Nilpotent Groups

This article describes the concept of the nilpotent group and some properties of the nilpotent groups.Li Dailu - Qingdao University of Science and Technology, ChinaLiang Xiquan - Qingdao University of Science and Technology, ChinaMen Yanhong - Qingdao University of Science and Technology, ChinaGrzegorz Bancerek. The fundamental properties of natural numbers. Formalized Mathematics, 1(1):41-46, 1990.Grzegorz Bancerek and Krzysztof Hryniewiecki. Segments of natural numbers and finite sequences. Formalized Mathematics, 1(1):107-114, 1990.Czesław Byliński. Functions and their basic properties. Formalized Mathematics, 1(1):55-65, 1990.Czesław Byliński. Functions from a set to a set. Formalized Mathematics, 1(1):153-164, 1990.Wojciech A. Trybulec. Classes of conjugation. Normal subgroups. Formalized Mathematics, 1(5):955-962, 1990.Wojciech A. Trybulec. Groups. Formalized Mathematics, 1(5):821-827, 1990.Wojciech A. Trybulec. Subgroup and cosets of subgroups. Formalized Mathematics, 1(5):855-864, 1990.Wojciech A. Trybulec. Commutator and center of a group. Formalized Mathematics, 2(4):461-466, 1991.Wojciech A. Trybulec and Michał J. Trybulec. Homomorphisms and isomorphisms of groups. Quotient group. Formalized Mathematics, 2(4):573-578, 1991.Katarzyna Zawadzka. Solvable groups. Formalized Mathematics, 5(1):145-147, 1996

Basic Properties of Periodic Functions

In this article we present definitions, basic properties and some examples of periodic functions according to [5].Li Bo - Qingdao University of Science and Technology, ChinaLi Dailu - Qingdao University of Science and Technology, ChinaMen Yanhong - Qingdao University of Science and Technology, ChinaLiang Xiquan - Qingdao University of Science and Technology, ChinaGrzegorz Bancerek. The fundamental properties of natural numbers. Formalized Mathematics, 1(1):41-46, 1990.Grzegorz Bancerek. The ordinal numbers. Formalized Mathematics, 1(1):91-96, 1990.Czesław Byliński. Functions and their basic properties. Formalized Mathematics, 1(1):55-65, 1990.Czesław Byliński. Some basic properties of sets. Formalized Mathematics, 1(1):47-53, 1990.Chuanzhang Chen. Mathematical Analysis. Higher Education Press, Beijing, 1978.Krzysztof Hryniewiecki. Basic properties of real numbers. Formalized Mathematics, 1(1):35-40, 1990.Andrzej Trybulec. Binary operations applied to functions. Formalized Mathematics, 1(2):329-334, 1990.Zinaida Trybulec. Properties of subsets. Formalized Mathematics, 1(1):67-71, 1990.Peng Wang and Bo Li. Several differentiation formulas of special functions. Part V. Formalized Mathematics, 15(3):73-79, 2007, doi:10.2478/v10037-007-0009-4.Edmund Woronowicz. Relations and their basic properties. Formalized Mathematics, 1(1):73-83, 1990.Yuguang Yang and Yasunari Shidama. Trigonometric functions and existence of circle ratio. Formalized Mathematics, 7(2):255-263, 1998

Innovation through urban diversity and achieving comprehensive sustainable urbanism from a community-oriented approach

In current urban studies focused on pathways of sustainable urbanism, the main discourse is on the development of all-inclusive approaches or comprehensive methods that could encompass multiple factors of sustainability. As a result, a bigger emphasis has been given to other dimensions of “cultural” or “governance” (also known as institutional). In more recent years, more scholarly research studies refer to governance as the fourth pillar of sustainability, and more research studies analyse the multi-dimensional methods of socio-cultural, socio-economic, etc. These factors all have major impacts on the social, environmental, and economic aspects of the city. As a result, this research study aims to test the notion of “sustainable urbanism” from a behavioural perspective, which includes market development behavior and political economy (or mixed economy). This study promotes urban diversity from the multiplicity of diversity across the physical environment, social and economic systems, cultural attributes, people backgrounds, professions and sectors, etc. This then leads to discussions on innovation through urban diversity. Hence, it is important to evaluate how diversity promotes sustainable urbanism through specific economic and social systems in contemporary city development scenarios. The findings from this study feed into matters of urban diversity and urban innovation, and towards pathways of sustainable urbanism

GPU-based ultra-fast direct aperture optimization for online adaptive radiation therapy

Online adaptive radiation therapy (ART) has great promise to significantly reduce normal tissue toxicity and/or improve tumor control through real-time treatment adaptations based on the current patient anatomy. However, the major technical obstacle for clinical realization of online ART, namely the inability to achieve real-time efficiency in treatment re-planning, has yet to be solved. To overcome this challenge, this paper presents our work on the implementation of an intensity modulated radiation therapy (IMRT) direct aperture optimization (DAO) algorithm on graphics processing unit (GPU) based on our previous work on CPU. We formulate the DAO problem as a large-scale convex programming problem, and use an exact method called column generation approach to deal with its extremely large dimensionality on GPU. Five 9-field prostate and five 5-field head-and-neck IMRT clinical cases with 5\times5 mm2 beamlet size and 2.5\times2.5\times2.5 mm3 voxel size were used to evaluate our algorithm on GPU. It takes only 0.7~2.5 seconds for our implementation to generate optimal treatment plans using 50 MLC apertures on an NVIDIA Tesla C1060 GPU card. Our work has therefore solved a major problem in developing ultra-fast (re-)planning technologies for online ART

External Field Effect on Electronic and Vibrational Properties of Carotenoids

Resonance Raman spectroscopy is one of the most popular molecular spectroscopy methods. Using this technique, carbon-carbon (CC) vibration can be well investigated and give much information on π-conjugated system. The CC vibration has a strong dependence on the π-electron band gap, which is able to be characterized by absorption spectroscopy. Electron-phonon coupling will strongly influence the Raman intensity and shows sensitivity to the surrounding environment. Through electron-phonon coupling, CC vibration can be modulated by the π-electron band gap. In this review, we present our absorption and Raman-scattering observations of all-trans-beta-carotene from different environment such as temperature, pressure, solvents, and phase transition. The relationship between π-electron band gap and CC vibration is studied. With the decreasing of the π-electron energy, the modulation on CC vibration and electron-phonon coupling is enhanced leading to spectral red-shift. This review has significant importance in understanding the resonance Raman scattering, structural properties of polyenes, and the physics behind it. The results are also valuable for the development of fine-quality photoelectric device