UNITARILY INVARIANT NORMS ON FINITE VON NEUMANN ALGEBRAS

John von Neumann’s 1937 characterization of unitarily invariant norms on the n × n matrices in terms of symmetric gauge norms on Cn had a huge impact on linear algebra. In 2008 his results were extended to Ifactor von Neumann algebras by J. Fang, D. Hadwin, E. Nordgren and J. Shen. There already have been many important applications. The factor von Neumann algebras are the atomic building blocks from which every von Neumann algebra can be built. My work, which includes a new proof of the Ifactor case, extends von Neumann’s results to an arbitrary finite von Neumann algebra on a separable Hilberts space. A major tool is the theory of direct integrals. The main idea is to associate to a von Neumann algebra R a measure space (Λ, λ) and a group G (R) of invertible measure-preserving transformations on L∞ (Λ, λ). Then we show that there is a one-to-one correspondence between the unitarily invariant norms on R and the normalized G (R)-symmetric gauge norms on (Λ, λ)

Vertex Arboricity of Toroidal Graphs with a Forbidden Cycle

The vertex arboricity $a(G)$ of a graph $G$ is the minimum $k$ such that $V(G)$ can be partitioned into $k$ sets where each set induces a forest. For a planar graph $G$, it is known that $a(G)\leq 3$. In two recent papers, it was proved that planar graphs without $k$-cycles for some $k\in\{3, 4, 5, 6, 7\}$ have vertex arboricity at most 2. For a toroidal graph $G$, it is known that $a(G)\leq 4$. Let us consider the following question: do toroidal graphs without $k$-cycles have vertex arboricity at most 2? It was known that the question is true for k=3, and recently, Zhang proved the question is true for $k=5$. Since a complete graph on 5 vertices is a toroidal graph without any $k$-cycles for $k\geq 6$ and has vertex arboricity at least three, the only unknown case was k=4. We solve this case in the affirmative; namely, we show that toroidal graphs without 4-cycles have vertex arboricity at most 2.Comment: 8 pages, 2 figure

Elimination of D-band in Raman spectra of double-wall carbon nanotubes by oxidation

In this Letter, we present an in situ Raman spectroscopy study of oxidation-induced changes in the structure and composition of double-wall carbon nanotubes (DWCNTs). Above 480 °C, the intensity of the D band decreases to less than 0.01% of the G band intensity, when measured using the 780 nm laser excitation. The D band was absent from the Raman spectra recorded with the 514.5 nm excitation. Thermogravimetric analysis and high-resolution transmission electron microscopy are used to explain the observed results. We conclude that oxidation provides a purification method for the DWCNT which leads to a sample containing tubes having nearly clean surfaces without disordered carbon

Concentration Flow Cells for Efficient Salinity Energy Recovery with Carbonized Peat Moss and Molybdenum Disulfide Electrodes

The globally extractable salinity gradient (SG) energy from the mixing of seawater and river water is estimated to be 3% of worldwide electricity consumption. Here we applied carbonized peat moss (CPM) electrodes and molybdenum disulfide (MoS2) electrodes to a concentration flow cell that is capable of harvesting SG energy based on the electrode (pseudo-)capacitance together with the Donnan potential. The CPM electrodes were made from the visually inexhaustible peat moss by a facile pyrolysis process. With two identical CPM electrodes and a cation-exchange membrane, the cell produced a peak power density of 5.33 W m-2 and an average power density of 950 mW m-2, the highest ever reported for CDLE-based techniques, using synthetic seawater (30 g L-1 NaCl) and river water (1 g L-1 NaCl). The cells with MoS2 electrodes and an anion-exchange membrane, although produced slightly lower power density with a peak power density of 5.21 W/m2 and an average power density of 0.76 W/m2, harvested more energy during each cycle with an energy density of 66.59 J/m2. The excellent performance of the concentration flow cells was a result of the superior properties of the electrode materials (the macroporous structure of CPM electrodes and the significantly expanded interlayer spacing of MoS2), the assistance of Donnan potential, and the double-channel structure of the cell. Both electrodes were durable as they could extract energy from highly saline water (300 g L-1 NaCl) and still worked well after 100 cycles. This study provides a new method to efficiently and continuously harvest SG energy without an external charge

An Archetypal Study on William Faulkner’s Absalom, Absalom!

This paper attempts to analyze Faulkner’s novel from archetypal perspective with a focus on Biblical allusions in the novel Absalom, Absalom. My purpose is to induce a kind of pattern in Faulkner’s writings which reveals the artist’s capability to assimilate archetypes as well as displace them. His unique method of using archetypes remarkably foregrounds the themes of his fictions and marks him as an innovative and talented writer

Novel Two-dimensional Nanomaterials and Their Gas Sensing Properties

Graphene, an atomic thin two-dimensional (2D) material with C atoms arranged in a honeycomb lattice, has sparked an unprecedented research interest across various scientific communities since its initial mechanical isolation in 2004. The linear energy dispersion with respect to the momentum within 1 eV around the Fermi level at the high symmetric K (Dirac) points in the Brillouin zone renders graphene a wonder material for scientists. However, graphene’s semimetallic nature significantly limits its high-end applications, e.g., in digital logic circuits. Therefore, continued efforts in opening the band gap for graphene and in searching for novel 2D semiconducting materials are rewarding. Various methods have been proposed for generating band gaps in graphene and other related 2D nanomaterials; however, few can be utilized to tune the band gap over a wide range on the same device and many are realized at the cost of severe degradation of carrier mobility. Recently, a new graphene-based crystalline structure, graphene monoxide (GMO), has been discovered based on electron diffraction observations during in situ thermal reduction of multilayer graphene oxide (GO) under vacuum in a transmission electron microscope (TEM) chamber. Supported by infrared spectroscopy and first-principles calculations, the new 2D material was identified as a two-phase hybrid containing GMO domains that evolve in the graphene matrix. GMO extends the electronic property of a graphene derivative into the semiconductor world, enabling potential applications for nanoelectronics. Another route to address the graphene band gap bottleneck is to search for new 2D nanomaterial candidates, among which 2D transition metal dichalcogenides (e.g., MoS2) and black phosphorus (BP) are attracting significant attention. Although both are layered structures and have a tunable band gap, a higher carrier mobility and a wider band gap ranging from 0.3 eV for bulk-like BP to 1.8 eV of monolayer BP make BP an outstanding candidate for future electronic applications. Conductance-based nanoscale gas sensors based on these 2D nanomaterials are attractive due to their superior sensitivity/selectivity and relatively low cost. Experimental studies have shown that in general semiconducting materials exhibit better sensitivity than insulating/metallic materials. Thus, it is crucial to understand the gas sensing mechanism of semiconducting materials and to gain better insights into the performance enhancement. This thesis aims to explore the fundamental properties of novel 2D nanomaterials and to understand their gas sensing performance. Various GMO properties were calculated using density functional theory (DFT)-based techniques. Infrared (IR) spectra of GMO were calculated for both pure GMO and GMO domains embedded into the graphene matrix to facilitate its identification during formation. GMO has three IR active modes that are distinctive from those of graphene and GO. The electronic and mechanical properties of GMO were predicted to illuminate its potential applications in semiconductor devices. The band gap of GMO can be tuned over a wide range from 0 to 1.35 eV. The capability of heat removal in intrinsic GMO was also simulated with and without planar lattice strains and compared with that of graphene and silicon. GMO exhibits a superior thermal conductivity (\u3e3,000 Wm-1K-1), 80% of that of graphene along the armchair direction for large lateral sample sizes (\u3e5 µm). The magnetic properties of zigzag graphene nanoribbons (ZGNRs) induced by GMO domains (or epoxy pair chains) were investigated. The epoxy pair chains can generate finite spin moments in ZGNRs irrespective of the spin coupling between ribbon edges. The gas sensing properties of selected 2D nanomaterials were characterized both theoretically and experimentally. First, we developed statistical thermodynamics models with the gas binding energy from DFT calculations as the only input to characterize the monolayer gas adsorption density on graphene and BP thin films. Our statistical thermodynamics models can successfully predict the gas adsorption density with high accuracy compared with experimental data. Second, an analytical model was established to interpret why semiconducting materials are preferred for gas sensing applications using a BP thin film-based gas sensors as an example. The sensitivity model suggests that the optimum thickness of BP thin film is from several to 10+ nm, corresponding with a band gap of 0.3 to 0.6 eV. Third, van der Pauw and Hall measurements were performed to obtain the sheet resistance, the carrier concentration, and the carrier mobility for thermally-reduced GO (TRGO) at various temperatures to illuminate relative contributions from the carrier concentration and the carrier mobility to the sheet resistance change upon gas adsorption, which suggests that the conductance change upon gas adsorption mainly results from the carrier concentration change. Finally, the sensitivity enhancement from the nanocrystalline particles deposited on the surface of graphene-base materials was also investigated