Short interval asymptotics for a class of arithmetic functions

Summary We provide a general asymptotic formula which permits applications to sums like <InlineEquation ID=IE"1"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"2"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"3"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"4"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"5"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"6"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"7"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"8"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"9"><EquationSource Format="TEX"><![CDATA[$]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>$]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation> \sum_{x< n\le x+y} \big(d(n)\big)^2, \quad \sum_{x< n\le x+y} d(n^3),\quad \sum_{x< n\le x+y}\big(r(n)\big)^2, \quad \sum_{x< n\le x+y}r(n^3), $$ where $d(n)$ and $r(n)$ are the usual arithmetic functions (number of divisors, sums of two squares), and $y$ is small compared to~$x$.     

Load balancing via random local search in closed and open systems

In this paper, we analyze the performance of random load resampling and migration strategies in parallel server systems. Clients initially attach themselves to an arbitrary server, but may switch servers independently at random instants of time in an attempt to improve their service rate. This approach to load balancing contrasts with traditional approaches where clients make smart server selections upon arrival (e.g., Join-the-Shortest-Queue policy and variants thereof). Load resampling is particularly relevant in scenarios where clients cannot predict the load of a server before being actually attached to it. An important example is in wireless spectrum sharing where clients try to share a set of frequency bands in a distributed manner. We first analyze the natural Random Local Search (RLS) strategy. Under this strategy, after sampling a new server randomly, clients only switch to it if their service rate is improved. In closed systems, where the client population is fixed, we derive tight estimates of the time it takes under RLS strategy to balance the load across servers. We then study open systems where clients arrive according to a random process and leave the system upon service completion. In this scenario, we analyze how client migrations within the system interact with the system dynamics induced by client arrivals and departures. We compare the load-aware RLS strategy to a load-oblivious strategy in which clients just randomly switch server without accounting for the server loads. Surprisingly, we show that both load-oblivious and load-aware strategies stabilize the system whenever this is at all possible. We use large-system asymptotics to characterize system performance, and augment this with simulations, which suggest that the average client sojourn time under the load-oblivious strategy is not considerably reduced when clients apply smarter load-aware strategies.     

Q-superlinear convergence of the iterates in primal-dual interior-point methods

Sufficient conditions are given for the Q-superlinear convergence of the iterates produced by primal-dual interior-point methods for linear complementarity problems. It is shown that those conditions are satisfied by several well known interior-point methods. In particular it is shown that the iteration sequences produced by the simplified predictor–corrector method of Gonzaga and Tapia, the simplified largest step method of Gonzaga and Bonnans, the LPF+ algorithm of Wright, the higher order methods of Wright and Zhang, Potra and Sheng, and Stoer, Wechs and Mizuno are Q-superlinearly convergent. Received: February 9, 2000 / Accepted: February 20, 2001?Published online May 3, 2001     

Diophantine equations with products of consecutive members of binary recurrences

We prove a finiteness result for the number of solutions of a Diophantine equation of the form \(u_n u_{n+1}\cdots u_{n+k}\pm 1 =\pm u_m^2\), where \(\{ u_n\}_{n\ge 1}\) is a binary recurrent sequence whose characteristic equation has roots which are real quadratic units.     

Changes in dynamics of SRE-RNA on binding to the VTS1p-SAM domain studied by 13C NMR relaxation

RNA recognition by proteins is often accompanied by significant changes in RNA dynamics in addition to conformational changes. However, there are very few studies which characterize the changes in molecular motions in RNA on protein binding. We present a quantitative (13)C NMR relaxation study of the changes in RNA dynamics in the pico-nanosecond time scale and micro-millisecond time scale resulting from interaction of the stem-loop SRE-RNA with the VTS1p-SAM domain. (13)C relaxation rates of the protonated carbons of the nucleotide base and anomeric carbons have been analyzed by employing the model-free formalism, for a fully (13)C/(15)N-labeled sample of the SRE-RNA in the free and protein-bound forms. In the free RNA, the nature of molecular motions are found to be distinctly different in the stem and the loop region. On binding to the protein, the nature of motions becomes more homogeneous throughout the RNA, with many residues showing increased flexibility at the aromatic carbon sites, while the anomeric carbon sites become more rigid. Surprisingly, we also observe indications of a slow collective motion of the RNA in the binding pocket of the protein. The observation of increased motions on binding is interesting in the context of growing evidence that binding does not always lead to motional restrictions and the resulting entropy gain could favor the free energy of association.     

Cover Picture: Supramolecular Self-Assembly as a Tool To Preserve the Electronic Purity of Perylene Diimide Chromophores (Angew. Chem. Int. Ed. 12/2023)

Organic semiconductors are promising for efficient, printable optoelectronics. However, strong excited-state quenching due to uncontrolled aggregation limits their use in devices. We report on the self-assembly of a supramolecular pseudo-cube formed from six perylene diimides (PDIs). The rigid, shape-persistent cage sets the distance and orientation of the PDIs and suppresses intramolecular rotations and vibrations, leading to non-aggregated, monomer-like properties in solution and the solid state, in contrast to the fast fluorescence quenching in the free ligand. The stabilized excited state and electronic purity in the cage enables the observation of delayed fluorescence due to a bright excited multimer, acting as excited-state reservoir in a rare case of benign inter-chromophore interactions in the cage. We show that self-assembly provides a powerful tool for retaining and controlling the electronic properties of chromophores, and to bring molecular electronics devices within reach.     

Visible-Light-Induced Control over Reversible Single-Chain Nanoparticle Folding

We introduce a class of single-chain nanoparticles (SCNPs) that respond to visible light (λ_max=415 nm) with complete unfolding from their compact structure into linear chain analogues. The initial folding is achieved by a simple esterification reaction of the polymer backbone constituted of acrylic acid and polyethylene glycol carrying monomer units, introducing bimane moieties, which allow for the photochemical unfolding, reversing the ester-bond formation. The compaction and the light driven unfolding proceed cleanly and are readily followed by size exclusion chromatography (SEC) and diffusion ordered NMR spectroscopy (DOSY), monitoring the change in the hydrodynamic radius (R_H). Importantly, the folding reaction and the light-induced unfolding are reversible, supported by the high conversion of the photo cleavage. As the unfolding reaction occurs in aqueous systems, the system holds promise for controlling the unfolding of SCNPs in biological environments.     

Molecular dynamics simulations of heptyl phosphonic acid: a potential polymer component for fuel cell polymer membrane

Atomistic molecular dynamics simulations have been performed on heptyl phosphonic acid (HPA) to understand the dynamic hydrogen bonding network in the liquid phase. HPA is a phosphonic-acid functionalized alkane (heptane) and a model compound for one of the promising polymers for high temperature (>100 degrees C) fuel cell polymer electrolyte membranes. For the simulation, a force field for this molecule has been generated with the help of quantum chemical calculations and optimized by simplex algorithm. The force field has been validated against experimentally measured properties, for example, density and self-diffusion constant. From molecular dynamics simulations conducted at different temperatures, we have confirmed the hypothesis of dynamic hydrogen bond network formation in this material.