Any monotone property of 3-uniform hypergraphs is weakly evasive

© 2014 Elsevier B.V. For a Boolean function f, let D(f) denote its deterministic decision tree complexity, i.e., minimum number of (adaptive) queries required in worst case in order to determine f. In a classic paper, Rivest and Vuillemin [11] show that any non-constant monotone property P:{0,1}(n2)→{0,1} of n-vertex graphs has D(P)=Ω(n2).We extend their result to 3-uniform hypergraphs. In particular, we show that any non-constant monotone property P:{0,1}(n3)→{0,1} of n-vertex 3-uniform hypergraphs has D(P)=Ω(n3).Our proof combines the combinatorial approach of Rivest and Vuillemin with the topological approach of Kahn, Saks, and Sturtevant [6]. Interestingly, our proof makes use of Vinogradov's Theorem (weak Goldbach Conjecture), inspired by its recent use by Babai et al. [1] in the context of the topological approach. Our work leaves the generalization to k-uniform hypergraphs as an intriguing open question

Networked fairness in cake cutting

We introduce a graphical framework for fair division in cake cutting, where comparisons between agents are limited by an underlying network structure. We generalize the classical fairness notions of envy-freeness and proportionality to this graphical setting. Given a simple undirected graph G, an allocation is envy-free on G if no agent envies any of her neighbor's share, and is proportional on G if every agent values her own share no less than the average among her neighbors, with respect to her own measure. These generalizations open new research directions in developing simple and efficient algorithms that can produce fair allocations under specific graph structures. On the algorithmic frontier, we first propose a moving-knife algorithm that outputs an envy-free allocation on trees. The algorithm is significantly simpler than the discrete and bounded envy-free algorithm recently designed in [Aziz and Mackenzie, 2016a] for complete graphs. Next, we give a discrete and bounded algorithm for computing a proportional allocation on descendant graphs, a class of graphs by taking a rooted tree and connecting all its ancestor-descendant pairs

Trapezoidal Current Modulation for Bidirectional High-Step-Ratio Modular DC–DC Converters

Modular dc-dc converter (MDCC) has been proposed for high step-ratio interconnection in dc grid applications. To further optimize the performance of MDCC, this paper presents a trapezoidal current modulation with bidirectional power flow ability. By giving all the sub-module (SM) capacitors an equal duty to withstand the stack dc voltage, their voltages are balanced without additional feedback control. Moreover, based on soft-switching performance and circulating current analysis, three-level and two-level operation modes featured with high efficiency conversion and large power transmission, respectively, are introduced. The control schemes of both modes are designed to minimize the conduction losses. Besides, the SM capacitor voltage ripples with different switching patterns are compared and the option for ripple minimization is presented. A full-scale case study is provided to introduce the design process and device selection of the MDCC. The experimental tests based on a downscaled prototype are finally presented to validate the theoretical analysis

Study on structural optimization of submerged nozzle of continuous casting mold by water model experiment

In this paper, by changing the shape of the inner wall of the nozzle, a water model experiment was carried out on the slab with the section of 200 / mm × 1 200 / mm. The results show that under the same immersion depth and the same angle, the improved No.1 nozzle and the improved No.2 nozzle have swirling flow, which reduced the dead region and the internal longitudinal flow velocity, and the swirl ratio generated in improved No. 2 nozzle is more obvious than improved No. 1 nozzle. The impact depth of the velocity at the outlet of the improved No.2 nozzle is obviously lower than that of the original nozzle and the improved No.1 nozzle

Tripartite-to-Bipartite Entanglement Transformation by Stochastic Local Operations and Classical Communication and the Structure of Matrix Spaces

© 2018, Springer-Verlag GmbH Germany, part of Springer Nature. We study the problem of transforming a tripartite pure state to a bipartite one using stochastic local operations and classical communication (SLOCC). It is known that the tripartite-to-bipartite SLOCC convertibility is characterized by the maximal Schmidt rank of the given tripartite state, i.e. the largest Schmidt rank over those bipartite states lying in the support of the reduced density operator. In this paper, we further study this problem and exhibit novel results in both multi-copy and asymptotic settings, utilizing powerful results from the structure of matrix spaces. In the multi-copy regime, we observe that the maximal Schmidt rank is strictly super-multiplicative, i.e. the maximal Schmidt rank of the tensor product of two tripartite pure states can be strictly larger than the product of their maximal Schmidt ranks. We then provide a full characterization of those tripartite states whose maximal Schmidt rank is strictly super-multiplicative when taking tensor product with itself. Notice that such tripartite states admit strict advantages in tripartite-to-bipartite SLOCC transformation when multiple copies are provided. In the asymptotic setting, we focus on determining the tripartite-to-bipartite SLOCC entanglement transformation rate. Computing this rate turns out to be equivalent to computing the asymptotic maximal Schmidt rank of the tripartite state, defined as the regularization of its maximal Schmidt rank. Despite the difficulty caused by the super-multiplicative property, we provide explicit formulas for evaluating the asymptotic maximal Schmidt ranks of two important families of tripartite pure states by resorting to certain results of the structure of matrix spaces, including the study of matrix semi-invariants. These formulas turn out to be powerful enough to give a sufficient and necessary condition to determine whether a given tripartite pure state can be transformed to the bipartite maximally entangled state under SLOCC, in the asymptotic setting. Applying the recent progress on the non-commutative rank problem, we can verify this condition in deterministic polynomial time

Analysis and Criterion for Inherent Balance Capability in Modular Multilevel DC–AC–DC Converters

Modular multilevel dc-ac-dc converters (MMDAC) have emergedrecently for high step-ratio connectionsin medium voltage distribution systems.Extended phase-shiftmodulation has been proposed and was found to create the opportunity for inherent balance of SM capacitor voltages. This letter presents fundamentalanalysis leading toclear criterion for the inherent balancecapability in MMDAC. A sufficient and necessary condition,with associated assumptions,to guarantee this capability isestablished. Using the mathematics of circulant matrices, this condition is simplified to a co-prime criterion which gives rise to practical guidance for the design of an MMDAC. Experimentson down-scaled prototypesand simulations on full-scale examples both provide verification of the analysis and criterion for the inherent balance capability of MMDAC