Generating facets for the cut polytope of a graph by triangular elimination

The cut polytope of a graph arises in many fields. Although much is known about facets of the cut polytope of the complete graph, very little is known for general graphs. The study of Bell inequalities in quantum information science requires knowledge of the facets of the cut polytope of the complete bipartite graph or, more generally, the complete k-partite graph. Lifting is a central tool to prove certain inequalities are facet inducing for the cut polytope. In this paper we introduce a lifting operation, named triangular elimination, applicable to the cut polytope of a wide range of graphs. Triangular elimination is a specific combination of zero-lifting and Fourier-Motzkin elimination using the triangle inequality. We prove sufficient conditions for the triangular elimination of facet inducing inequalities to be facet inducing. The proof is based on a variation of the lifting lemma adapted to general graphs. The result can be used to derive facet inducing inequalities of the cut polytope of various graphs from those of the complete graph. We also investigate the symmetry of facet inducing inequalities of the cut polytope of the complete bipartite graph derived by triangular elimination.Comment: 19 pages, 1 figure; filled details of the proof of Theorem 4, made many other small change

Determination of Condition for Electrolytic Extractions of Nitrides in Steel by Means of Potentiostat

The polarizing characteristics of synthesized nitrides, Fe_3N, Fe_4N, Mn_3N_2 Mo_2N, MoN, Cr_2N CrN, VN, Si_3N_4 AlN, TiN, ZrN and NbN, were studied by means of potentiostat in order to determine the condition for the electrolytic extractions of nitrides from iron and steel. From the results of polarization curves and natural electrode potentials of iron and nitrides, some useful informations were obtained as to the electrolyte and the electrolytic potential for the electrolytic extractions of nitrides contained in iron and steel ; that is, if a suitable electrolytic potential is applied, both acidic and neutral solutions can be used as an electrolyte for the extractions except nitrides of Al, Fe and Mn. For the Fe-nitride, only neutral electrolyte is useful, while for the nitrides of Al and Mn, there seems to be no favorable electrolyte

Structural Diagrams and Phase Reactions of the Quaternary 12%Cr-Fe-C-N System

The isothermal diagrams of the 12% Cr-Fe-C-N system in the composition range up to 0.4% carbon and 0.3% nitrogen were studied at temperatures from 1300℃ to 700℃. Resorting to the consideration of phase relationship, the phase reaction of 12%Cr-Fe-C-N system was clarified and the sectional diagrams were constructed at fixed contents of 0.1% and 0.2% nitrogen and carbon, respectively. In Fe-Cr-C-N system, the quaternary peritecto-eutectoid reaction (chemical formula) exists at about 780℃, representing by the hexahedron of five-phase region consisting of α, γ, Cr_C_6, Cr_7C_3 and Cr_2N. The Cr_7C_3 type carbide detected in the present work is expressed by the formula (Fe_2C_5)C_N_

Solubility of Nitrogen in Austenitic Iron under High Nitrogen Pressure and Thermodynamic Properties of Iron-Nitrogen Interstitial Solid Solution

Studies on the equilibrium between austenitic iron and nitrogen under various conditions of the maximum pressure of 920 kg/cm^2 and the temperature range from 950°to 1300℃ were carried out by using a high temperature and high pressure equipment. It was shown that the concentration of nitrogen in austenite deviated from Sieverts\u27 law with increasing pressure. The casue for such discrepancy was considered thermodynamically and statistically. That is, the experimental result can reasonably be explained from the geometrical consideration that the chemical potential of a nitrogen atom in austenite, in case of a high nitrogen content deviates remarkably from that in an idial random interstitial solid solution, since each intestitial atom added excludes other interstitial atoms from the seven adjacent sites. Thus, the activity of nitrogen in austenitic iron can be expressed by the following equations : log a_N=log N_N/1-9N_N+ 3.34-427/T for the standard state of nitrogen gas 1 atm pressure, a_N=f^_ log a_N=log N_N/1-9N_N for the standard state of infinitely dilute solid solution

Constitutional Diagrams of Iron-Rich Corner in 18% Cr-Fe-C-N System

The constitutional diagrams of 18%Cr-Fe-C-N quaternary system were examined up to 0.5% of both carbon and nitrogen in the temperature range of 1300°to 700℃, and the sectional diagrams were described at fixed contents of 0.1 and 0.2% nitrogen, and 0.1, 0.2 and 0.3% carbon. In the present range of composition, ferrite and austenite existed in equilibrium in the range of 1300°to 1100℃, and the austenite-promoting effects of carbon and nitrogen were very nearly equivalent and additive. The minimum concentration of carbon plus nitrogen required to obtain a completely austenitic structure was about 0.4%. Below 1100℃, Cr_C_6, Cr_7C_3 and Cr_2N precipitated and the phase regions recognized were γ+Cr_C_6, γ+Cr_7C_3, γ+Cr_2N, γ+Cr_C_6+Cr_2N, α+γ+Cr_C_6, α+γ+Cr_2N, α+γ+Cr_C_6+Cr_2N, α+Cr_C_6+Cr_2N, α+Cr_C_6 and α+Cr_2N. It was presumed that the four-phase region of α+γ+Cr_C_6+Cr_2N arose from the eutectoid reaction γ→α+Cr_C_6+Cr_2N. The compositions of Cr_C_6 and Cr_2N were generally represented by (Fe_7Cr_)C_6 and (Fe_Cr_)C_N_, respectively

Microstructures and Nitrides of Fe-Cr-N Ternary System

The microstructures and the characters of nitrides in various alloys of the Fe-Cr-N system in the range of 0.5 to 40% chromium with up to about 1% nitrogen were studied by optical- and electron-microscope and by X-ray diffraction and chemical analysis of the extracted residue. The results are summarized as follows : (1) The nitrides detected were Fe_4N, CrN and Cr_2N. (2) In the content of about 0.1% N, the changes in microstructure with the lowering of temperature were as follows : 0.5% Cr : γ→α+γ→α+γ+CrN→α+CrN+Fe_4N ; about 0.5～6% Cr : γ→α+γ→α+γ+CrN→α+CrN ; abaut 6～7% Cr : γ→α+γ→α+γ+Cr_2N→α+Cr_2N+CrN ; about 7～13% Cr : γ→α+γ→α+γ+Cr_2N→α+Cr_2N ; about 13～26% Cr : α+γ→α+γ+Cr_2N→α+Cr_2N ; above about 26% Cr : α→α+Cr_2N. (3) In the case of alloys containing chromium above about 13%, the minimum content of nitrogen to obtain a completely austenitic structure increased with the chromium content ; for example, the minimum nitrogen contents in 13, 18 and 16% Cr alloys were about 0.1, 0.4 and 0.8%, respectively. The microstructures of these alloys revealed such a change as γ→γ+Cr_2N→α+γ+Cr_2N→α→Cr_2N. (4) The very fine acicular phase was observed within ferrite in the Fe-Cr-N alloys, which contained chromium above about 15% and were quenched in water from above about 1100℃. It was confirmed that this phase was Cr_2N precipitated during cooling clue to a remarkable decrease of the solubility of nitrogen in ferrite, and that the suppression of this precipitation was very difficult

Enhancement of Power Generation and Organic Removal in Double Anode Chamber Designed Dual-Chamber Microbial Fuel Cell (DAC-DCMFC)

One of the important factors in enhancing the performance of microbial fuel cells (MFCs) is reactor design and configuration. Therefore, this study was conducted to evaluate the regressors and their operating parameters affecting the double anode chamber–designed dual-chamber microbial fuel cell (DAC-DCMFC) performance. Its primary design consists of two anode chamber compartments equipped with a separator and cathode chamber. The DAC-DCMFCs were parallelly operated over 8 days (60 days after the acclimation period). They were intermittently pump-fed with the different organic loading rates (OLRs), using chemically enriched sucrose as artificial wastewater. The applied OLRs were adjusted at low, medium, and high ranges from 0.4 kg.m−3.d−1 to 2.5 kg.m−3.d−1. The reactor types were type 1 and type 2 with different cathode materials. The pH, temperature, oxidation-reduction potential (ORP), optical density 600 (OD600), chemical oxygen demand (COD), and total organic carbon (TOC) were measured, using standard analytical instruments. In general, the power production achieved a maximum of 866 ± 44 mW/m2, with a volumetric power density of 5.15 ± 0.26 W/m3 and coulombic efficiency of 84%. Two-stage COD and TOC removal at medium OLR achieved a range of 60–80%. Medium OLR is the recommended level to enhance power production and organic removal in DAC-DCMFC. The separated anode chambers into two parts in a dual anode chamber microbial fuel cell adjusted by various organic loadings expressed a preferable comprehension in the integrated MFCs for wastewater treatment