In this paper, we consider the Schrodinger equation with quadratic potential i partial derivative/partial derivative t u = -Delta u + |x|(2) u in Rn+1, u(x,0) = f(x) is an element of L-2(R-n). Using Herrnite functions...
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In this paper, we consider the Schrodinger equation with quadratic potential i partial derivative/partial derivative t u = -Delta u + |x|(2) u in Rn+1, u(x,0) = f(x) is an element of L-2(R-n). Using Herrnite functions and some other classical tools, we give an elementary proof of the Kato-type smoothing estimate: for i not equal j not equal k, delta is an element of [0,1], and n >= 3, integral(2 pi)(0) integral(Rn) |u(x,t)|(2)/(x(i)(2) + x(j)(2) + x(k)(2))(delta) dxdt <= C parallel to f parallel to(2)(2). This is equivalent to proving a uniform L-2(R-n) boundedness result for a family of singularized Hermite projection kernels. As an application of the above estimate, we also prove the R-9 collapsing variable-type Strichartz estimate integral(2 pi)(0) integral(R3) |u(x,x,x,t)|(2) dxdt <= C parallel to(-Delta + |x|(2))f parallel to(2)(2) where x is an element of R-3.
In this paper, we investigate bounded traveling waves of the generalized nonlinear Klein-Gordon model equations by using bifurcation theory of planar dynamical systems to study the effects of horizontal singular strai...
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In this paper, we investigate bounded traveling waves of the generalized nonlinear Klein-Gordon model equations by using bifurcation theory of planar dynamical systems to study the effects of horizontal singular straight lines in nonlinear wave equations. Besides the well-known smooth traveling wave solutions and the non-smooth ones, four kinds of new bounded singular traveling wave solution are found for the first time. These singular traveling wave solutions are characterized by discontinuous second-order derivatives at some points, even though their first-order derivatives are continuous. Obviously, they are different from the singular traveling wave solutions such as compactons, cuspons, peakons. Their implicit expressions are also studied in this paper. These new interesting singular solutions, which are firstly founded, enrich the results on the traveling wave solutions of nonlinear equations. It is worth mentioning that the nonlinear equations with horizontal singular straight lines may have abundant and interesting new kinds of traveling wave solution.
In this work, a novel method based on ultrasonic-assisted nebulization extraction coupled with solid phase extraction (UANE-SPE) and determined by high performance liquid chromatography was developed for the determina...
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In this work, a novel method based on ultrasonic-assisted nebulization extraction coupled with solid phase extraction (UANE-SPE) and determined by high performance liquid chromatography was developed for the determination of triterpenoids in root of Euphorbia pekinensis Rupr. The experimental conditions for the UANE-SPE, such as type of extraction solvent, sample amount, type and amount of sorbent, extraction time and volume of the elution solvent, were examined and optimized. The method was successfully applied to determine euphol and tirucallol in the dried root of E. pekinensis Rupr. The recoveries of the analytes were in the range of 89.1-102.0 %. The limits of detection were 12 mu g g(-1) for tirucallol and 10 mu g g(-1) for euphol. The extraction yields obtained by the proposed method are higher than those obtained by the conventional extraction methods, such as reflux and ultrasonic-assisted extraction. Compared with the traditional methods, the proposed method can reduce the consumption of the labor, shorten the sample preparation time and increase the efficiencies in the extraction of active constituents from plant materials.
Recently, the quality of the work of the city appearance and environmental sanitation has been constantly improved, however, the assessment of the work has not been standardized and institutionalized and the existing ...
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Recently, the quality of the work of the city appearance and environmental sanitation has been constantly improved, however, the assessment of the work has not been standardized and institutionalized and the existing professional assessment and social evaluation are certain one-sided. In that case, it is an inexorable trend to construct the multiple evaluation index system of the city appearance and environmental sanitation. This paper studies the expert evaluation index of the city appearance and environmental sanitation using Delphi method, questionnaire survey, AHP and many other methods. Moreover, it sets up an expert evaluation system of city appearance and environmental sanitation, which has good practicability and maneuverability and can reflect the development trend of the city appearance and environmental sanitation.
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