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In this study, we derive the Drazin inverse (A+εB)D{\left(A+\varepsilon B)}^{D} of the complex matrix A+εBA+\varepsilon B with Ind(A+εB)>1{\rm{Ind}}\left(A+\varepsilon B)\gt 1 and Ind(A)=k{\rm{Ind}}\left(A)=k and the group inverse (A+εB)#{\left(A+\varepsilon B)}^{\#} of the complex matrix A+εBA+\varepsilon B with Ind(A+εB)=1{\rm{Ind}}\left(A+\varepsilon B)=1 and Ind(A)=k{\rm{Ind}}\left(A)=k when εB\varepsilon B is viewed as the perturbation of AA. If the dual Drazin inverse (DDGI) A^DDGI{\widehat{A}}^{{\rm{DDGI}}} of A^\widehat{A} is considered as a notation. We calculate (A+εB)D−A^DDGI{\left(A+\varepsilon B)}^{D}-{\widehat{A}}^{{\rm{DDGI}}} and (A+εB)#−A^DDGI{\left(A+\varepsilon B)}^{\#}-{\widehat{A}}^{{\rm{DDGI}}} and obtain ‖(A+εB)D−A^DDGI‖P∈O(ε2)\Vert {\left(A+\varepsilon B)}^{D}-{\widehat{A}}^{{\rm{DDGI}}}{\Vert }_{P}\in {\mathcal{O}}\left({\varepsilon }^{2}) and ‖(A+εB)#−A^DDGI‖P∈O(ε2)\Vert {\left(A+\varepsilon B)}^{\#}-{\widehat{A}}^{{\rm{DDGI}}}{\Vert }_{P}\in {\mathcal{O}}\left({\varepsilon }^{2}). Meanwhile, we give some examples to verify these conclusions.

In this paper, we define a QLY total order $ \overset {Q}\leq $ ≤ Q over $ \mathbb {D}_m $ D m to compare the magnitude of dual vectors. Then we consider the QLY least-squares problem and give its compact formula. Meanwhile, by comparing with a least-squares and the least-squares minimal-norm solutions, we can investigate a QLY least-squares and the QLY least-squares minimal-norm of linear dual least-squares problems. In particular, in the presence of a least-squares solution, we can get a QLY least-squares solution to be more accurate than a least-squares solution under the QLY total order. [ABSTRACT FROM AUTHOR]

For the safe and efficient utilization of hydrogen-enriched natural gas combustion in industrial gas-fired boilers, the present study adopted a combination of numerical simulation and field tests to investigate its adaptability. Firstly, the combustion characteristics of hydrogen-enriched natural gas with different hydrogen blending ratios and equivalence ratios were evaluated by using the Chemkin Pro platform. Secondly, a field experimental study was carried out based on the WNS2-1.25-Q gas-fired boiler to investigate the boiler's thermal efficiency, heat loss, and pollutant emissions after hydrogen addition. The results show that at the same equivalence ratio, with the hydrogen blending ratio increasing from 0% to 25%, the laminar flame propagation speed of the fuel increases, the extinction strain rate rises, and the combustion limit expands. The laminar flame propagation speed of premixed methane/air gas reaches the maximum value when the equivalence ratio is 1.0, and the combustion intensity of the flame is the highest at this time. In the field tests, as the hydrogen blending ratio increases from 0% to nearly 10% with the increasing excess air ratio, the boiler's thermal efficiency decreases as well as the NOx emission. This indicates that there exists a tradeoff between the boiler thermal efficiency and NOx emission in practice. [ABSTRACT FROM AUTHOR]

The 06Cr18Ni11Ti (321) stainless steel has good corrosion resistance and mechanical properties.However, the addition of Ti leads to the formation of Ti N inclusions, which deteriorates the condition of continuous casting and causes the clogging.The formation of clogging in the submerged entry nozzle (SEN) of 321 stainless steel was studied by scanning electron microscopy (SEM) , energy disperse spectroscope (EDS) and thermodynamic calculation.According to the change of components, three layers in sample were defined:the refractory materials layer, the reaction layer and metal layer from the outer layer to the inner layer of SEN.The results show that the molten steel corrodes the refractory materials of SEN and the high-melting-point slag phase forms.Then Ti N inclusions attach to the inner layer and form curved chain or reticular structure with slag phase, which causes the clogging of SEN.Resultsof thermodynamic calculation indicate that the equilibrium solubility of titanium and nitrogen increases with temperature.There is high Ca/Al proportion in Ca O-Al2O3-Ti O2-5%MgO phase.So controlling the compositions of slag is needed to prevent clogging.

In this study, glass-fiber, grafted by epoxide-terminated hyperbranched polymer (GF-HBPE), was incorporated into epoxy resins for reinforcement purpose. The effects of GF-HBPE content on mechanical properties of the resulting epoxy-based composites, such as tensile strength, percentage elongation at break, flexural strength, and impact strength, were investigated. The experimental results revealed that GF-HBPE substantially outperformed impact resistance in both tensile and flexural tests. For instance, the tensile strength, percentage elongation at break, flexural strength, and impact strength of the epoxy composite with 1 wt% GF-HBPE increase by about 23.6%, 125%, 26%, and 74.5%, respectively, compared to the unmodified epoxy thermoset.