Load Balancing in Hypergraphs

Consider a simple locally finite hypergraph on a countable vertex set, where each edge represents one unit of load which should be distributed among the vertices defining the edge. An allocation of load is called balanced if load cannot be moved from a vertex to another that is carrying less load. We analyze the properties of balanced allocations of load. We extend the concept of balancedness from finite hypergraphs to their local weak limits in the sense of Benjamini and Schramm (Electron J Probab 6(23):13, 2001) and Aldous and Steele (in: Probability on discrete structures. Springer, Berlin, pp 1–72, 2004). To do this, we define a notion of unimodularity for hypergraphs which could be considered an extension of unimodularity in graphs. We give a variational formula for the balanced load distribution and, in particular, we characterize it in the special case of unimodular hypergraph Galton–Watson processes. Moreover, we prove the convergence of the maximum load under some conditions. Our work is an extension to hypergraphs of Anantharam and Salez (Ann Appl Probab 26(1):305–327, 2016), which considered load balancing in graphs, and is aimed at more comprehensively resolving conjectures of Hajek (IEEE Trans Inf Theory 36(6):1398–1414, 1990).     

Triple-quantum magic angle spinning (27)Al NMR of aluminum hydroxides

We show that (27)Al triple-quantum magic angle spinning (3Q-MAS) experiments alleviate the second-order quadrupolar broadening to reveal the structure-building units of nonequivalent aluminum octahedra in the most extensively studied aluminum hydroxides, namely, gibbsite, bayerite, and boehmite. Further, aided by ab initio calculations of the electric field gradient tensors, the 3Q-MAS/MAS results are shown to lead to the assignment of (27)Al isotropic resonances to the aluminum positions in their X-ray-determined structures. The present work paves the way for future studies on various structurally transformed materials derived from these basic aluminum hydroxides.     

Depletion and pair interactions of proteins in polymer solutions

We study the depletion, pair interaction, and phase behavioral characteristics of proteins in polymer solutions. We use a McMillan-Mayer-like approach [W. G. McMillan, Jr. and J. E. Mayer, J. Chem. Phys. 13, 276 (1945)] to suggest that the depletion characteristics should be studied at an effective polymer concentration which is a function of both the average polymer and the protein concentrations. In the protein limit, we show that the volume of the polymer depletion layers exceeds the size of the proteins, leading to effective polymer concentrations typically in the semidilute and concentrated regimes even when the average polymer concentrations are in the dilute regimes. We propose an approximate approach that accounts for the multibody depletion overlaps, and use an accurate numerical solution of polymer mean-field theory to address depletion characteristics in these regimes which are characterized by both the importance of polymer interactions as well as the curvature of the proteins relative to the correlation length of polymers. We show that the depletion characteristics of the protein-polymer mixture can be quite different when viewed in this framework, and this can have profound consequences for the phase behavior of the mixture. Our theoretical predictions for the phase diagram match semiquantitatively with published experimental results.     

Crystal Structures and Packing of 2,4,6-tris(4-Fluorophenoxy)-1,3,5-triazine and 2,4,6-tris(3,4-Dimethylphenoxy)-1,3,5-triazine. New Materials for Octupolar Nonlinear Optics

The crystal structures and packing of 2,4,6-tris(4-fluorophenoxy)-1,3,5-triazine and 2,4,6-tris(3,4-dimethylphenoxy)-1,3,5-triazine are discussed. These structures have been determined as a continuation of a series of octupolar NLO materials we have been investigating. The crystal structures are characterized by C–H...F and C–H... hydrogen bonds, respectively. A characteristic of these triazine structures is the presence of dimeric Piedfort Units (PU) that are extended into more elaborate two-dimensional (2-D) networks. The structure of the fluoro derivative is compared with that of the corresponding unsubstituted and chloro/bromo-substituted derivatives. The structure of the dimethyl triazine is compared with that of the corresponding 4-methyl derivative. The noncentrosymmetric nature of the dimethyl derivative was confirmed by a powder SHG signal at 1.064 m of the order of 0.5 × KDP. Interestingly, the dimethyl derivative studied here is isostructural with the corresponding 4-methyl triazine. This H/Me isostructurality is shown to be an uncommon phenomenon by an analysis with the CSD.     

Studies on the reaction of unsymmetrical trifluoromethyl 1,2-phenylenediamine with various ketones leading to novel fluorinated heterocycles

Unsymmetrical 1,2-phenylenediamine on reaction with various ketones resulted in a number of fluorinated heterocycles such as benzimidazoles, quinoxalines and spiro benzimidazoles in high yields. The role of substituents in diamine in its reaction with various ketones and on the nature of product formation has been studied in detail.     

Evaluation of protein, peptide, and amino acid retention on C5 hydride-based stationary phases

A pentyl (C5) stationary phase is synthesized on a hydride surface using both 100 and 300 A silica particles. The aqueous normal phase properties of these new phases are evaluated using amino acids at high concentrations of ACN in the mobile phase containing 0.1% formic acid. The RP properties of the materials are tested by using peptides and proteins in order to evaluate the effect of pore size on retention. Comparisons of retention as well as efficiencies and peak symmetry with a C18 phase are used to determine the influence of the hydrophobic properties of the bonded material. An evaporative light scattering detector (ELSD) is used in all phases of the study and compared with UV detector for gradient elution of proteins.     

(t-Bu)2PN=P(i-BuNCH2CH2)3N: new efficient ligand for palladium-catalyzed C-N couplings of aryl and heteroaryl bromides and chlorides and for vinyl bromides at room temperature

By employing Pd(OAc)2, Cs2CO3, or NaOH, and the new ligand (t-Bu)2PN=P(i-BuNCH2CH2)3N (3a), an electronically diverse array of aryl bromides and chlorides possessing base-sensitive substituents (nitro, ester, and keto) provide coupling products with bulky aryl amines in good to excellent yields. Aryl halides possessing other functional groups including cyano, amino, trifluoromethyl, and phenol, coupled with equal ease, producing highly functionalized amines in good to excellent yields. Moreover, an aryl chloro group can be preserved in the presence of a bromo substituent under our reaction conditions. BOC-protected amines also participated efficiently. Heterocyclic bromides and chlorides underwent clean couplings with amines in excellent yields. An important strength of our protocol is the use of lower palladium loadings than those reported earlier, without compromising yields. The air-stable palladium complex (eta3-cinnamyl)PdCl.(3a) (5) was also employed successfully in C-N coupling reactions while the crotyl analogue was less efficacious. The 3a/Pd(OAc)2 catalyst system promotes, for the first time, efficient coupling of vinyl bromides with a variety of amines to produce imines and enamines at room temperature.     

How large delays build up in a GI/G/1 queue

LetW _k denote the waiting time of customerk, k 0, in an initially empty GI/G/1 queue. Fixa> 0. We prove weak limit theorems describing the behaviour ofW _k /n, 0kn, given W_n >na. LetX have the distribution of the difference between the service and interarrival distributions. We consider queues for which Cramer type conditions hold forX, and queues for whichX has regularly varying positive tail.The results can also be interpreted as conditional limit theorems, conditional on large maxima in the partial sums of random walks with negative drift.Research supported by the NSF under Grant NCR 8710840 and under the PYI Award NCR 8857731.