A method of slope reliability analysis was developed by imposing a state equation on the limit equilibrium theory, given the basis of a fixed safety factor technique. Among the many problems of reliability analysis, t...A method of slope reliability analysis was developed by imposing a state equation on the limit equilibrium theory, given the basis of a fixed safety factor technique. Among the many problems of reliability analysis, the most important problem is to find a performance function. We have created a new method of building a limit state equation for planar slip surfaces by applying the mathematical cusp catastrophe theory. This new technique overcomes the defects in the traditional rigid limit equilibrium theory and offers a new way for studying the reliability problem of planar slip surfaces. Consequently, we applied the technique to a case of an open-pit mine and compared our results with that of the traditional approach. From the results we conclude that both methods are essentially consistent, but the reliability index calculated by the traditional model is lower than that from the catastrophic model. The catastrophe model takes into consideration two possible situations of a slope being in the limit equilibrium condition, i.e., it may or may not slip. In the traditional method, however, a slope is definitely considered as slipping when it meets the condition of a limit equilibrium. We conclude that the catastrophe model has more actual and instructive importance compared to the traditional model.展开更多
Cusp displacement catastrophe theory can be introduced to propose a new method about instability failure of the interbed for gas storage cavern in bedded salt in solution mining.We can calculate initial fracture drawi...Cusp displacement catastrophe theory can be introduced to propose a new method about instability failure of the interbed for gas storage cavern in bedded salt in solution mining.We can calculate initial fracture drawing pace of this interbed to obtain 2D and 3D gas storage shapes at this time.Moreover,Stability evaluation of strength reduction finite element method(FEM)based on this catastrophe theory can used to evaluate this interbed stability after initial fracture.A specific example is simulated to obtain the influence of the interbed depth,cavern internal pressure,and cavern building time on stability safety factor(SSF).The results indicate:the value of SSF will be lower with the increase of cavern building time in solution mining and the increase of interbed depth and also this value remains a rise with the increase of cavern internal pressure Especially,we can conclude that the second-fracture of the interbed may take place when this pressure is lower than 6 MPa or after 6 days later of the interbed after initial fracture.According to above analysis,some effective measures,namely elevating the tube up to the top of the interbed,or changing the circulation of in-and-out lines,can be introduced to avoid the negative effects when the secondfracture of the interbed may occur.展开更多
For the (2 + 1)-dimensional nonlinear dispersive Boussinesq equation, by using the bifurcation theory of planar dynamical systems to study its corresponding traveling wave system, the bifurcations and phase portraits ...For the (2 + 1)-dimensional nonlinear dispersive Boussinesq equation, by using the bifurcation theory of planar dynamical systems to study its corresponding traveling wave system, the bifurcations and phase portraits of the regular system are obtained. Under different parametric conditions, various sufficient conditions to guarantee the existence of analytical and non-analytical solutions of the singular system are given by using singular traveling wave theory. For certain special cases, some explicit and exact parametric representations of traveling wave solutions are derived such as analytical periodic waves and non-analytical periodic cusp waves. Further, two-dimensional wave plots of analytical periodic solutions and non-analytical periodic cusp wave solutions are drawn to visualize the dynamics of the equation.展开更多
During the monitoring engineering of landslides, the monitoring data of accumulated displacement are usually affected by the external factors. Therefore, the displacement curve always has step-like character, which br...During the monitoring engineering of landslides, the monitoring data of accumulated displacement are usually affected by the external factors. Therefore, the displacement curve always has step-like character, which brings some difficulties to the accurate prediction of landslides. In order to solve this problem, based on the wavelet analysis and cusp catastrophe, a new kind of analysis method is proposed in this article. First, Fourier transform method can be used to extract the frequency component of the curve of monitoring displacement. Second, the wavelet transform was adopted to inspect the breakpoints of signals, which can be used to analyze the cause of the occurrence of the step-like character in the curve of landslide monitoring. Based on the cusp catastrophe theory, a nonlinear dynamic model was established to conduct the simulation calculation of time forecasting of landslides. According to a case study of landslide, the periodical rainfall and reservoir level fluctuation are the main factors leading to the step-like changes in the curve of monitoring displacement. In addition, the results of simulation calculation are in agreement with the fact of local failure of landslides. This method can provide a new analysis way for the time prediction of landslides.展开更多
Solar wind charge exchange produces emissions in the soft X-ray energy range which can enable the study of near-Earth space regions such as the magnetopause,the magnetosheath and the polar cusps by remote sensing tech...Solar wind charge exchange produces emissions in the soft X-ray energy range which can enable the study of near-Earth space regions such as the magnetopause,the magnetosheath and the polar cusps by remote sensing techniques.The Solar wind Magnetosphere Ionosphere Link Explorer(SMILE)and Lunar Environment heliospheric X-ray Imager(LEXI)missions aim to obtain soft Xray images of near-Earth space thanks to their Soft X-ray Imager(SXI)instruments.While earlier modeling works have already simulated soft X-ray images as might be obtained by SMILE SXI during its mission,the numerical models used so far are all based on the magnetohydrodynamics description of the space plasma.To investigate the possible signatures of ion-kinetic-scale processes in soft Xray images,we use for the first time a global hybrid-Vlasov simulation of the geospace from the Vlasiator model.The simulation is driven by fast and tenuous solar wind conditions and purely southward interplanetary magnetic field.We first produce global X-ray images of the dayside near-Earth space by placing a virtual imaging satellite at two different locations,providing meridional and equatorial views.We then analyze regional features present in the images and show that they correspond to signatures in soft X-ray emissions of mirrormode wave structures in the magnetosheath and flux transfer events(FTEs)at the magnetopause.Our results suggest that,although the time scales associated with the motion of those transient phenomena will likely be significantly smaller than the integration time of the SMILE and LEXI imagers,mirror-mode structures and FTEs can cumulatively produce detectable signatures in the soft X-ray images.For instance,a local increase by 30%in the proton density at the dayside magnetopause resulting from the transit of multiple FTEs leads to a 12%enhancement in the line-of-sight-and time-integrated soft X-ray emissivity originating from this region.Likewise,a proton density increase by 14%in the magnetosheath associated with mirror-mode structures can resu展开更多
According to the engineering features of subgrade cave roof in karst region, the clamped beam model of subgrade cave roof in karst region was set up. Based on the catastrophe theory, the cusp catastrophe model for bea...According to the engineering features of subgrade cave roof in karst region, the clamped beam model of subgrade cave roof in karst region was set up. Based on the catastrophe theory, the cusp catastrophe model for bearing capacity of subgrade cave roof and safe thickness of subgrade cave roof in karst region was established. The necessary instability conditions of subgrade cave roof were deduced, and then the methods to determine safe thickness of cave roofs under piles and bearing capacity of subgrade cave roof were proposed. At the same time, a practical engineering project was applied to verifying this method, which has been proved successfu1ly. At last, the major factors that affect the stability on cave roof under pile in karst region were deeply discussed and some results in quality were acquired.展开更多
By means of CUSP model of catastrophe theory. this paper has studied thephysics process of rockburst occured on circular chamber. The present paper has nolonly described the instability process of rockburst more deepl...By means of CUSP model of catastrophe theory. this paper has studied thephysics process of rockburst occured on circular chamber. The present paper has nolonly described the instability process of rockburst more deeply. but also got the crilicaldepth of plastic softening area of chamber that is valuable in the controlling engineering of rockburst. the chamber displacement jump and energy liberation have been derived. the influence of rock parameters on the rockburst has been discussed .展开更多
Background: Perimembranous ventricular septal defect combined with right coronary cusp bulge generally should be treated with surgical thoracotomy, owing to the potential aortic regurgitation. However, the minimally i...Background: Perimembranous ventricular septal defect combined with right coronary cusp bulge generally should be treated with surgical thoracotomy, owing to the potential aortic regurgitation. However, the minimally invasive method of transcatheter closure has always attracted the attention of cardiologists and patients. The present study aimed to apply transcatheter occlusion in treating ventricular septal defect with right coronary cusp bulge and further evaluate the clinical effect through follow-up. Materials and methods: A total of 40 children diagnosed as having a ventricular septal defect with right coronary cusp bulge, examined using transthoracic echocardiography and cardiovascular angiography, were enrolled in this study. The ventricular septal defects were closed by placing occluders through transcatheter occlusion treatment. During the operation process, the children underwent angiography and transthoracic echocardiography examinations to check the position of the occlude and the extent of aortic regurgitation. The influence of occlusion on the conduction system was evaluated using a surface electrocardiogram. The children were followed up after their procedures. Results: All 40 patients were immediately and successfully occluded. Three patients with filament residual shunts were observed during the operations. No major surgical complications occurred during the perioperative period. During the follow-up period, the positions of all the occluders were good, the residual shunts in the three patients disappeared, and no new or aggravated aortic regurgitation occurred. Electrocardiogram did not reveal any atrioventricular blocks. Only one patient suffered from an incomplete right bundle branch block. Conclusions: Children diagnosed with ventricular septal defect combined with right coronary cusp bulge could be considered for transcatheter occlusion. With appropriate indications and methods, the effect may be favorable.展开更多
In order to detect fault exactly and quickly, cusp catastrophe theory is used to interpret 3D coal seismic data in this paper. By establishing a cusp model, seismic signal is transformed into standard form of cusp cat...In order to detect fault exactly and quickly, cusp catastrophe theory is used to interpret 3D coal seismic data in this paper. By establishing a cusp model, seismic signal is transformed into standard form of cusp catastrophe and catastrophe parameters, including time-domain catastrophe potential, time-domain catastrophe time, frequency-domain catastrophe potential and frequency- domain degree, are calculated. Catastrophe theory is used in 3D seismic structural interpretation in coal mine. The results show that the position of abnormality of the catastrophe parameter profile or curve is related to the location of fault, and the cusp catastrophe theory is effective to automatically pick up geology information and improve the interpretation precision in 3D seismic data.展开更多
Frequently observed throat auroras have been suggested to be correspondent to indentations on the subsolar magnetopause,but how these indentations can be generated is unknown yet.Based on analyzing the detailed observ...Frequently observed throat auroras have been suggested to be correspondent to indentations on the subsolar magnetopause,but how these indentations can be generated is unknown yet.Based on analyzing the detailed observational features of throat aurora,a conceptual model for generation of throat aurora is proposed.This model suggests that precipitation of a north-south aligned stripy diffuse aurora can lead to an ionospheric conductivity enhancement and thus produce a polarization electric field in dusk-to-dawn direction in the ionosphere.After mapping to the magnetosphere along closed field lines,this electric field can guide a magnetopause reconnection to develop inward the magnetosphere and result in a throat aurora.Because this model can comprehensively explain all the observational results that have been presented up to now,we argue that the assumption of ionospheric polarization electric field affecting magnetopause reconnection should be true and be worthy of further investigations.展开更多
Let f be a fixed Maass form for SL_3(Z)with Fourier coefficients A_(f)(m,n).Let g be a Maass cusp form for SL_2(G)with Laplace eigenvalue(1/4)+k^(2) and Fourier coefficientλ_(g)(n),or a holomorphic cusp form of even ...Let f be a fixed Maass form for SL_3(Z)with Fourier coefficients A_(f)(m,n).Let g be a Maass cusp form for SL_2(G)with Laplace eigenvalue(1/4)+k^(2) and Fourier coefficientλ_(g)(n),or a holomorphic cusp form of even weight k.Denote by S_(X)(f×g,α,β)a smoothly weighted sum of A_(f)(1,n)λ_(g)(n)e(αn~β)for X 0 are fixed real numbers.The subject matter of the present paper is to prove non-trivial bounds for a sum of S_(X)(f×g,α,β)over g as k tends to∞with X.These bounds for average provide insight for the corresponding resonance barriers toward the Hypothesis S as proposed by Iwaniec,Luo,and Sarnak.展开更多
Based on the physical analysis that the soil moisture and vegetation depend mainly on the precipitation and evaporation as well as the growth, decay and consumption of vegetation a nonlinear dynamic coupled system of ...Based on the physical analysis that the soil moisture and vegetation depend mainly on the precipitation and evaporation as well as the growth, decay and consumption of vegetation a nonlinear dynamic coupled system of soil moisture-vegetation is established. Using this model, the stabilities of the steady states of vegetation are analyzed. This paper focuses on the research of the vegetation catastrophe point which represents the transition between aridness and wetness to a great extent. It is shown that the catastrophe point of steady states of vegetation depends mainly on the rainfall P and saturation value υ0, which is selected to balance the growth and decay of vegetation. In addition, when the consumption of vegetation remains constant, the analytic solution of the vegetation equation is obtained.展开更多
Abstract Let λf（n） be the n-th normalized Fourier coefficient of a holomorphic Hecke eigenform f（z） ∈ Sk（F）, In this paper, we established nontrivial estimates for ∑n≤x λf（n^i）λf（n^j）,where 1≤ij≤4.
Wavelets are applied to a regression model with an additive stationary noise. By checking the empirical wavelet coefficients with significantly large absolute values across fine scale levels, the jump points are detec...Wavelets are applied to a regression model with an additive stationary noise. By checking the empirical wavelet coefficients with significantly large absolute values across fine scale levels, the jump points are detected first. Then the cusp points are identified by checking the wavelet coefficients with significantly large absolute values which are secondly large only to the previous wavelet coefficient across fine scale levels. All estimators are shown to be consistent.展开更多
基金financial support from Changjiang Scholars and Innovative Research Team in University, and research project of ‘SUST Spring Bud’
文摘A method of slope reliability analysis was developed by imposing a state equation on the limit equilibrium theory, given the basis of a fixed safety factor technique. Among the many problems of reliability analysis, the most important problem is to find a performance function. We have created a new method of building a limit state equation for planar slip surfaces by applying the mathematical cusp catastrophe theory. This new technique overcomes the defects in the traditional rigid limit equilibrium theory and offers a new way for studying the reliability problem of planar slip surfaces. Consequently, we applied the technique to a case of an open-pit mine and compared our results with that of the traditional approach. From the results we conclude that both methods are essentially consistent, but the reliability index calculated by the traditional model is lower than that from the catastrophic model. The catastrophe model takes into consideration two possible situations of a slope being in the limit equilibrium condition, i.e., it may or may not slip. In the traditional method, however, a slope is definitely considered as slipping when it meets the condition of a limit equilibrium. We conclude that the catastrophe model has more actual and instructive importance compared to the traditional model.
文摘Cusp displacement catastrophe theory can be introduced to propose a new method about instability failure of the interbed for gas storage cavern in bedded salt in solution mining.We can calculate initial fracture drawing pace of this interbed to obtain 2D and 3D gas storage shapes at this time.Moreover,Stability evaluation of strength reduction finite element method(FEM)based on this catastrophe theory can used to evaluate this interbed stability after initial fracture.A specific example is simulated to obtain the influence of the interbed depth,cavern internal pressure,and cavern building time on stability safety factor(SSF).The results indicate:the value of SSF will be lower with the increase of cavern building time in solution mining and the increase of interbed depth and also this value remains a rise with the increase of cavern internal pressure Especially,we can conclude that the second-fracture of the interbed may take place when this pressure is lower than 6 MPa or after 6 days later of the interbed after initial fracture.According to above analysis,some effective measures,namely elevating the tube up to the top of the interbed,or changing the circulation of in-and-out lines,can be introduced to avoid the negative effects when the secondfracture of the interbed may occur.
文摘For the (2 + 1)-dimensional nonlinear dispersive Boussinesq equation, by using the bifurcation theory of planar dynamical systems to study its corresponding traveling wave system, the bifurcations and phase portraits of the regular system are obtained. Under different parametric conditions, various sufficient conditions to guarantee the existence of analytical and non-analytical solutions of the singular system are given by using singular traveling wave theory. For certain special cases, some explicit and exact parametric representations of traveling wave solutions are derived such as analytical periodic waves and non-analytical periodic cusp waves. Further, two-dimensional wave plots of analytical periodic solutions and non-analytical periodic cusp wave solutions are drawn to visualize the dynamics of the equation.
基金supported by the National Natural Science Foundation of China (Nos. 40202028, 50609026)Postdoctors Foundation of China (No. 20060400256)Excellent Young Teacher Science and Technology Program of Faculty of Engi-neering, China University of Geosciences
文摘During the monitoring engineering of landslides, the monitoring data of accumulated displacement are usually affected by the external factors. Therefore, the displacement curve always has step-like character, which brings some difficulties to the accurate prediction of landslides. In order to solve this problem, based on the wavelet analysis and cusp catastrophe, a new kind of analysis method is proposed in this article. First, Fourier transform method can be used to extract the frequency component of the curve of monitoring displacement. Second, the wavelet transform was adopted to inspect the breakpoints of signals, which can be used to analyze the cause of the occurrence of the step-like character in the curve of landslide monitoring. Based on the cusp catastrophe theory, a nonlinear dynamic model was established to conduct the simulation calculation of time forecasting of landslides. According to a case study of landslide, the periodical rainfall and reservoir level fluctuation are the main factors leading to the step-like changes in the curve of monitoring displacement. In addition, the results of simulation calculation are in agreement with the fact of local failure of landslides. This method can provide a new analysis way for the time prediction of landslides.
基金the European Research Council for starting grant 200141-QuESpace,with which the Vlasiator model was developedconsolidator grant 682068-PRESTISSIMO awarded for further development of Vlasiator and its use in scientific investigations+4 种基金Academy of Finland grant numbers 338629-AERGELC’H,339756-KIMCHI,336805-FORESAIL,and 335554-ICT-SUNVACThe Academy of Finland also supported this work through the PROFI4 grant(grant number 3189131)support from the NASA grants,80NSSC20K1670 and 80MSFC20C0019the NASA GSFC FY23 IRADHIF funds。
文摘Solar wind charge exchange produces emissions in the soft X-ray energy range which can enable the study of near-Earth space regions such as the magnetopause,the magnetosheath and the polar cusps by remote sensing techniques.The Solar wind Magnetosphere Ionosphere Link Explorer(SMILE)and Lunar Environment heliospheric X-ray Imager(LEXI)missions aim to obtain soft Xray images of near-Earth space thanks to their Soft X-ray Imager(SXI)instruments.While earlier modeling works have already simulated soft X-ray images as might be obtained by SMILE SXI during its mission,the numerical models used so far are all based on the magnetohydrodynamics description of the space plasma.To investigate the possible signatures of ion-kinetic-scale processes in soft Xray images,we use for the first time a global hybrid-Vlasov simulation of the geospace from the Vlasiator model.The simulation is driven by fast and tenuous solar wind conditions and purely southward interplanetary magnetic field.We first produce global X-ray images of the dayside near-Earth space by placing a virtual imaging satellite at two different locations,providing meridional and equatorial views.We then analyze regional features present in the images and show that they correspond to signatures in soft X-ray emissions of mirrormode wave structures in the magnetosheath and flux transfer events(FTEs)at the magnetopause.Our results suggest that,although the time scales associated with the motion of those transient phenomena will likely be significantly smaller than the integration time of the SMILE and LEXI imagers,mirror-mode structures and FTEs can cumulatively produce detectable signatures in the soft X-ray images.For instance,a local increase by 30%in the proton density at the dayside magnetopause resulting from the transit of multiple FTEs leads to a 12%enhancement in the line-of-sight-and time-integrated soft X-ray emissivity originating from this region.Likewise,a proton density increase by 14%in the magnetosheath associated with mirror-mode structures can resu
基金Project(50378036) supported by the National Natural Science Foundation of China
文摘According to the engineering features of subgrade cave roof in karst region, the clamped beam model of subgrade cave roof in karst region was set up. Based on the catastrophe theory, the cusp catastrophe model for bearing capacity of subgrade cave roof and safe thickness of subgrade cave roof in karst region was established. The necessary instability conditions of subgrade cave roof were deduced, and then the methods to determine safe thickness of cave roofs under piles and bearing capacity of subgrade cave roof were proposed. At the same time, a practical engineering project was applied to verifying this method, which has been proved successfu1ly. At last, the major factors that affect the stability on cave roof under pile in karst region were deeply discussed and some results in quality were acquired.
文摘By means of CUSP model of catastrophe theory. this paper has studied thephysics process of rockburst occured on circular chamber. The present paper has nolonly described the instability process of rockburst more deeply. but also got the crilicaldepth of plastic softening area of chamber that is valuable in the controlling engineering of rockburst. the chamber displacement jump and energy liberation have been derived. the influence of rock parameters on the rockburst has been discussed .
基金supported by the National Science Foundation of China(Grant number 81470443)
文摘Background: Perimembranous ventricular septal defect combined with right coronary cusp bulge generally should be treated with surgical thoracotomy, owing to the potential aortic regurgitation. However, the minimally invasive method of transcatheter closure has always attracted the attention of cardiologists and patients. The present study aimed to apply transcatheter occlusion in treating ventricular septal defect with right coronary cusp bulge and further evaluate the clinical effect through follow-up. Materials and methods: A total of 40 children diagnosed as having a ventricular septal defect with right coronary cusp bulge, examined using transthoracic echocardiography and cardiovascular angiography, were enrolled in this study. The ventricular septal defects were closed by placing occluders through transcatheter occlusion treatment. During the operation process, the children underwent angiography and transthoracic echocardiography examinations to check the position of the occlude and the extent of aortic regurgitation. The influence of occlusion on the conduction system was evaluated using a surface electrocardiogram. The children were followed up after their procedures. Results: All 40 patients were immediately and successfully occluded. Three patients with filament residual shunts were observed during the operations. No major surgical complications occurred during the perioperative period. During the follow-up period, the positions of all the occluders were good, the residual shunts in the three patients disappeared, and no new or aggravated aortic regurgitation occurred. Electrocardiogram did not reveal any atrioventricular blocks. Only one patient suffered from an incomplete right bundle branch block. Conclusions: Children diagnosed with ventricular septal defect combined with right coronary cusp bulge could be considered for transcatheter occlusion. With appropriate indications and methods, the effect may be favorable.
文摘In order to detect fault exactly and quickly, cusp catastrophe theory is used to interpret 3D coal seismic data in this paper. By establishing a cusp model, seismic signal is transformed into standard form of cusp catastrophe and catastrophe parameters, including time-domain catastrophe potential, time-domain catastrophe time, frequency-domain catastrophe potential and frequency- domain degree, are calculated. Catastrophe theory is used in 3D seismic structural interpretation in coal mine. The results show that the position of abnormality of the catastrophe parameter profile or curve is related to the location of fault, and the cusp catastrophe theory is effective to automatically pick up geology information and improve the interpretation precision in 3D seismic data.
基金supported by the National Key R & D Program of China (Grant No. 2018YFC1407303)the National Natural Science Foundation of China (Grant Nos. 41774174, 41704159, 41831072 & 41431072)
文摘Frequently observed throat auroras have been suggested to be correspondent to indentations on the subsolar magnetopause,but how these indentations can be generated is unknown yet.Based on analyzing the detailed observational features of throat aurora,a conceptual model for generation of throat aurora is proposed.This model suggests that precipitation of a north-south aligned stripy diffuse aurora can lead to an ionospheric conductivity enhancement and thus produce a polarization electric field in dusk-to-dawn direction in the ionosphere.After mapping to the magnetosphere along closed field lines,this electric field can guide a magnetopause reconnection to develop inward the magnetosphere and result in a throat aurora.Because this model can comprehensively explain all the observational results that have been presented up to now,we argue that the assumption of ionospheric polarization electric field affecting magnetopause reconnection should be true and be worthy of further investigations.
文摘Let f be a fixed Maass form for SL_3(Z)with Fourier coefficients A_(f)(m,n).Let g be a Maass cusp form for SL_2(G)with Laplace eigenvalue(1/4)+k^(2) and Fourier coefficientλ_(g)(n),or a holomorphic cusp form of even weight k.Denote by S_(X)(f×g,α,β)a smoothly weighted sum of A_(f)(1,n)λ_(g)(n)e(αn~β)for X 0 are fixed real numbers.The subject matter of the present paper is to prove non-trivial bounds for a sum of S_(X)(f×g,α,β)over g as k tends to∞with X.These bounds for average provide insight for the corresponding resonance barriers toward the Hypothesis S as proposed by Iwaniec,Luo,and Sarnak.
基金Many thanks are due to the Ministry of Science and Technology of China for support through the special public welfare project under grant 2002DIB20070to the National Natural Science Foundation of China for Grant No.40305006
文摘Based on the physical analysis that the soil moisture and vegetation depend mainly on the precipitation and evaporation as well as the growth, decay and consumption of vegetation a nonlinear dynamic coupled system of soil moisture-vegetation is established. Using this model, the stabilities of the steady states of vegetation are analyzed. This paper focuses on the research of the vegetation catastrophe point which represents the transition between aridness and wetness to a great extent. It is shown that the catastrophe point of steady states of vegetation depends mainly on the rainfall P and saturation value υ0, which is selected to balance the growth and decay of vegetation. In addition, when the consumption of vegetation remains constant, the analytic solution of the vegetation equation is obtained.
基金Supported by National Natural Science Foundation of China(Grant No.11101249)
文摘Abstract Let λf（n） be the n-th normalized Fourier coefficient of a holomorphic Hecke eigenform f（z） ∈ Sk（F）, In this paper, we established nontrivial estimates for ∑n≤x λf（n^i）λf（n^j）,where 1≤ij≤4.
文摘Wavelets are applied to a regression model with an additive stationary noise. By checking the empirical wavelet coefficients with significantly large absolute values across fine scale levels, the jump points are detected first. Then the cusp points are identified by checking the wavelet coefficients with significantly large absolute values which are secondly large only to the previous wavelet coefficient across fine scale levels. All estimators are shown to be consistent.
