A periodic isoperimetric problem related to the unique games conjecture

We prove the endpoint case of a conjecture of Khot and Moshkovitz related to the unique games conjecture, less a small error. Let n ≥ 2. Suppose a subset Ω of n‐dimensional Euclidean space satisfies ?Ω = Ω^c and Ω + v = Ω^c (up to measure zero sets) for every standard basis vector . For any and for any q ≥ 1, let and let . For any x ∈ ?Ω, let N(x) denote the exterior normal vector at x such that ‖N(x)‖₂ = 1. Let . Our main result shows that B has the smallest Gaussian surface area among all such subsets Ω, less a small error: In particular, Standard arguments extend these results to a corresponding weak inequality for noise stability. Removing the factor 6 × 10^?9 would prove the endpoint case of the Khot‐Moshkovitz conjecture. Lastly, we prove a Euclidean analogue of the Khot and Moshkovitz conjecture. The full conjecture of Khot and Moshkovitz provides strong evidence for the truth of the unique games conjecture, a central conjecture in theoretical computer science that is closely related to the P versus NP problem. So, our results also provide evidence for the truth of the unique games conjecture. Nevertheless, this paper does not prove any case of the unique games conjecture.     

Ultrasonic assisted synthesis of nanocrystalline cellulose as support and reducing agent for Ag nanoparticles: green synthesis and novel effective nanocatalyst for degradation of organic dyes

In this study, nanocrystalline cellulose (NCC) prepared from microcrystalline cellulose using high‐intensity ultrasonication as mechanical method without any chemical treatment. The obtained NCC with around 30–50 nm diameters, utilized as support, reducing and stabilizing agent for in‐situ green and eco‐friendly synthesis of silver nanoparticles (Ag NPs). The catalytic activity of composite was examined for degradation of environmental pollutants. The structure of as‐synthesized composite (Ag@NCC) was characterized by ultraviolet–visible spectroscopy (UV–vis), field emission scanning electron microscopy (FE‐SEM); Transmission electron microscopy (TEM); Energy dispersive spectroscopy (EDS), Fourier transform infrared spectroscopy (FT‐IR), X‐ray diffraction (XRD) and thermogravimetric analysis (TGA). The results of the catalytic reaction experiments showed that spherically shaped silver nanoparticles of around 20 nm distributed on the surface of nanocellulose demonstrated high catalytic efficiency towards the removal of methyl orange (MO) and 4‐nitrophenol (4‐NP).     

Cooling of Al-Cu-Fe-Cr-Ni high entropy alloy with different size

Cooling rate is critical in synthesizing nanoparticles (NPs), which determines the microstructure and the corresponding mechanical, thermal and electrical properties. This research, as the most initiative one, studies microstructure formation of three different sized high entropy alloy (HEA) NPs under three different cooling rates, employing molecular dynamics (MD). Through analysis of potential energy, “common neighbor analysis”, radial distribution function, and also the mean square displacement, it is found that phase transition temperature is independent of HEA NP size, and as cooling rate decrease, more amorphous atoms transform to fcc and hcp orders, which can alternate the mechanical and thermodynamic properties of the final structure. The Cr atoms are found to aggregate into one cluster inside the NP and also try to migrate to the surface of the HEA NP, due to the large diffusivity. This research provides new insights in the size dependency of the nanoparticles, which may motivate more applications in which the strong size dependency is not desirable.     

Chiral Separations in Normal‐phase Liquid Chromatography: Updating a Screening Strategy with a Chlorine‐containing Polysaccharide‐based Selector

The screening conditions of an existing column and mobile phase selection strategy for chiral compounds in normal‐phase high performance liquid chromatography (NP‐HPLC) were tested for their applicability on Chiralpak IC, which is a chiral stationary phase (CSP) made by immobilising cellulosic tris (3,5‐dichlorophenyl‐carbamate) on silica gel. In this study, the applicability of the optimization steps of the existing separation strategy was examined using 36 compounds representing the three possible resolution situations that occur after screening. The cumulative number of separated compounds is 27 (75.0 %), and the cumulative number of baseline separated compounds is 19 (52.8 %).     

Optimal media planning for multi-products in segmented market

Promotion is an essential stage of any product life cycle. All firms either concerned with single or multiple products promote their products. Advertising is the most dominated form of promotion for most of the products. Our research focuses on firms which produces a multiple range of products and advertise them using available media options. The budget constraint forms the major restriction. The existing research in media allocation of multiple products ignores the market segmentation and formulates the media plan assuming all segments alike. In this paper we formulate a media planning problem for allocating the available budget in multiple media that are found suitable for the advertising of multiple products considering marketing segmentation aspect of advertising. The objective is to maximize the total reach in all the segments. Goal programming technique is used to solve the problem. A case study is presented to show the real life application of the model.     

Enantioresolution of 2-methoxy-2-(1-naphtyl)propionic acid using diastereomeric salt formation with chiral phenylethylamine

An enantioresolution of 2-methoxy-2-(1-naphtyl)propionic acid (MαNP acid) using the diastereomeric salt with chiral (R)-phenylethylamine was achieved to give enantiopure (R)-MαNP acid in 29% yield with >99% ee based on rac-MαNP acid. X-ray crystallographic analysis of diastereomeric salt revealed that (R)-MαNP acid was tightly arranged by four independent hydrogen bonds and one CH–π interaction with (R)-phenylethylamine.     

On deciding the non‐emptiness of 2SAT polytopes with respect to First Order Queries

This paper is concerned with techniques for identifying simple and quantified lattice points in 2SAT polytopes. 2SAT polytopes generalize the polyhedra corresponding to Boolean 2SAT formulas, Vertex‐Packing (Covering, Partitioning) and Network flow problems; they find wide application in the domains of Program verification (Software Engineering) and State‐Space search (Artificial Intelligence). Our techniques are based on the symbolic elimination strategy called the Fourier‐Motzkin elimination procedure and thus have the advantages of being extremely simple (from an implementational perspective) and incremental. We also provide a characterization of a 2SAT polytope in terms of its extreme points and derive some interesting hardness results for associated optimization problems. Finally, we provide a brief discussion on the maximum size of a subdeterminant of the linear system representing a 2SAT polytope; this parameter plays a vital role in deriving analytical bounds on the size of the search space for checking whether the polyhedron includes a lattice point. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

求解课程表问题的分支定界算法

本通过对中学排课程表问题的特征分析，给出了基于分支定界法的优化算法，数值试验表明这是解决一般编排中学课程表问题的有效算法。     

工时依赖于开工时间的单台机排序问题

此文考察工时依赖于开工时间的排序问题．文章证明：(1)即使准奋时间完全相同，判断工时与开工时间相关的单台机排序问题是否有可行解也是NP完全的．(2)即使只存在两个不同的截止期，判断工时与开工时间相关的单台机排序问题是否有可行解仍然是NP完全的．(3)当所有工件有相同截止期时，问题是否有可行解可在多项式时间内判定．