86,117 research outputs found
The number of independent sets in a graph with small maximum degree
Let be the number of independent sets in a graph . We show
that if has maximum degree at most then
(where is vertex degree, is the number of isolated
vertices in and is the complete bipartite graph with vertices
in one partition class and in the other), with equality if and only if each
connected component of is either a complete bipartite graph or a single
vertex. This bound (for all ) was conjectured by Kahn.
A corollary of our result is that if is -regular with then with
equality if and only if is a disjoint union of copies of
. This bound (for all ) was conjectured by Alon and Kahn and
recently proved for all by the second author, without the characterization
of the extreme cases.
Our proof involves a reduction to a finite search. For graphs with maximum
degree at most the search could be done by hand, but for the case of
maximum degree or , a computer is needed.Comment: Article will appear in {\em Graphs and Combinatorics
Number of states for nucleons in a single- shell
In this paper we obtain number of states with a given spin and a given
isospin for systems with three and four nucleons in a single- orbit, by
using sum rules of six- and nine- symbols obtained in earlier works.Comment: to be published in Physical Review
Wood-Inspired Morphologically Tunable Aligned Hydrogel for High-Performance Flexible All-Solid-State Supercapacitors
Oriented microstructures are widely found in various biological systems for multiple functions. Such anisotropic structures provide low tortuosity and sufficient surface area, desirable for the design of high-performance energy storage devices. Despite significant efforts to develop supercapacitors with aligned morphology, challenges remain due to the predefined pore sizes, limited mechanical flexibility, and low mass loading. Herein, a wood-inspired flexible all-solid-state hydrogel supercapacitor is demonstrated by morphologically tuning the aligned hydrogel matrix toward high electrode-materials loading and high areal capacitance. The highly aligned matrix exhibits broad morphological tunability (47–12 µm), mechanical flexibility (0°–180° bending), and uniform polypyrrole loading up to 7 mm thick matrix. After being assembled into a solid-state supercapacitor, the areal capacitance reaches 831 mF cm−2 for the 12 µm matrix, which is 259% times of the 47 µm matrix and 403% times of nonaligned matrix. The supercapacitor also exhibits a high energy density of 73.8 µWh cm−2, power density of 4960 µW cm−2, capacitance retention of 86.5% after 1000 cycles, and bending stability of 95% after 5000 cycles. The principle to structurally design the oriented matrices for high electrode material loading opens up the possibility for advanced energy storage applications
Building stock dynamics and its impacts on materials and energy demand in China
China hosts a large amount of building stocks, which is nearly 50 billion square meters. Moreover, annual new construction is growing fast, representing half of the world's total. The trend is expected to continue through the year 2050. Impressive demand for new residential and commercial construction, relative shorter average building lifetime, and higher material intensities have driven massive domestic production of energy intensive building materials such as cement and steel. This paper developed a bottom-up building stock turnover model to project the growths, retrofits and retirements of China's residential and commercial building floor space from 2010 to 2050. It also applied typical material intensities and energy intensities to estimate building materials demand and energy consumed to produce these building materials. By conducting scenario analyses of building lifetime, it identified significant potentials of building materials and energy demand conservation. This study underscored the importance of addressing building material efficiency, improving building lifetime and quality, and promoting compact urban development to reduce energy and environment consequences in China
Direct Measurement of Quantum Dot Spin Dynamics using Time-Resolved Resonance Fluorescence
We temporally resolve the resonance fluorescence from an electron spin
confined to a single self-assembled quantum dot to measure directly the spin's
optical initialization and natural relaxation timescales. Our measurements
demonstrate that spin initialization occurs on the order of microseconds in the
Faraday configuration when a laser resonantly drives the quantum dot
transition. We show that the mechanism mediating the optically induced
spin-flip changes from electron-nuclei interaction to hole-mixing interaction
at 0.6 Tesla external magnetic field. Spin relaxation measurements result in
times on the order of milliseconds and suggest that a magnetic field
dependence, due to spin-orbit coupling, is sustained all the way down to 2.2
Tesla.Comment: An additional EPAPS file in PDF format is available for download at
the publications section of our website
http://www.amop.phy.cam.ac.uk/amop-ma
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