A Unified Approach to the Proportional Relation for Discrete-Time Single-Server Queues

It is well known that a simple relation called proportional relation holds for some queueing models, that is, the stationary queue length distribution of one system can be expressed as the product of a constant and the distribution of another system which is different only in the buffer capacity. Recently, the proportional relation has been verified for various discrete-time single-server queues with correlated arrivals, where it has been also shown that the proportional constant can be expressed in terms of the distribution of one system. This implies that the stationary queue length distribution of one system can be completely expressed in terms of the distribution of the other system. In this paper, we consider a generalized model of discrete-time single-server queue, which covers all previous ones, and give a simple and unified proof to the proportional relation as well as the expression of the proportional constant.     

Quantum scattering in one-dimensional systems satisfying the minimal length uncertainty relation

In quantum gravity theories, when the scattering energy is comparable to the Planck energy the Heisenberg uncertainty principle breaks down and is replaced by the minimal length uncertainty relation. In this paper, the consequences of the minimal length uncertainty relation on one-dimensional quantum scattering are studied using an approach involving a recently proposed second-order differential equation. An exact analytical expression for the tunneling probability through a locally-periodic rectangular potential barrier system is obtained. Results show that the existence of a non-zero minimal length uncertainty tends to shift the resonant tunneling energies to the positive direction. Scattering through a locally-periodic potential composed of double-rectangular potential barriers shows that the first band of resonant tunneling energies widens for minimal length cases when the double-rectangular potential barrier is symmetric but narrows down when the double-rectangular potential barrier is asymmetric. A numerical solution which exploits the use of Wronskians is used to calculate the transmission probabilities through the Pöschl–Teller well, Gaussian barrier, and double-Gaussian barrier. Results show that the probability of passage through the Pöschl–Teller well and Gaussian barrier is smaller in the minimal length cases compared to the non-minimal length case. For the double-Gaussian barrier, the probability of passage for energies that are more positive than the resonant tunneling energy is larger in the minimal length cases compared to the non-minimal length case. The approach is exact and applicable to many types of scattering potential.     

Interactions of Puerarin with Micelles: p<Emphasis Type="Italic">K</Emphasis><Subscript>a</Subscript> Shifts and Thermodynamics

In this paper, the effect of varying hydrophobic chain lengths on the acid-base equilibrium of an isoflavone, puerarin, in cationic micelles has been studied. The interaction of puerarin with three types of micelles of identical positively charged head groups and varying tail lengths, viz., DTAB, TTAB and CTAB, shows that there is a shift in the apparent pK _a in the direction that favors the displacement of the acid-base equilibrium of puerarin towards the base form. The relative location depth of puerarin in micelles is determined according to their dielectric values (D) taking into account that lower D values correspond to deeper location. It is evident that the reduction of polarity favors the deprotonation of puerarin. The isothermal titration calorimetry measurements show that the effect of puerarin on the thermodynamic properties of micellization is enhanced in the order CTAB>TTAB>DTAB and the results illustrate that puerarin has more affinity for micelles with greater chain length.     

A SLOW RELAXATION MODE OF POLYMER CHAINS IN A SEMIDILUTE SOLUTION

Dust-free semidilute and concentrated polystyrene(PS)solutions in different solvents were prepared by slow evaporation and in situ anionic polymerization,which removes the effects of troublesome artifacts such as dust contamination and concentration gradient.The dynamics was reexamined by a combination of static and dynamic laser light scattering.In benzene and toluene(good solvents for PS),only one fast diffusive mode of polymer chains can be observed when the concentration(c)is up to 20%,which is attri...     

Chain Length Distributions of Polyolefins Made in Stopped‐Flow Reactors for Non‐Instantaneous Site Activation

We developed an analytical solution to describe how the chain length distribution (CLD) of polymers made with coordination polymerization catalysts vary as a function of time for very short polymerizations considering non‐instantaneous site activation. This solution is an extension of our previous analytical expression for instantaneous site activation. We validated the analytical solution with dynamic Monte Carlo simulation and obtained excellent agreement. Simulation results indicate that, unless the catalyst activation rate is much lower than the propagation rate, it will have only a minor effect on the initial shape of the CLD of polymers made in stopped‐flow reactors (SFR). We also show how incorrect polymerization kinetic parameters may be estimated when assuming instantaneous site activation when this hypothesis is not applicable to the polymerization data under investigation.

     

Halving circular arcs in normed planes

Studying the relation between the length of a chord of the unit circle and the length of the arc corresponding to it, some new characterizations of the Euclidean plane among all normed planes are obtained. All these results yield characterizations of inner product spaces in higher dimensions. Supported by the National Natural Science Foundation of China, grant number 10671048.     

Generalized Pleijel identity

A family of identities primarily associated with isoperimetric inequalities for planar convex domains was discovered by Pleijel in 1956. We call these identities classical Pleijel identities. R. V. Ambartzumian gave combinatorial proof of these identities and pointed out that they can be applied to find chord length distribution functions for convex domains. In the classical Pleijel identities integration is over the measure in the space \(\mathbb{G}\) of lines which is invariant with respect to the all Euclidean motions. In the present paper they are considered for any locally-finite measure in the space \(\mathbb{G}\). These identities are applied to find the so-called orientation-dependent chord length distribution (or density) functions for bounded convex domains.     

Controlled Genetic Encoding of Unnatural Amino Acids in a Protein Nanopore

Conventional protein engineering methods for modifying protein nanopores are typically limited to 20 natural amino acids, which restrict the diversity of the nanopores in structure and function. To enrich the chemical environment inside the nanopore, we employed the genetic code expansion (GCE) technique to site-specifically incorporate the unnatural amino acid (UAA) into the sensing region of aerolysin nanopores. This approach leveraged the efficient pyrrolysine-based aminoacyl-tRNA synthetase-tRNA pair for a high yield of pore-forming protein. Both molecular dynamics (MD) simulations and single-molecule sensing experiments demonstrated that the conformation of UAA residues provided a favorable geometric orientation for the interactions of target molecules and the pore. This rationally designed chemical environment enabled the direct discrimination of multiple peptides containing hydrophobic amino acids. Our work provides a new framework for endowing nanopores with unique sensing properties that are difficult to achieve using classical protein engineering approaches.     

Real-Time In Situ Volatile Organic Compound Sensing by a Dual-Emissive Polynuclear Ln-MOF with Pronounced LnIII Luminescence Response

Lanthanide metal–organic frameworks (Ln-MOFs) are promising for luminescence detection of volatile organic compound (VOC) vapors, but usually suffer from the silent or quenched Ln³⁺ emission. Herein, we report a new dual-emissive Eu-MOF composed of the coordinatively unsaturated Eu₉ clusters that afford abundant open metal sites to form a confined “binding pocket” to facilitate the preconcentration and recognition of VOCs. Single-crystal structural analyses reveal that specific analytes can replace the OH oscillators in the first coordination sphere of Eu³⁺ and form a unique hydrogen-bonding second-sphere adduct tying adjacent Eu₉ clusters together to minimize their nonradiative vibrational decay. With the promoted Eu³⁺ luminescence, the MOF realizes real-time in situ visual sensing of THF vapor (<1 s) and shows a quantitative ratiometric response to the vapor pressure with a limit of detection down to 17.33 Pa. Also, it represents a top-performing ratiometric luminescent thermometer.     

Effects of Heavy Iodine Atoms and π-Expanded Conjugation on Charge Transfer Dynamics in Carboxylic Acid BODIPY Derivatives as Triplet Photosensitizers

Borondipyrromethene (BODIPY) chromophores are composed of a functional-COOH group at meso position with or without a biphenyl ring, and their compounds with heavy iodine atoms at −2, −6 positions of the BODIPY indacene core were synthesized. The photophysical properties of the compounds were studied with steady-state absorption and fluorescence measurements. It was observed that the absorption band is significantly red-shifted, and fluorescence signals are quenched in the presence of iodine atoms. In addition to that, it was indicated that the biphenyl ring does not affect the spectral shifting in the absorption as well as fluorescence spectra. In an attempt to investigate the effect of π-expanded biphenyl moieties and heavy iodine atoms on charge transfer dynamics, femtosecond transient absorption spectroscopy measurements were carried out in the environment of the tetrahydrofuran (THF) solution. Based on the performed ultrafast pump-probe spectroscopy, BODIPY compounds with iodine atoms lead to intersystem crossing (ISC) and ISC rates were determined as 150 ps and 180 ps for iodine BODIPY compounds with and without π-expanded biphenyl moieties, respectively. According to the theoretical results, the charge transfer in the investigated compounds mostly appears to be intrinsic local excitations, corresponding to high photoluminescence efficiency. These experimental findings are useful for the design and study of the fundamental photochemistry of organic triplet photosensitizers.