groups

Given two finite p-local finite groups and a fusion preserving morphism between their Sylow subgroups, we study the question of extending it to a continuous map between their classifying spaces. The results depend on the construction of the wreath product of p-local finite groups which is also used to study p-local permutation representations.     

Noise enhanced persistence in a biochemical regulatory network with feedback control

We find that discrete noise of inhibiting (signal) molecules can greatly delay the extinction of plasmids in a plasmid replication system: a prototypical biochemical regulatory network. We calculate the probability distribution of the metastable state of the plasmids and show in this example that the reaction rate equations may fail in predicting the average number of regulated molecules even when this number is large, and the time is much shorter than the mean extinction time.     

Observation of soliton tunneling phenomena and soliton ejection

We study, theoretically and experimentally, the nonlinear dynamics of a wave packet launched inside a trap potential. Increasing the power of the wave packet transforms its dynamics from linear tunneling through a potential barrier, to soliton tunneling, and eventually, above a well-defined threshold, to the ejection of a soliton from the potential trap.     

Scheduling identical jobs and due-window on uniform machines

We extend a classical common due-window assignment problem to a setting of parallel uniform machines. Jobs are assumed to have identical processing times. The objective is minimum earliness, tardiness, due-window starting time, and due-window size. We focus on the case of two machines. Despite the many (12) candidate schedules for optimality, an efficient constant time solution is introduced.     

Minimum weighted number of tardy jobs on an m-machine flow-shop with a critical machine

We study a flow-shop problem, where each of the jobs is limited to no more than two operations. One of these operations is common for all the jobs, and is performed on the same (”critical”) machine. Reflecting many applications, jobs are assumed to be processed in blocks on the critical machine. All the jobs share a common due-date, and the objective is minimum weighted number of tardy jobs. We prove that the problem is NP-hard. Then we formulate the problem as an integer program, and introduce a pseudo-polynomial dynamic programming algorithm, proving that the problem is NP-hard in the ordinary sense. We also propose an efficient heuristic, which is shown numerically to produce very close-to-optimal schedules. Finally, we show that the special case of identical weights is polynomially solvable.