Glasses are known to possess low-frequency excess modes beyond the Debye prediction.For decades,it has been assumed that evolution of low-frequency density of excess modes D(ω) with frequency ω follows a power-law s...Glasses are known to possess low-frequency excess modes beyond the Debye prediction.For decades,it has been assumed that evolution of low-frequency density of excess modes D(ω) with frequency ω follows a power-law scaling:D(ω)~ω~γ.However,it remains debated on the value of γ at low frequencies below the first phonon-like mode in finitesize glasses.Early simulation studies reported γ=4 at low frequencies in two-(2D),three-(3D),and four-dimensional(4D)glasses,whereas recent observations in 2D and 3D glasses suggested γ=3.5 in a lower-frequency regime.It is uncertain whether the low-frequency scaling of D(ω)~ω^(3.5) could be generalized to 4D glasses.Here,we conduct numerical simulation studies of excess modes at frequencies below the first phonon-like mode in 4D model glasses.It is found that the system size dependence of D(ω) below the first phonon-like mode varies with spatial dimensions:D(ω) increases in2D glasses but decreases in 3D and 4D glasses as the system size increases.Furthermore,we demonstrate that the ω^(3.5)scaling,rather than the ω~4 scaling,works in the lowest-frequency regime accessed in 4D glasses,regardless of interaction potentials and system sizes examined.Therefore,our findings in 4D glasses,combined with previous results in 2D and 3D glasses,suggest a common low-frequency scaling of D(ω)~ ω^3.5) below the first phonon-like mode across different spatial dimensions,which would inspire further theoretical studies.展开更多
A new four-dimensional quadratic smooth autonomous chaotic system is presented in this paper, which can exhibit periodic orbit and chaos under the conditions on the system parameters. Importantly, the system can gener...A new four-dimensional quadratic smooth autonomous chaotic system is presented in this paper, which can exhibit periodic orbit and chaos under the conditions on the system parameters. Importantly, the system can generate one-, two-, three- and four-scroll chaotic attractors with appropriate choices of parameters. Interestingly, all the attractors are generated only by changing a single parameter. The dynamic analysis approach in the paper involves time series, phase portraits, Poincare maps, a bifurcation diagram, and Lyapunov exponents, to investigate some basic dynamical behaviours of the proposed four-dimensional system.展开更多
In this paper, we propose a novel four-dimensional autonomous chaotic system. Of particular interest is that this novel system can generate one-, two, three- and four-wing chaotic attractors with the variation of a si...In this paper, we propose a novel four-dimensional autonomous chaotic system. Of particular interest is that this novel system can generate one-, two, three- and four-wing chaotic attractors with the variation of a single parameter, and the multi-wing type of the chaotic attractors can be displayed in all directions. The system is simple with a large positive Lyapunov exponent and can exhibit some interesting and complicated dynamical behaviours. Basic dynamical properties of the four-dimensional chaotic system, such as equilibrium points, the Poincare map, the bifurcation diagram and the Lyapunov exponents are investigated by using either theoretical analysis or numerical method. Finally, a circuit is designed for the implementation of the multi-wing chaotic attractors. The electronic workbench observations axe in good agreement with the numerical simulation results.展开更多
The problem addressed is the exact determination of the operator norm and lower bound of four-dimensional generalized Hausdorff matrices on the double sequence spaces L_(p).A Hardy type formulae is found as an exact v...The problem addressed is the exact determination of the operator norm and lower bound of four-dimensional generalized Hausdorff matrices on the double sequence spaces L_(p).A Hardy type formulae is found as an exact value for their operator norm and a Copson type formulae is established as a lower estimate for their lower bound.Further,exact values are found for the operator norm and lower bound of the transpose of generalized Hausdorff matrices.展开更多
The eigenvalue space of the canonical four-dimensional Chua＇s circuit which can realize every eigenvalue for fourdimensional system is studied in this paper. First, the analytical relations between the circuit parame...The eigenvalue space of the canonical four-dimensional Chua＇s circuit which can realize every eigenvalue for fourdimensional system is studied in this paper. First, the analytical relations between the circuit parameters and the eigenvalues of the system are established, and therefore all the circuit parameters can be determined explicitly by any given set of eigenvalues. Then, the eigenvalue space of the circuit is investigated in two cases by the nonlinear elements used. According to the types of the eigenvalues, some novel hyperchaotic attractors are presented. Further, the dynamic behaviours of the circuit are studied by the bifurcation diagrams and the Lyapunov spectra of the eigenvalues.展开更多
We introduce the Dirac equation in four-dimensional gravity which is a generally covariant form. We choose the suitable variable and solve the corresponding equation. To solve such equation and to obtain the correspon...We introduce the Dirac equation in four-dimensional gravity which is a generally covariant form. We choose the suitable variable and solve the corresponding equation. To solve such equation and to obtain the corresponding bispinor, we employ the factorization method which introduces the associated Laguerre polynomial. The asso- ciated Laguerre polynomials help us to write the Dirac equation of four-dimensional gravity in the form of the shape invariance equation. Thus we write the shape invariance condition with respect to the secondary quantum number. Finally, we obtain the spinor wave function and achieve the corresponding stability of condition for the four-dimensional gravity system.展开更多
Dynamic structuralcolors can change in response todifferent environmental stimuli.This ability remains effectiveeven when the size of the speciesresponsible for the structural coloris reduced to a few micrometers,prov...Dynamic structuralcolors can change in response todifferent environmental stimuli.This ability remains effectiveeven when the size of the speciesresponsible for the structural coloris reduced to a few micrometers,providing a promising sensingmechanism for solving microenvironmentalsensing problems inmicro-robotics and microfluidics.However, the lack of dynamicstructural colors that can encoderapidly, easily integrate, and accuratelyreflect changes in physical quantities hinders their use in microscale sensing applications. Herein, we present a 2.5-dimensionaldynamic structural color based on nanogratings of heterogeneous materials, which were obtained by interweaving a pH-responsive hydrogelwith an IP-L photoresist. Transverse gratings printed with pH-responsive hydrogels elongated the period of longitudinal grating in the swollenstate, resulting in pH-tuned structural colors at a 45° incidence. Moreover, the patterned encoding and array printing of dynamic structuralcolors were achieved using grayscale stripe images to accurately encode the periods and heights of the nanogrid structures. Overall, dynamicstructural color networks exhibit promising potential for applications in information encryption and in situ sensing for microfluidic chips.展开更多
Intensity-modulated particle therapy(IMPT)with carbon ions is comparatively susceptible to various uncertainties caused by breathing motion,including range,setup,and target positioning uncertainties.To determine relat...Intensity-modulated particle therapy(IMPT)with carbon ions is comparatively susceptible to various uncertainties caused by breathing motion,including range,setup,and target positioning uncertainties.To determine relative biological effectiveness-weighted dose(RWD)distributions that are resilient to these uncertainties,the reference phase-based four-dimensional(4D)robust optimization(RP-4DRO)and each phase-based 4D robust optimization(EP-4DRO)method in carbon-ion IMPT treatment planning were evaluated and compared.Based on RWD distributions,4DRO methods were compared with 4D conventional optimization using planning target volume(PTV)margins(PTV-based optimization)to assess the effectiveness of the robust optimization methods.Carbon-ion IMPT treatment planning was conducted in a cohort of five lung cancer patients.The results indicated that the EP-4DRO method provided better robustness(P=0.080)and improved plan quality(P=0.225)for the clinical target volume(CTV)in the individual respiratory phase when compared with the PTV-based optimization.Compared with the PTV-based optimization,the RP-4DRO method ensured the robustness(P=0.022)of the dose distributions in the reference breathing phase,albeit with a slight sacrifice of the target coverage(P=0.450).Both 4DRO methods successfully maintained the doses delivered to the organs at risk(OARs)below tolerable levels,which were lower than the doses in the PTV-based optimization(P<0.05).Furthermore,the RP-4DRO method exhibited significantly superior performance when compared with the EP-4DRO method in enhancing overall OAR sparing in either the individual respiratory phase or reference respiratory phase(P<0.05).In general,both 4DRO methods outperformed the PTV-based optimization in terms of OAR sparing and robustness.展开更多
基金the support from the National Natural Science Foundation of China(Grant Nos.12374202 and 12004001)Anhui Projects(Grant Nos.2022AH020009,S020218016,and Z010118169)Hefei City(Grant No.Z020132009)。
文摘Glasses are known to possess low-frequency excess modes beyond the Debye prediction.For decades,it has been assumed that evolution of low-frequency density of excess modes D(ω) with frequency ω follows a power-law scaling:D(ω)~ω~γ.However,it remains debated on the value of γ at low frequencies below the first phonon-like mode in finitesize glasses.Early simulation studies reported γ=4 at low frequencies in two-(2D),three-(3D),and four-dimensional(4D)glasses,whereas recent observations in 2D and 3D glasses suggested γ=3.5 in a lower-frequency regime.It is uncertain whether the low-frequency scaling of D(ω)~ω^(3.5) could be generalized to 4D glasses.Here,we conduct numerical simulation studies of excess modes at frequencies below the first phonon-like mode in 4D model glasses.It is found that the system size dependence of D(ω) below the first phonon-like mode varies with spatial dimensions:D(ω) increases in2D glasses but decreases in 3D and 4D glasses as the system size increases.Furthermore,we demonstrate that the ω^(3.5)scaling,rather than the ω~4 scaling,works in the lowest-frequency regime accessed in 4D glasses,regardless of interaction potentials and system sizes examined.Therefore,our findings in 4D glasses,combined with previous results in 2D and 3D glasses,suggest a common low-frequency scaling of D(ω)~ ω^3.5) below the first phonon-like mode across different spatial dimensions,which would inspire further theoretical studies.
文摘A new four-dimensional quadratic smooth autonomous chaotic system is presented in this paper, which can exhibit periodic orbit and chaos under the conditions on the system parameters. Importantly, the system can generate one-, two-, three- and four-scroll chaotic attractors with appropriate choices of parameters. Interestingly, all the attractors are generated only by changing a single parameter. The dynamic analysis approach in the paper involves time series, phase portraits, Poincare maps, a bifurcation diagram, and Lyapunov exponents, to investigate some basic dynamical behaviours of the proposed four-dimensional system.
文摘In this paper, we propose a novel four-dimensional autonomous chaotic system. Of particular interest is that this novel system can generate one-, two, three- and four-wing chaotic attractors with the variation of a single parameter, and the multi-wing type of the chaotic attractors can be displayed in all directions. The system is simple with a large positive Lyapunov exponent and can exhibit some interesting and complicated dynamical behaviours. Basic dynamical properties of the four-dimensional chaotic system, such as equilibrium points, the Poincare map, the bifurcation diagram and the Lyapunov exponents are investigated by using either theoretical analysis or numerical method. Finally, a circuit is designed for the implementation of the multi-wing chaotic attractors. The electronic workbench observations axe in good agreement with the numerical simulation results.
文摘The problem addressed is the exact determination of the operator norm and lower bound of four-dimensional generalized Hausdorff matrices on the double sequence spaces L_(p).A Hardy type formulae is found as an exact value for their operator norm and a Copson type formulae is established as a lower estimate for their lower bound.Further,exact values are found for the operator norm and lower bound of the transpose of generalized Hausdorff matrices.
基金Project supported by the National Natural Science Foundation of China (Grant No. 50877007)
文摘The eigenvalue space of the canonical four-dimensional Chua＇s circuit which can realize every eigenvalue for fourdimensional system is studied in this paper. First, the analytical relations between the circuit parameters and the eigenvalues of the system are established, and therefore all the circuit parameters can be determined explicitly by any given set of eigenvalues. Then, the eigenvalue space of the circuit is investigated in two cases by the nonlinear elements used. According to the types of the eigenvalues, some novel hyperchaotic attractors are presented. Further, the dynamic behaviours of the circuit are studied by the bifurcation diagrams and the Lyapunov spectra of the eigenvalues.
文摘We introduce the Dirac equation in four-dimensional gravity which is a generally covariant form. We choose the suitable variable and solve the corresponding equation. To solve such equation and to obtain the corresponding bispinor, we employ the factorization method which introduces the associated Laguerre polynomial. The asso- ciated Laguerre polynomials help us to write the Dirac equation of four-dimensional gravity in the form of the shape invariance equation. Thus we write the shape invariance condition with respect to the secondary quantum number. Finally, we obtain the spinor wave function and achieve the corresponding stability of condition for the four-dimensional gravity system.
基金supported by the National Natural Science Foundation of China(Grant No.61925307).
文摘Dynamic structuralcolors can change in response todifferent environmental stimuli.This ability remains effectiveeven when the size of the speciesresponsible for the structural coloris reduced to a few micrometers,providing a promising sensingmechanism for solving microenvironmentalsensing problems inmicro-robotics and microfluidics.However, the lack of dynamicstructural colors that can encoderapidly, easily integrate, and accuratelyreflect changes in physical quantities hinders their use in microscale sensing applications. Herein, we present a 2.5-dimensionaldynamic structural color based on nanogratings of heterogeneous materials, which were obtained by interweaving a pH-responsive hydrogelwith an IP-L photoresist. Transverse gratings printed with pH-responsive hydrogels elongated the period of longitudinal grating in the swollenstate, resulting in pH-tuned structural colors at a 45° incidence. Moreover, the patterned encoding and array printing of dynamic structuralcolors were achieved using grayscale stripe images to accurately encode the periods and heights of the nanogrid structures. Overall, dynamicstructural color networks exhibit promising potential for applications in information encryption and in situ sensing for microfluidic chips.
基金supported by National Key Research and Development Program of China(No.2022YFC2401503)National Natural Science Foundation of China(Nos.11875299,61631001,U1532264,and 12005271).
文摘Intensity-modulated particle therapy(IMPT)with carbon ions is comparatively susceptible to various uncertainties caused by breathing motion,including range,setup,and target positioning uncertainties.To determine relative biological effectiveness-weighted dose(RWD)distributions that are resilient to these uncertainties,the reference phase-based four-dimensional(4D)robust optimization(RP-4DRO)and each phase-based 4D robust optimization(EP-4DRO)method in carbon-ion IMPT treatment planning were evaluated and compared.Based on RWD distributions,4DRO methods were compared with 4D conventional optimization using planning target volume(PTV)margins(PTV-based optimization)to assess the effectiveness of the robust optimization methods.Carbon-ion IMPT treatment planning was conducted in a cohort of five lung cancer patients.The results indicated that the EP-4DRO method provided better robustness(P=0.080)and improved plan quality(P=0.225)for the clinical target volume(CTV)in the individual respiratory phase when compared with the PTV-based optimization.Compared with the PTV-based optimization,the RP-4DRO method ensured the robustness(P=0.022)of the dose distributions in the reference breathing phase,albeit with a slight sacrifice of the target coverage(P=0.450).Both 4DRO methods successfully maintained the doses delivered to the organs at risk(OARs)below tolerable levels,which were lower than the doses in the PTV-based optimization(P<0.05).Furthermore,the RP-4DRO method exhibited significantly superior performance when compared with the EP-4DRO method in enhancing overall OAR sparing in either the individual respiratory phase or reference respiratory phase(P<0.05).In general,both 4DRO methods outperformed the PTV-based optimization in terms of OAR sparing and robustness.
