Sharing : a synergy of natural forces, existing urban conditions and human characteristics

Sharing is a synergy which produce a combine effort greater than their separate efforts. It based on the dependency of independent material(s), object(s) or structure(s), and can generate more possibilities. There always be one dominant, one compliant, one gives the direction, one follows the order. Sometimes the direction is selectional and compliance can be multiple. By following the characteristic of objects, with the superposition of selected orders, the result can be complex and diverse. When design complient to the natural forces and existing urban condition, then meeting different human characteristics, taking advantage of the restrictions, carefully making small changes that can lead to big differences should lead to the future of architecture

Classification of Symmetry-Protected Phases for Interacting Fermions in Two Dimensions

Recently, it has been shown that two-dimensional bosonic symmetry-protected topological(SPT) phases with on-site unitary symmetry $G$ can be completely classified by the group cohomology class $H^3(G, \mathrm{U}(1))$. Later, group super-cohomology class was proposed as a partial classification for SPT phases of interacting fermions. In this work, we revisit this problem based on the mathematical framework of $G$-extension of unitary braided tensor category(UBTC) theory. We first reproduce the partial classifications given by group super-cohomology, then we show that with an additional $H^1(G, \mathbb{Z}_2)$ structure, a complete classification of SPT phases for two-dimensional interacting fermion systems for a total symmetry group $G\times\mathbb{Z}_2^f$ can be achieved. We also discuss the classification of interacting fermionic SPT phases protected by time-reversal symmetry.Comment: references added; published versio

Exact and Approximate Range Mode Query Data Structures in Practice

We conduct an experimental study on the range mode problem. In the exact version of the problem, we preprocess an array A, such that given a query range [a, b], the most frequent element in A[a, b] can be found efficiently. For this problem, our most important finding is that the strategy of using succinct data structures to encode more precomputed information not only helped Chan et al. (Linear-space data structures for range mode query in arrays, Theory of Computing Systems, 2013) improve previous results in theory but also helps us achieve the best time/space tradeoff in practice; we even go a step further to replace more components in their solution with succinct data structures and improve the performance further. In the approximate version of this problem, a (1+?)-approximate range mode query looks for an element whose occurrences in A[a,b] is at least F_{a,b}/(1+?), where F_{a,b} is the frequency of the mode in A[a,b]. We implement all previous solutions to this problems and find that, even when ? = 1/2, the average approximation ratio of these solutions is close to 1 in practice, and they provide much faster query time than the best exact solution. These solutions achieve different useful time-space tradeoffs, and among them, El-Zein et al. (On Approximate Range Mode and Range Selection, 30th International Symposium on Algorithms and Computation, 2019) provide us with one solution whose space usage is only 35.6% to 93.8% of the cost of storing the input array of 32-bit integers (in most cases, the space cost is closer to the lower end, and the average space cost is 20.2 bits per symbol among all datasets). Its non-succinct version also stands out with query support at least several times faster than other O(n/?)-word structures while using only slightly more space in practice