A simple derivation of the MacWilliams identity for linear ordered codes and orthogonal arrays

In (Can J Math 51(2):326–346, 1999), Martin and Stinson provide a generalized MacWilliams identity for linear ordered orthogonal arrays and linear ordered codes (introduced by Rosenbloom and Tsfasman (Prob Inform Transm 33(1):45–52, 1997) as “codes for the m-metric”) using association schemes. We give an elementary proof of this generalized MacWilliams identity using group characters and use it to derive an explicit formula for the dual type distribution of a linear ordered code or orthogonal array.      

Optical and electrical characterization of TiO2 nanotube arrays on titanium substrate

Novel oriented aligned TiO₂ nanotube (TN) arrays were fabricated by anodizing titanium foil in 0.5% HF electrolyte solution. It is indicated that the sizes of the TNs greatly depended on the applied voltages to some extent. The electrical properties of the TN arrays were characterized by current-voltage (I-V) measurements. It exhibits a nonlinear, asymmetric I-V characterization, which can be explained that there exists an n-type semiconductor/metal Schottky barrier diode between TN arrays and titanium substrate interface. The absorption edges shift towards shorter wavelengths with the decrease of the anodizing voltages, which is attributed to the quantum size effects. At room temperature, a novel wide PL band consisting of four overlapped peaks was observed in the photoluminescence (PL) measurements of the TN arrays. Such peaks were proposed to be resulted from the direct transition X₁ → X₂/X₁, indirect transition Γ₁ → X₂/X₁, self-trapped excitons and oxygen vacancies, respectively.     

Frequency permutation arrays

Motivated by recent interest in permutation arrays, we introduce and investigate the more general concept of frequency permutation arrays (FPAs). An FPA of length n = mλ and distance d is a set T of multipermutations on a multiset of m symbols, each repeated with frequency λ, such that the Hamming distance between any distinct x,y ∈ T is at least d. Such arrays have potential applications in powerline communication. In this article, we establish basic properties of FPAs, and provide direct constructions for FPAs using a range of combinatorial objects, including polynomials over finite fields, combinatorial designs, and codes. We also provide recursive constructions, and give bounds for the maximum size of such arrays. © 2006 Wiley Periodicals, Inc. J Combin Designs 14: 463–478, 2006     

Interferometric array generation

We report the formation of square, rectangular as well as hexagonal arrays of small light spots in one single setup using Michelson and Mach–Zehnder interferometers in tandem. The geometry of arrays can be altered easily online, by changing the relative orientations of the mirrors. The arrays could be scanned over large longitudinal distances and could be compressed to give large spot density. The expression for the resultant intensity distribution for the arrays has been worked out and the computed pattern is compared with the experimental data.     

Atomic Latin squares of order eleven

A Latin square is pan‐Hamiltonian if the permutation which defines row i relative to row j consists of a single cycle for every i ≠ j. A Latin square is atomic if all of its conjugates are pan‐Hamiltonian. We give a complete enumeration of atomic squares for order 11, the smallest order for which there are examples distinct from the cyclic group. We find that there are seven main classes, including the three that were previously known. A perfect 1‐factorization of a graph is a decomposition of that graph into matchings such that the union of any two matchings is a Hamiltonian cycle. Each pan‐Hamiltonian Latin square of order n describes a perfect 1‐factorization of K_n,n, and vice versa. Perfect 1‐factorizations of K_n,n can be constructed from a perfect 1‐factorization of K_n+1. Six of the seven main classes of atomic squares of order 11 can be obtained in this way. For each atomic square of order 11, we find the largest set of Mutually Orthogonal Latin Squares (MOLS) involving that square. We discuss algorithms for counting orthogonal mates, and discover the number of orthogonal mates possessed by the cyclic squares of orders up to 11 and by Parker's famous turn‐square. We find that the number of atomic orthogonal mates possessed by a Latin square is not a main class invariant. We also define a new sort of Latin square, called a pairing square, which is mapped to its transpose by an involution acting on the symbols. We show that pairing squares are often orthogonal mates for symmetric Latin squares. Finally, we discover connections between our atomic squares and Franklin's diagonally cyclic self‐orthogonal squares, and we correct a theorem of Longyear which uses tactical representations to identify self‐orthogonal Latin squares in the same main class as a given Latin square. © 2003 Wiley Periodicals, Inc.     

Core–Shell NiO@Ni‐P Hybrid Nanosheet Array for Synergistically Enhanced Oxygen Evolution Electrocatalysis: Experimental and Theoretical Insights

Cost‐effective and highly‐efficient electrocatalysts for the oxygen evolution reaction are crucial for electrolytic hydrogen production. Here, we report core–shell NiO@Ni‐P nanosheet arrays as a high‐performance 3D catalyst for water oxidation electrocatalysis. Such nanoarrays demand overpotentials of 292 and 350 mV to drive geometrical catalytic current densities of 10 and 100 mA cm^?2, respectively, with an activity superior to its NiO and Ni‐P counterparts. Notably, this catalyst also shows a high long‐term electrochemical durability with a Faradaic efficiency of 98.1 %. Density functional theory calculation reveals that the superior activity benefits from the synergistic effect between NiO and Ni‐P.     

TiN纳米管的制备及其作为SERS基底的应用

采用阳极氧化法在钛片表面制备TiO₂纳米管阵列,并经氨气氮化得到TiN纳米管阵列,通过XRD、SEM、XPS、UV-Vis对基底的组成、形貌和光学性能进行了表征,并以罗丹明6G（R6G）为探针分子,对TiN基底的SERS活性进行检测。结果显示,TiO₂纳米管经800℃以上高温可转化为TiN纳米管,纳米管直径在100~200 nm,排列规整有序。氮化后的样品在500~600 nm处出现TiN的等离子体共振吸收峰,该基底显示出较高的SERS活性,对R6G分子的检测极限可达~10^-7 mol·L^-1,增强因子达~10⁵。     

Band gap modulation of transition-metal dichalcogenide MX2 nanosheets by in-plane strain

The electronic properties of quasi-two-dimensional honeycomb structures of MX₂ nanosheets (M=Mo, W and X=S, Se) subjected to in-plane biaxial strain have been investigated using first-principles calculations. We demonstrate that the band gap of MX₂ nanosheets can be widely tuned by applying tensile or compressive strain, and these ultrathin materials undergo a universal reversible semiconductor-metal transition at a critical strain. Compared to WX₂, MoX₂ need a smaller critical tensile strain for the band gap close, and MSe₂ need a smaller critical compressive strain than MS₂. Taking bilayer MoS₂ as an example, the variation of the band structures was studied and the semiconductor-metal transition involves a slightly different physical mechanism between tensile and compressive strain. The ability to tune the band gap of MX₂ nanosheets in a controlled fashion over a wide range of energy opens up the possibility for its usage in a range of application.