A Structural Entropy Measurement Principle of Propositional Formulas in Conjunctive Normal Form

The satisfiability (SAT) problem is a core problem in computer science. Existing studies have shown that most industrial SAT instances can be effectively solved by modern SAT solvers while random SAT instances cannot. It is believed that the structural characteristics of different SAT formula classes are the reasons behind this difference. In this paper, we study the structural properties of propositional formulas in conjunctive normal form (CNF) by the principle of structural entropy of formulas. First, we used structural entropy to measure the complex structure of a formula and found that the difficulty solving the formula is related to the structural entropy of the formula. The smaller the compressing information of a formula, the more difficult it is to solve the formula. Secondly, we proposed a λ-approximation strategy to approximate the structural entropy of large formulas. The experimental results showed that the proposed strategy can effectively approximate the structural entropy of the original formula and that the approximation ratio is more than 92%. Finally, we analyzed the structural properties of a formula in the solution process and found that a local search solver tends to select variables in different communities to perform the next round of searches during a search and that the structural entropy of a variable affects the probability of the variable being flipped. By using these conclusions, we also proposed an initial candidate solution generation strategy for a local search for SAT, and the experimental results showed that this strategy effectively improves the performance of the solvers CCAsat and Sparrow2011 when incorporated into these two solvers.     

QK乘子代数上的Corona定理和Wolff定理

本文给出对数K-Carleson测度的一个新特征,并以此为工具研究Q_K空间的乘子代数M(Q_K),给出乘子代数M(Q_K)的某些特征描述.利用对数K-Carleson测度及Q_K空间的一个新特征,建立乘子代数M(Q_K)上的Corona定理和Wolff定理.     

A Generalization of Lappan’s Theorem to Higher Dimensional Complex Projective Space

In this paper, the authors discuss a generalization of Lappan’s theorem to higher dimensional complex projective space and get the following result: Let f be a holomorphic mapping of ? into Pⁿ(C), and let H₁, · · · , H_q be hyperplanes in general position in Pⁿ(C).Assume that sup {(1 ? |z|²)f^?(z) : z ∈ q[ j=1 f^?1(H_j )o < ∞,if q ≥ 2n² + 3, then f is normal.     

Metamorphic InAs(Sb)/InGaAs/InAlAs nanoheterostructures grown on GaAs for efficient mid-IR emitters

High-efficiency semiconductor lasers and light-emitting diodes operating in the 3–5?μm mid-infrared (mid-IR) spectral range are currently of great demand for a wide variety of applications, in particular, gas sensing, noninvasive medical tests, IR spectroscopy etc. III-V compounds with a lattice constant of about 6.1?Å are traditionally used for this spectral range. The attractive idea to fabricate such emitters on GaAs substrates by using In(Ga,Al)As compounds is restricted by either the minimum operating wavelength of ～8?μm in case of pseudomorphic AlGaAs-based quantum cascade lasers or requires utilization of thick metamorphic In_xAl_1-xAs buffer layers (MBLs) playing a key role in reducing the density of threading dislocations (TDs) in an active region, which otherwise result in a strong decay of the quantum efficiency of such mid-IR emitters. In this review we present the results of careful investigations of employing the convex-graded In_xAl_1-xAs MBLs for fabrication by molecular beam epitaxy on GaAs (001) substrates of In(Ga,Al)As heterostructures with a combined type-II/type-I InSb/InAs/InGaAs quantum well (QW) for efficient mid-IR emitters (3–3.6?μm). The issues of strain relaxation, elastic stress balance, efficiency of radiative and non-radiative recombination at T?=?10–300?K are discussed in relation to molecular beam epitaxy (MBE) growth conditions and designs of the structures. A wide complex of techniques including in-situ reflection high-energy electron diffraction, atomic force microscopy (AFM), scanning and transmission electron microscopies, X-ray diffractometry, reciprocal space mapping, selective area electron diffraction, as well as photoluminescence (PL) and Fourier-transformed infrared spectroscopy was used to study in detail structural and optical properties of the metamorphic QW structures. Optimization of the growth conditions (the substrate temperature, the As₄/III ratio) and elastic strain profiles governed by variation of an inverse step in the In content profile between the MBL and the InAlAs virtual substrate results in decrease in the TD density (down to 3?×?10⁷ cm^?2), increase of the thickness of the low-TD-density near-surface MBL region to 250–300?nm, the extremely low surface roughness with the RMS value of 1.6–2.4?nm, measured by AFM, as well as rather high 3.5?μm-PL intensity at temperatures up to 300?K in such structures. The obtained results indicate that the metamorphic InSb/In(Ga,Al)As QW heterostructures of proper design, grown under the optimum MBE conditions, are very promising for fabricating the efficient mid-IR emitters on a GaAs platform.     

A review of recent advances in catalytic combustion of VOCs on perovskite-type catalysts

Along with the rapid development of industry, VOCs gradually move into the spotlight, and now become a kind of harmful environmental pollutants that cannot be overlooked. This paper introduces the hazards of VOCs and the common catalytic combustion catalysts, noble metal catalysts and non-noble metal catalysts, for the elimination of VOCs. Perovskite catalysts, as one of the non-noble catalysts, play an important role in the field of catalytic combustion in recent years. According to the classification of elements doping in perovskites, the research achievements in the past five years were analyzed and reviewed. In addition, this paper also analyzes and elaborates the reaction kinetics and QSAR/QSPR models for the introduction of structural properties and reaction mechanisms.     

Uniqueness in Harper’s vertex-isoperimetric theorem

For a set $A\subseteq Q_n={\left\{0,1\right\}}^n$ the $t$-neighbourhood of $A$ is $N^t\left(A\right)=\left\{x:d\left(x,A\right)\le t\right\}$, where $d$ denotes the usual graph distance on $Q_n$. Harper’s vertex-isoperimetric theorem states that among the subsets $A\subseteq Q_n$ of given size, the size of the $t$-neighbourhood is minimised when $A$ is taken to be an initial segment of the simplicial order. Aubrun and Szarek asked the following question: if $A\subseteq Q_n$ is a subset of given size for which the sizes of both $N^t\left(A\right)$ and $N^t\left(A^c\right)$ are minimal for all $t>0$, does it follow that $A$ is isomorphic to an initial segment of the simplicial order?Our aim is to give a counterexample. Surprisingly it turns out that there is no counterexample that is a Hamming ball, meaning a set that lies between two consecutive exact Hamming balls, i.e. a set $A$ with $B\left(x,r\right)\subseteq A\subseteq B\left(x,r+1\right)$ for some $x\in Q_n$. We go further to classify all the sets $A\subseteq Q_n$ for which the sizes of both $N^t\left(A\right)$ and $N^t\left(A^c\right)$ are minimal for all $t>0$ among the subsets of $Q_n$ of given size. We also prove that, perhaps surprisingly, if $A\subseteq Q_n$ for which the sizes of $N\left(A\right)$ and $N\left(A^c\right)$ are minimal among the subsets of $Q_n$ of given size, then the sizes of both $N^t\left(A\right)$ and $N^t\left(A^c\right)$ are also minimal for all $t>0$ among the subsets of $Q_n$ of given size. Hence the same classification also holds when we only require $N\left(A\right)$ and $N\left(A^c\right)$ to have minimal size among the subsets $A\subseteq Q_n$ of given size.