Reflection and transmission coefficients of a thin bed

The study of thin-bed seismic response is an important part in lithologic and methane reservoir modeling, critical for predicting their physical attributes and/or elastic parameters. The complex propagator matrix for the exact reflections and transmissions of thin beds limits their application in thin-bed inversion. Therefore, approximation formulas with a high accuracy and a relatively simple form are needed for thin-bed seismic analysis and inversion. We have derived thin-bed reflection and transmission coefficients, defined in terms of displacements, and approximated them to be in a quasi-Zoeppritz matrix form under the assumption that the middle layer has a very thin thickness. We have verified the approximation accuracy through numerical calculation and concluded that the errors in PP-wave reflection coefficients RPP are generally smaller than 10% when the thin-bed thicknesses are smaller than one-eighth of the PP-wavelength. The PS-wave reflection coefficients RPS have lower approximation accuracy than RPP for the same ratios of thicknesses to their respective wavelengths, and the RPS approximation is not acceptable for incident angles approaching the critical angles (when they exist) except in the case of extremely strong impedance difference. Errors in phase for the RPP and RPS approximation are less than 10% for the cases of thicknesses less than one-tenth of the wavelengths. As expected, a thinner middle layer and a weaker impedance difference would result in higher approximation accuracy

A Remark on Soliton Equation of Mean Curvature Flow

In this short note, we consider self-similar immersions $F: \mathbb{R}^n \to \mathbb{R}^{n+k}$ of the Graphic Mean Curvature Flow of higher co-dimension. We show that the following is true: Let $F(x) = (x,f(x)), x \in \mathbb{R}^{n}$ be a graph solution to the soliton equation $$\bar{H}(x) + F^{\bot}(x) = 0.$$ Assume $\sup_{\mathbb{R}^{n}}|Df(x)| \le C_{0} < + \infty$. Then there exists a unique smooth function $f_{\infty}: \mathbb{R}^{n}\to \mathbb{R}^k$ such that $$f_{\infty}(x) = \lim_{\lambda \to \infty}f_{\lambda}(x)$$ and $$f_{\infty}(r x)=r f_{\infty}(x)$$ for any real number $r\not= 0$, where $$f_{\lambda}(x) = \lambda^{-1}f(\lambda x).$$Comment: 6 page

Modular transformation and twist between trigonometric limits of sl(n)sl(n) elliptic R-matrix

We study the modular transformation of ${\bf Z}_n$-symmetric elliptic R-matrix and construct the twist between the trigonometric degeneracy of the elliptic R-matrix.Comment: 8 pages, latex, reference revise

Fire resistance of steel beam to square CFST column composite joints using RC slabs: Experiments and numerical studies

In this paper, experimental investigation and numerical simulation of steel beam to square concrete-filled steel tube (CFST) column composite joints that use reinforced concrete (RC) slabs subjected to localized and global fire conditions are presented. Eight joints were tested under the ISO 834 fire standard, and the effect of different parameters including the load ratio of beams, the beam-to-column ratio of linear stiffness, and different fire scenarios was studied during testing. The failure patterns and the thermal responses of the structural members including the temperature distribution, axial displacement of columns, vertical deflection of the beam ends, and fire resistance of the joints were recorded and discussed. The results show that tube buckling of the square CFST columns, flange buckling of the steel beams, and separation between the top flange of the steel beams and the RC slabs were the primary failure patterns of this type of joint. Moreover, the temperatures of structural members within the connection zone were lower than those in the other regions. Compared with other factors, the load ratio of the beams demonstrated a significant influence on the displacement of the structural members and the fire resistance of the joints. A three-dimensional finite element analysis (FEA) model was built to simulate the fire performance of this type of composite joint. The simulation results were compared to the test results in terms of failure patterns, temperature distributions, displacements, and fire resistances, and good agreement in general was achieved. Finally, the FEA model was adopted to examine the effect of parameters on the fire resistance of the composite joints with axial and flexural constraints applied at the ends of the beam