There are several shape measures for quantitative variables, some of which can also be applied to ordinal variables. In quantitative variables, symmetry, peakedness, and kurtosis are essential properties to evaluate t...There are several shape measures for quantitative variables, some of which can also be applied to ordinal variables. In quantitative variables, symmetry, peakedness, and kurtosis are essential properties to evaluate the deviation from assumptions, particularly normality. They aid in selecting the most appropriate method for estimating parameters and testing hypotheses. Initially, these properties serve a descriptive role in qualitative variables. Once defined, they can be considered to check for non-compliance with assumptions and to propose modifications for testing procedures. The objective of this article is to present three measures of the shape of the distribution of a qualitative variable. The concepts of qualitative asymmetry and peakedness are defined. The measurement of the first concept involves calculating the average frequency difference between qualitative categories matched by frequency homogeneity or proximity. For the second concept, the peak-to-shoulder difference and the qualitative percentile kurtosis are taken into consideration. This last measurement is a less effective option than the peak-to-shoulder difference to measure peakedness. A simulated example of the application of these three measures is given and the paper closes with some conclusions and suggestions.展开更多
There are several shape measures for quantitative variables, some of which can also be applied to ordinal variables. In quantitative variables, symmetry, peakedness, and kurtosis are essential properties to evaluate t...There are several shape measures for quantitative variables, some of which can also be applied to ordinal variables. In quantitative variables, symmetry, peakedness, and kurtosis are essential properties to evaluate the deviation from assumptions, particularly normality. They aid in selecting the most appropriate method for estimating parameters and testing hypotheses. Initially, these properties serve a descriptive role in qualitative variables. Once defined, they can be considered to check for non-compliance with assumptions and to propose modifications for testing procedures. The objective of this article is to present three measures of the shape of the distribution of a qualitative variable. The concepts of qualitative asymmetry and peakedness are defined. The measurement of the first concept involves calculating the average frequency difference between qualitative categories matched by frequency homogeneity or proximity. For the second concept, the peak-to-shoulder difference and the qualitative percentile kurtosis are taken into consideration. This last measurement is a less effective option than the peak-to-shoulder difference to measure peakedness. A simulated example of the application of these three measures is given and the paper closes with some conclusions and suggestions.展开更多
The minimum entropy deconvolution is considered as one of the methods for decomposing non-Gaussian linear processes. The concept of peakedness of a system response sequence is presented and its properties are studied....The minimum entropy deconvolution is considered as one of the methods for decomposing non-Gaussian linear processes. The concept of peakedness of a system response sequence is presented and its properties are studied. With the aid of the peakedness, the convergence theory of the minimum entropy deconvolution is established. The problem of the minimum entropy deconvolution of multi-dimensional non-Gaussian linear random processes is first investigated and the corresponding theory is given. In addition, the relation between the minimum entropy deconvolution and parameter method is discussed.展开更多
We use qualitative analysis and numerical simulation to study peaked traveling wave solutions of CH-γ and CH equations. General expressions of peakon and periodic cusp wave solutions are obtained. Some previous resul...We use qualitative analysis and numerical simulation to study peaked traveling wave solutions of CH-γ and CH equations. General expressions of peakon and periodic cusp wave solutions are obtained. Some previous results become our special cases.展开更多
In this paper the qualitative analysis methods of planar autonomous systems and numerical simu-lation are used to investigate the peaked wave solutions of CH-r equation. Some explicit expressions of peakedsolitary wav...In this paper the qualitative analysis methods of planar autonomous systems and numerical simu-lation are used to investigate the peaked wave solutions of CH-r equation. Some explicit expressions of peakedsolitary wave solutions and peaked periodic wave solutions are obtained, and some of their relationships arerevealed. Why peaked points are generated is discussed.展开更多
As an intense picosecond laser pulse irradiates a hydrocarbon target,the protons therein can be accelerated by the radiation pressure as well as the sheath field behind the target.We investigate the effect of the lase...As an intense picosecond laser pulse irradiates a hydrocarbon target,the protons therein can be accelerated by the radiation pressure as well as the sheath field behind the target.We investigate the effect of the laser and hydrocarbon target parameters on proton acceleration with two/threedimensional particle-in-cell simulations.It is found that the resulting two-ion species plasma can generate a multiple peaked charge-separation field that accelerates the protons.In particular,a smaller carbon-to-hydrogen ratio,as well as the thinner and/or lower density of the target,leads to a larger sheath field and thus proton beams with a larger cutoff energy and smoother energy spectrum.These results may be useful in achieving high-flux quasi-monoenergetic proton beams by properly designing the hydrocarbon target.展开更多
By constructing auxiliary differential equations, we obtain peaked solitary wave solutions of the generalized Camassa-Holm equation, including periodic cusp waves expressed in terms of elliptic functions.
It is well-known that the celebrated Camassa-Holm equation has the peaked solitary waves,which have been not reported for other mainstream models of shallow water waves.In this letter,the closed-form solutions of peak...It is well-known that the celebrated Camassa-Holm equation has the peaked solitary waves,which have been not reported for other mainstream models of shallow water waves.In this letter,the closed-form solutions of peaked solitary waves of the KdV equation,the BBM equation and the Boussinesq equation are given for the first time.All of them have either a peakon or an anti-peakon.Each of them exactly satisfies the corresponding Rankine-Hogoniot jump condition and could be understood as weak solution.Therefore,the peaked solitary waves might be common for most of shallow water wave models,no matter whether or not they are integrable and/or admit breaking-wave solutions.展开更多
By qualitative analysis method, a sufficient condition for the existence of peaked periodic wave solutions to the Broer–Kaup equation is given. Some exact explicit expressions of peaked periodic wave solutions are al...By qualitative analysis method, a sufficient condition for the existence of peaked periodic wave solutions to the Broer–Kaup equation is given. Some exact explicit expressions of peaked periodic wave solutions are also presented.展开更多
We study peaked wave solutions of a generalized Hyperelastic-rod wave equation describing waves in compressible hyperelastic-rods by using the bifurcation theory of planar dynamical systems and numerical simulation me...We study peaked wave solutions of a generalized Hyperelastic-rod wave equation describing waves in compressible hyperelastic-rods by using the bifurcation theory of planar dynamical systems and numerical simulation method. The existence domain of the peaked solitary waves are found. The analytic expressions of peaked solitary wave solutions are obtained. Our numerical simulation and qualitative results are identical.展开更多
We study a two-component Novikov system,which is integrable and can be viewed as a twocomponent generalization of the Novikov equation with cubic nonlinearity.The primary goal of this paper is to understand how multi-...We study a two-component Novikov system,which is integrable and can be viewed as a twocomponent generalization of the Novikov equation with cubic nonlinearity.The primary goal of this paper is to understand how multi-component equations,nonlinear dispersive terms and other nonlinear terms affect the dispersive dynamics and the structure of the peaked solitons.We establish the local well-posedness of the Cauchy problem in Besov spaces B^s/p,r with 1 p,r+∞,s>max{1+1/p,3/2}and Sobolev spaces H^s(R)with s>3/2,and the method is based on the estimates for transport equations and new invariant properties of the system.Furthermore,the blow-up and wave-breaking phenomena of solutions to the Cauchy problem are studied.A blow-up criterion on solutions of the Cauchy problem is demonstrated.In addition,we show that this system admits single-peaked solitons and multi-peaked solitons on the whole line,and the single-peaked solitons on the circle,which are the weak solutions in both senses of the usual weak form and the weak Lax-pair form of the system.展开更多
In this paper,we investigate the orbital stability of the peaked solitons(peakons)for the modified Camassa–Holm equation with cubic nonlinearity.We consider a minimization problem with an appropriately chosen constra...In this paper,we investigate the orbital stability of the peaked solitons(peakons)for the modified Camassa–Holm equation with cubic nonlinearity.We consider a minimization problem with an appropriately chosen constraint,from which we establish the orbital stability of the peakons under H^(1)∩W^(1,4)norm.展开更多
文摘There are several shape measures for quantitative variables, some of which can also be applied to ordinal variables. In quantitative variables, symmetry, peakedness, and kurtosis are essential properties to evaluate the deviation from assumptions, particularly normality. They aid in selecting the most appropriate method for estimating parameters and testing hypotheses. Initially, these properties serve a descriptive role in qualitative variables. Once defined, they can be considered to check for non-compliance with assumptions and to propose modifications for testing procedures. The objective of this article is to present three measures of the shape of the distribution of a qualitative variable. The concepts of qualitative asymmetry and peakedness are defined. The measurement of the first concept involves calculating the average frequency difference between qualitative categories matched by frequency homogeneity or proximity. For the second concept, the peak-to-shoulder difference and the qualitative percentile kurtosis are taken into consideration. This last measurement is a less effective option than the peak-to-shoulder difference to measure peakedness. A simulated example of the application of these three measures is given and the paper closes with some conclusions and suggestions.
文摘There are several shape measures for quantitative variables, some of which can also be applied to ordinal variables. In quantitative variables, symmetry, peakedness, and kurtosis are essential properties to evaluate the deviation from assumptions, particularly normality. They aid in selecting the most appropriate method for estimating parameters and testing hypotheses. Initially, these properties serve a descriptive role in qualitative variables. Once defined, they can be considered to check for non-compliance with assumptions and to propose modifications for testing procedures. The objective of this article is to present three measures of the shape of the distribution of a qualitative variable. The concepts of qualitative asymmetry and peakedness are defined. The measurement of the first concept involves calculating the average frequency difference between qualitative categories matched by frequency homogeneity or proximity. For the second concept, the peak-to-shoulder difference and the qualitative percentile kurtosis are taken into consideration. This last measurement is a less effective option than the peak-to-shoulder difference to measure peakedness. A simulated example of the application of these three measures is given and the paper closes with some conclusions and suggestions.
基金Project supported by the National Natural Science Foundation of China.
文摘The minimum entropy deconvolution is considered as one of the methods for decomposing non-Gaussian linear processes. The concept of peakedness of a system response sequence is presented and its properties are studied. With the aid of the peakedness, the convergence theory of the minimum entropy deconvolution is established. The problem of the minimum entropy deconvolution of multi-dimensional non-Gaussian linear random processes is first investigated and the corresponding theory is given. In addition, the relation between the minimum entropy deconvolution and parameter method is discussed.
文摘We use qualitative analysis and numerical simulation to study peaked traveling wave solutions of CH-γ and CH equations. General expressions of peakon and periodic cusp wave solutions are obtained. Some previous results become our special cases.
基金This work was supported by the National Natural Science Foundation of China(Grant No.10261008)the Natural Science Foundation of Yunnan Province(Grant No,2002A0002M).
文摘In this paper the qualitative analysis methods of planar autonomous systems and numerical simu-lation are used to investigate the peaked wave solutions of CH-r equation. Some explicit expressions of peakedsolitary wave solutions and peaked periodic wave solutions are obtained, and some of their relationships arerevealed. Why peaked points are generated is discussed.
基金the National Key R&D Program of China(No.2016YFA0401100)National Natural Science Foundation of China(Nos.12175154,11875092,and 12005149)+1 种基金the Natural Science Foundation of Top Talent of SZTU(Nos.2019010801001 and 2019020801001)The EPOCH code is used under UK EPSRC contract(EP/G055165/1 and EP/G056803/1).
文摘As an intense picosecond laser pulse irradiates a hydrocarbon target,the protons therein can be accelerated by the radiation pressure as well as the sheath field behind the target.We investigate the effect of the laser and hydrocarbon target parameters on proton acceleration with two/threedimensional particle-in-cell simulations.It is found that the resulting two-ion species plasma can generate a multiple peaked charge-separation field that accelerates the protons.In particular,a smaller carbon-to-hydrogen ratio,as well as the thinner and/or lower density of the target,leads to a larger sheath field and thus proton beams with a larger cutoff energy and smoother energy spectrum.These results may be useful in achieving high-flux quasi-monoenergetic proton beams by properly designing the hydrocarbon target.
基金Supported by the Nature Science Foundation of Shandong (No. 2004zx16,Q2005A01)
文摘By constructing auxiliary differential equations, we obtain peaked solitary wave solutions of the generalized Camassa-Holm equation, including periodic cusp waves expressed in terms of elliptic functions.
基金supported by the State Key Lab of Ocean Engineering(Grant No.GKZD010056-6)the National Natural Science Foundation of China(Grant Nos.10872129 and 11272209
文摘It is well-known that the celebrated Camassa-Holm equation has the peaked solitary waves,which have been not reported for other mainstream models of shallow water waves.In this letter,the closed-form solutions of peaked solitary waves of the KdV equation,the BBM equation and the Boussinesq equation are given for the first time.All of them have either a peakon or an anti-peakon.Each of them exactly satisfies the corresponding Rankine-Hogoniot jump condition and could be understood as weak solution.Therefore,the peaked solitary waves might be common for most of shallow water wave models,no matter whether or not they are integrable and/or admit breaking-wave solutions.
基金Supported by National Nature Science Foundation of China under Grant No.11102076Natural Science Fund for Colleges and Universities in Jiangsu Province under Grant No.15KJB110005
文摘By qualitative analysis method, a sufficient condition for the existence of peaked periodic wave solutions to the Broer–Kaup equation is given. Some exact explicit expressions of peaked periodic wave solutions are also presented.
基金This research was supported by National Natural Science Foundation of China (10571062)Natural Science Foundation of Yunnan (6Y147A).
文摘We study peaked wave solutions of a generalized Hyperelastic-rod wave equation describing waves in compressible hyperelastic-rods by using the bifurcation theory of planar dynamical systems and numerical simulation method. The existence domain of the peaked solitary waves are found. The analytic expressions of peaked solitary wave solutions are obtained. Our numerical simulation and qualitative results are identical.
基金supported by National Natural Science Foundation of China(Grant Nos.11631007,11471174 and 11471259)。
文摘We study a two-component Novikov system,which is integrable and can be viewed as a twocomponent generalization of the Novikov equation with cubic nonlinearity.The primary goal of this paper is to understand how multi-component equations,nonlinear dispersive terms and other nonlinear terms affect the dispersive dynamics and the structure of the peaked solitons.We establish the local well-posedness of the Cauchy problem in Besov spaces B^s/p,r with 1 p,r+∞,s>max{1+1/p,3/2}and Sobolev spaces H^s(R)with s>3/2,and the method is based on the estimates for transport equations and new invariant properties of the system.Furthermore,the blow-up and wave-breaking phenomena of solutions to the Cauchy problem are studied.A blow-up criterion on solutions of the Cauchy problem is demonstrated.In addition,we show that this system admits single-peaked solitons and multi-peaked solitons on the whole line,and the single-peaked solitons on the circle,which are the weak solutions in both senses of the usual weak form and the weak Lax-pair form of the system.
文摘In this paper,we investigate the orbital stability of the peaked solitons(peakons)for the modified Camassa–Holm equation with cubic nonlinearity.We consider a minimization problem with an appropriately chosen constraint,from which we establish the orbital stability of the peakons under H^(1)∩W^(1,4)norm.
