Easy knapsacks and the complexity of energy allocation problems in the smart grid

Motivated by the growing interest in the smart grid and intelligent energy management mechanisms we study two classes of domestic energy allocation problems. In the first case we work with a system that is tasked with scheduling the work on a number of appliances over a given time window. In the second one a collection of air conditioning appliances is used to control the temperature of a given domestic environment. Our framework for this case includes a simplified mechanism for modelling the heat exchange between the interior and the exterior of the given environment. We present various polynomial time algorithms and NP-hardness proofs. In particular the main result of the paper is a proof that although it is NP-hard to schedule the operation of a single air-conditioning unit, working at various temperature levels in a variable energy price regime, there is a polynomial time algorithm for controlling one such device working at a single temperature level, for houses with low thermal inertia. The algorithm analysis hinges on the properties of a polynomial time variant of the minimisation version of the knapsack problem which may be of independent interest.     

A note on the nucleolus for 2-convex TU games

For 2-convex n-person cooperative TU games, the nucleolus is determined as some type of constrained equal award rule. Its proof is based on Maschler, Peleg, and Shapley’s geometrical characterization for the intersection of the prekernel with the core. Pairwise bargaining ranges within the core are required to be in equilibrium. This system of non-linear equations is solved and its unique solution agrees with the nucleolus.     

Exact categories

We survey the basics of homological algebra in exact categories in the sense of Quillen. All diagram lemmas are proved directly from the axioms, notably the five lemma, the 3×3$3\times 3$-lemma and the snake lemma. We briefly discuss exact functors, idempotent completion and weak idempotent completeness. We then show that it is possible to construct the derived category of an exact category without any embedding into abelian categories and we sketch Deligne's approach to derived functors. The construction of classical derived functors with values in an abelian category painlessly translates to exact categories, i.e., we give proofs of the comparison theorem for projective resolutions and the horseshoe lemma. After discussing some examples we elaborate on Thomason's proof of the Gabriel–Quillen embedding theorem in an appendix.     

Fast Algorithms for Nonparametric Curve Estimation

Abstract

Naive implementations of local polynomial fits and kernel estimators require almost O(n ²) operations. In this article two fast O(n) algorithms for nonparametric local polynomial fitting are presented. They are based on updating normal equations. Numerical stability is guaranteed by controlling ill-conditioned situations for small bandwidths and data-tuned restarting of the updating procedure. Restarting at every output point results in a moderately fast but highly stable O(n ^7/5) algorithm. Applicability of algorithms is evaluated for estimation of regression curves and their derivatives. The idea is also applied to kernel estimators of regression curves and densities.     

Intrinsic relaxation rates of polarized Cs vapor in miniaturized cells

The intrinsic relaxation rates of the vector magnetization of cesium vapor enclosed in microfabricated atomic magnetometer cells are investigated. Two methods—the optically detected magnetic resonance and the ground-state Hanle effect—are used to carry out automated measurements in dependence on cell temperature and nitrogen buffer gas pressure. The experimental results are compared with expected contributions of the different relaxation processes and in this way allow the discrimination between them to help further optimization of cell design. The methods are compared in terms of basic features, data quality, and practical applicability.     

Strong optical nonlinearities of self-assembled polymorphic microstructures of phenylethynyl functionalized fluorenones

The polymorphs of a phenylethynyl functionalized fluorenone derivative, and their controlled self-assembly for microstructures with different morphologies have been studied. These polymorphic microcrystals exhibit very distinctive NLO properties, which are highly correlated to their electronic and supramolecular structures.     

Fast Exfoliation and Functionalisation of Two‐Dimensional Crystalline Carbon Nitride by Framework Charging

Two‐dimensional (2D) layered graphitic carbon nitride (gCN) nanosheets offer intriguing electronic and chemical properties. However, the exfoliation and functionalisation of gCN for specific applications remain challenging. We report a scalable one‐pot reductive method to produce solutions of single‐ and few‐layer 2D gCN nanosheets with excellent stability in a high mass yield (35 %) from polytriazine imide. High‐resolution imaging confirmed the intact crystalline structure and identified an AB stacking for gCN layers. The charge allows deliberate organic functionalisation of dissolved gCN, providing a general route to adjust their properties.     

X-ray structure of 14-phenyldiquino[3,4 b;4′,3′-e][1,4]thiazine<Superscript>(1)</Superscript>

The title compound 1 (C₂₄H₁₅N₃S) crystallizes in an orthorhombic unit cell, with a = 8.2355(3), b = 19.1499(8), and c = 11.3520(5) Å, Z = 2, and space group Pnma. The pentacyclic ring system is folded mainly along the S–N(2) axis and slightly along the C(3)–C(4) and C(3′)–C(4′) axes. The butterfly folding angle between the two quinoline planes is 48.96(3)^∘ and the angle between the planes of the two halves of the thiazine ring is 43.46(7)^∘. The central thiazine ring is in a boat conformation. The phenyl substituent is unexpectedly in an axial location with respect to the thiazine ring with the C–N⋅ < eqid1 > ⋅S angle of 114.0(1)^∘ and it is perpendicular to the bisecting plane of the pentacyclic ring system.     

Crystal and molecular structure of sodium picrate monohydrate

Na⁺ [C₆H₂(NO₂)₃O]^– H₂O,M _r=269.11, monoclinic,C2/m (No. 12),a=13.065(1),b=20.065(2),c=3.6957(4) Å,=90.75(1)°,V=968.7(1) ų,Z=4,D _m=1.848 g cm^–3 (by flotation),D _x=1.845 g cm^–3, (CuK)=1.54184 Å,(CuK)=17.5 cm^–1,F(000)=540,T=295 K,R=0.055,wR=0.060 for 879 observed reflections withI2.5(I). p]In the structure the picrate anion has exact twofold symmetry, the water molecule has exact mirror symmetry and the two sodium cations occupy special positions on inversion centers at (0, 1/2, 0) and (1/2, 1/2, 1/2). The phenyl ring together with the four substituent N and O atoms is strictly planar, but the phenyl ring is severely distorted from hexagonal symmetry, as is commonly observed in structures containing the picrate fragment. Both sodium ions are eight-coordinated with Na(1) and Na(2) in a cubic and distorted cubic arrangement, respectively. Na(1) coordinates to four charged phenoxide O atoms and four O atoms of nitro groups, and Na(2) to four O atoms of nitro groups and four water molecules. The water molecule donates two hydrogen bonds to symmetry-related O(2) atoms ofortho nitro groups. Two symmetry-related O(3) atoms ofortho nitro groups serve as acceptors in a very unusual asymmetric bifurcated C-HO hydrogen bond.