Although traditional gamma-gamma density(GGD)logging technology is widely utilized,its potential environmental risks have prompted the development of more environmentally friendly neutron-gamma density(NGD)logging tec...Although traditional gamma-gamma density(GGD)logging technology is widely utilized,its potential environmental risks have prompted the development of more environmentally friendly neutron-gamma density(NGD)logging technology.However,NGD measurements are influenced by both neutron and gamma radiations.In the logging environment,variations in the formation composition indicate different elemental compositions,which affect the neutron-gamma reaction cross-sections and gamma generation.Compared to traditional gamma sources such as Cs-137,these changes significantly affect the generation and transport of neutron-induced inelastic gamma rays and hinder accurate measurements.To address this,a novel method is proposed that incorporates the mass attenuation coefficient function to account for the effects of various lithologies and pore contents on gamma-ray attenuation,thereby achieving more accurate density measurements by clarifying the transport processes of inelastic gamma rays with varying energies and spatial distributions in varied logging environments.The proposed method avoids the complex correction of neutron transport and is verified through Monte Carlo simulations for its applicability across various lithologies and pore contents,demonstrating absolute density errors that are less than 0.02 g/cm^(3)in clean formations and indicating good accuracy.This study clarifies the NGD mechanism and provides theoretical guidance for the application of NGD logging methods.Further studies will be conducted on extreme environmental conditions and tool calibration.展开更多
The morphological distribution of absorbent in composites is equally important with absorbents for the overall electromagnetic properties,but it is often ignored.Herein,a comprehensive consideration including electrom...The morphological distribution of absorbent in composites is equally important with absorbents for the overall electromagnetic properties,but it is often ignored.Herein,a comprehensive consideration including electromagnetic component regulation,layered arrangement structure,and gradient concentration distribution was used to optimize impedance matching and enhance electromagnetic loss.On the microscale,the incorporation of magnetic Ni nanoparticles into MXene nanosheets(Ni@MXene)endows suitable intrinsic permittivity and permeability.On the macroscale,the layered arrangement of Ni@MXene increases the effective interaction area with electromagnetic waves,inducing multiple reflection/scattering effects.On this basis,according to the analysis of absorption,reflection,and transmission(A-R-T)power coefficients of layered composites,the gradient concentration distribution was constructed to realize the impedance matching at low-concentration surface layer,electromagnetic loss at middle concentration interlayer and microwave reflection at high-concentration bottom layer.Consequently,the layered gradient composite(LG5-10-15)achieves complete absorption coverage of X-band at thickness of 2.00-2.20 mm with RL_(min) of-68.67 dB at 9.85 GHz in 2.05 mm,which is 199.0%,12.6%,and 50.6%higher than non-layered,layered and layered descending gradient composites,respectively.Therefore,this work confirms the importance of layered gradient structure in improving absorption performance and broadens the design of high-performance microwave absorption materials.展开更多
结直肠癌(colorectal cancer, CRC)是全球范围内高发的恶性肿瘤之一,早期筛查是降低其发病率和死亡率的关键。近年来,基于表观遗传学的生物标志物研究为CRC筛查提供了新方向,其中SDC2 (Syndecan-2)基因甲基化检测因其高敏感性和特异性...结直肠癌(colorectal cancer, CRC)是全球范围内高发的恶性肿瘤之一,早期筛查是降低其发病率和死亡率的关键。近年来,基于表观遗传学的生物标志物研究为CRC筛查提供了新方向,其中SDC2 (Syndecan-2)基因甲基化检测因其高敏感性和特异性备受关注。现有研究表明,SDC2甲基化在CRC早期病变(如腺瘤和癌前息肉)中已呈现异常,且可通过非侵入性粪便DNA检测实现高效筛查,然而标准化检测流程的建立和成本效益的优化仍需进一步探索。本文综述了SDC2基因在CRC发生中的甲基化机制、检测技术的优化及其临床应用进展。未来,SDC2甲基化检测有望成为CRC早期筛查的重要工具。Colorectal cancer (CRC) is one of the highly prevalent malignant tumors worldwide, and early screening is the key to reducing its morbidity and mortality. In recent years, epigenetic-based biomarker studies have provided a new direction for CRC screening, among which the SDC2 (Syndecan-2) gene methylation assay has attracted much attention due to its high sensitivity and specificity. Existing studies have shown that SDC2 methylation has been aberrant in early CRC lesions (e.g., adenomas and precancerous polyps) and can be efficiently screened by non-invasive fecal DNA testing. Still, the establishment of a standardized testing process and the optimization of cost-effectiveness need to be further explored. This article reviews the methylation mechanism of the SDC2 gene in CRC development, the optimization of detection technology, and its clinical application progress. In the future, SDC2 methylation detection is expected to become an essential tool for early screening of CRC.展开更多
As the first gold mine discovered at the sea in China and the only coastal gold mine currently mined there,Sanshandao Gold Mine faces unique challenges.The mine's safety is under continual threat from its faulted ...As the first gold mine discovered at the sea in China and the only coastal gold mine currently mined there,Sanshandao Gold Mine faces unique challenges.The mine's safety is under continual threat from its faulted structure coupled with the overlying water.As the mining proceeds deeper,the risk of water inrush increases.The mine's maximum water yield reaches 15000 m3/day,which is attributable to water channels present in fault zones.Predominantly composed of soil–rock mixtures(SRM),these fault zones'seepage characteristics significantly impact water inrush risk.Consequently,investigating the seepage characteristics of SRM is of paramount importance.However,the existing literature mostly concentrates on a single stress state.Therefore,this study examined the characteristics of the permeability coefficient under three distinct stress states:osmotic,osmotic–uniaxial,and osmotic–triaxial pressure.The SRM samples utilized in this study were extracted from in situ fault zones and then reshaped in the laboratory.In addition,the micromechanical properties of the SRM samples were analyzed using computed tomography scanning.The findings reveal that the permeability coefficient is the highest under osmotic pressure and lowest under osmotic–triaxial pressure.The sensitivity coefficient shows a higher value when the rock block percentage ranges between 30%and 40%,but it falls below 1.0 when this percentage exceeds 50%under no confining pressure.Notably,rock block percentages of 40%and 60%represent the two peak points of the sensitivity coefficient under osmotic–triaxial pressure.However,SRM samples with a 40%rock block percentage consistently show the lowest permeability coefficient under all stress states.This study establishes that a power function can model the relationship between the permeability coefficient and osmotic pressure,while its relationship with axial pressure can be described using an exponential function.These insights are invaluable for developing water inrush prevention and control strategies in mining environments.展开更多
In order to solve the problem of the variable coefficient ordinary differen-tial equation on the bounded domain,the Lagrange interpolation method is used to approximate the exact solution of the equation,and the error...In order to solve the problem of the variable coefficient ordinary differen-tial equation on the bounded domain,the Lagrange interpolation method is used to approximate the exact solution of the equation,and the error between the numerical solution and the exact solution is obtained,and then compared with the error formed by the difference method,it is concluded that the Lagrange interpolation method is more effective in solving the variable coefficient ordinary differential equation.展开更多
BACKGROUND At present,the conventional methods for diagnosing cerebral edema in clinical practice are computed tomography(CT)and magnetic resonance imaging(MRI),which can evaluate the location and degree of peripheral...BACKGROUND At present,the conventional methods for diagnosing cerebral edema in clinical practice are computed tomography(CT)and magnetic resonance imaging(MRI),which can evaluate the location and degree of peripheral cerebral edema,but cannot realize quantification.When patients have symptoms of diffuse cerebral edema or high cranial pressure,CT or MRI often suggests that cerebral edema is lagging and cannot be dynamically monitored in real time.Intracranial pressure monitoring is the gold standard,but it is an invasive operation with high cost and complications.For clinical purposes,the ideal cerebral edema monitoring should be non-invasive,real-time,bedside,and continuous dynamic monitoring.The dis-turbance coefficient(DC)was used in this study to dynamically monitor the occu-rrence,development,and evolution of cerebral edema in patients with cerebral hemorrhage in real time,and review head CT or MRI to evaluate the development of the disease and guide further treatment,so as to improve the prognosis of patients with cerebral hemorrhage.AIM To offer a promising new approach for non-invasive adjuvant therapy in cerebral edema treatment.METHODS A total of 160 patients with hypertensive cerebral hemorrhage admitted to the Department of Neurosurgery,Second Affiliated Hospital of Xi’an Medical University from September 2018 to September 2019 were recruited.The patients were randomly divided into a control group(n=80)and an experimental group(n=80).Patients in the control group received conventional empirical treatment,while those in the experimental group were treated with mannitol dehydration under the guidance of DC.Subsequently,we compared the two groups with regards to the total dosage of mannitol,the total course of treatment,the incidence of complications,and prognosis.RESULTS The mean daily consumption of mannitol,the total course of treatment,and the mean hospitalization days were 362.7±117.7 mL,14.8±5.2 days,and 29.4±7.9 in the control group and 283.1±93.6 mL,11.8±4.2 days,and 23.9±8.3 in the experimental group(P<0.05).In the control group,there were 20 patients with pulmonary infection(25%),30 with electrolyte disturbance(37.5%),20 with renal impairment(25%),and 16 with stress ulcer(20%).In the experimental group,pulmonary infection occurred in 18 patients(22.5%),electrolyte disturbance in 6(7.5%),renal impairment in 2(2.5%),and stress ulcers in 15(18.8%)(P<0.05).According to the Glasgow coma scale score 6 months after discharge,the prognosis of the control group was good in 20 patients(25%),fair in 26(32.5%),and poor in 34(42.5%);the prognosis of the experimental group was good in 32(40%),fair in 36(45%),and poor in 12(15%)(P<0.05).CONCLUSION Using DC for non-invasive dynamic monitoring of cerebral edema demonstrates considerable clinical potential.It reduces mannitol dosage,treatment duration,complication rates,and hospital stays,ultimately lowering hospital-ization costs.Additionally,it improves overall patient prognosis,offering a promising new approach for non-invasive adjuvant therapy in cerebral edema treatment.展开更多
BACKGROUND Various stone factors can affect the net results of shock wave lithotripsy(SWL).Recently a new factor called variation coefficient of stone density(VCSD)is being considered to have an impact on stone free r...BACKGROUND Various stone factors can affect the net results of shock wave lithotripsy(SWL).Recently a new factor called variation coefficient of stone density(VCSD)is being considered to have an impact on stone free rates.AIM To assess the role of VCSD in determining success of SWL in urinary calculi.METHODS Charts review was utilized for collection of data variables.The patients were subjected to SWL,using an electromagnetic lithotripter.Mean stone density(MSD),stone heterogeneity index(SHI),and VCSD were calculated by generating regions of interest on computed tomography(CT)images.Role of these factors were determined by applying the relevant statistical tests for continuous and categorical variables and a P value of<0.05 was gauged to be statistically significant.RESULTS There were a total of 407 patients included in the analysis.The mean age of the subjects in this study was 38.89±14.61 years.In total,165 out of the 407 patients could not achieve stone free status.The successful group had a significantly lower stone volume as compared to the unsuccessful group(P<0.0001).Skin to stone distance was not dissimilar among the two groups(P=0.47).MSD was significantly lower in the successful group(P<0.0001).SHI and VCSD were both significantly higher in the successful group(P<0.0001).CONCLUSION VCSD,a useful CT based parameter,can be utilized to gauge stone fragility and hence the prediction of SWL outcomes.展开更多
Joint roughness coefficient(JRC)is the most commonly used parameter for quantifying surface roughness of rock discontinuities in practice.The system composed of multiple roughness statistical parameters to measure JRC...Joint roughness coefficient(JRC)is the most commonly used parameter for quantifying surface roughness of rock discontinuities in practice.The system composed of multiple roughness statistical parameters to measure JRC is a nonlinear system with a lot of overlapping information.In this paper,a dataset of eight roughness statistical parameters covering 112 digital joints is established.Then,the principal component analysis method is introduced to extract the significant information,which solves the information overlap problem of roughness characterization.Based on the two principal components of extracted features,the white shark optimizer algorithm was introduced to optimize the extreme gradient boosting model,and a new machine learning(ML)prediction model was established.The prediction accuracy of the new model and the other 17 models was measured using statistical metrics.The results show that the prediction result of the new model is more consistent with the real JRC value,with higher recognition accuracy and generalization ability.展开更多
Let f be a primitive holomorphic cusp form with even integral weight k≥2 for the full modular groupΓ=SL(2,Z)andλ_(sym^(j)f)(n)be the n-th coefficient of Dirichlet series of j-th symmetric L-function L(s,sym^(j)f)at...Let f be a primitive holomorphic cusp form with even integral weight k≥2 for the full modular groupΓ=SL(2,Z)andλ_(sym^(j)f)(n)be the n-th coefficient of Dirichlet series of j-th symmetric L-function L(s,sym^(j)f)attached to f.In this paper,we study the mean value distribution over a specific sparse sequence of positive integers of the following sum∑(a^(2)+b^(2)+c^(2)+d^(2)≤x(a,b,c,d)∈Z^(4))λ_(sym^(j))^(i)f(a^(2)+b^(2)+c^(2)+d^(2))where j≥2 is a given positive integer,i=2,3,4 andαis sufficiently large.We utilize Python programming to design algorithms for higher power conditions,combining Perron's formula,latest results of representations of natural integers as sums of squares,as well as analytic properties and subconvexity and convexity bounds of automorphic L-functions,to ensure the accuracy and verifiability of asymptotic formulas.The conclusion we obtained improves previous results and extends them to a more general settings.展开更多
基金supported by the National Natural Science Foundation of China(U23B20151 and 52171253).
文摘Although traditional gamma-gamma density(GGD)logging technology is widely utilized,its potential environmental risks have prompted the development of more environmentally friendly neutron-gamma density(NGD)logging technology.However,NGD measurements are influenced by both neutron and gamma radiations.In the logging environment,variations in the formation composition indicate different elemental compositions,which affect the neutron-gamma reaction cross-sections and gamma generation.Compared to traditional gamma sources such as Cs-137,these changes significantly affect the generation and transport of neutron-induced inelastic gamma rays and hinder accurate measurements.To address this,a novel method is proposed that incorporates the mass attenuation coefficient function to account for the effects of various lithologies and pore contents on gamma-ray attenuation,thereby achieving more accurate density measurements by clarifying the transport processes of inelastic gamma rays with varying energies and spatial distributions in varied logging environments.The proposed method avoids the complex correction of neutron transport and is verified through Monte Carlo simulations for its applicability across various lithologies and pore contents,demonstrating absolute density errors that are less than 0.02 g/cm^(3)in clean formations and indicating good accuracy.This study clarifies the NGD mechanism and provides theoretical guidance for the application of NGD logging methods.Further studies will be conducted on extreme environmental conditions and tool calibration.
基金support for this work by Key Research and Development Project of Henan Province(Grant.No.241111232300)the National Natural Science Foundation of China(Grant.No.52273085 and 52303113)the Open Fund of Yaoshan Laboratory(Grant.No.2024003).
文摘The morphological distribution of absorbent in composites is equally important with absorbents for the overall electromagnetic properties,but it is often ignored.Herein,a comprehensive consideration including electromagnetic component regulation,layered arrangement structure,and gradient concentration distribution was used to optimize impedance matching and enhance electromagnetic loss.On the microscale,the incorporation of magnetic Ni nanoparticles into MXene nanosheets(Ni@MXene)endows suitable intrinsic permittivity and permeability.On the macroscale,the layered arrangement of Ni@MXene increases the effective interaction area with electromagnetic waves,inducing multiple reflection/scattering effects.On this basis,according to the analysis of absorption,reflection,and transmission(A-R-T)power coefficients of layered composites,the gradient concentration distribution was constructed to realize the impedance matching at low-concentration surface layer,electromagnetic loss at middle concentration interlayer and microwave reflection at high-concentration bottom layer.Consequently,the layered gradient composite(LG5-10-15)achieves complete absorption coverage of X-band at thickness of 2.00-2.20 mm with RL_(min) of-68.67 dB at 9.85 GHz in 2.05 mm,which is 199.0%,12.6%,and 50.6%higher than non-layered,layered and layered descending gradient composites,respectively.Therefore,this work confirms the importance of layered gradient structure in improving absorption performance and broadens the design of high-performance microwave absorption materials.
文摘结直肠癌(colorectal cancer, CRC)是全球范围内高发的恶性肿瘤之一,早期筛查是降低其发病率和死亡率的关键。近年来,基于表观遗传学的生物标志物研究为CRC筛查提供了新方向,其中SDC2 (Syndecan-2)基因甲基化检测因其高敏感性和特异性备受关注。现有研究表明,SDC2甲基化在CRC早期病变(如腺瘤和癌前息肉)中已呈现异常,且可通过非侵入性粪便DNA检测实现高效筛查,然而标准化检测流程的建立和成本效益的优化仍需进一步探索。本文综述了SDC2基因在CRC发生中的甲基化机制、检测技术的优化及其临床应用进展。未来,SDC2甲基化检测有望成为CRC早期筛查的重要工具。Colorectal cancer (CRC) is one of the highly prevalent malignant tumors worldwide, and early screening is the key to reducing its morbidity and mortality. In recent years, epigenetic-based biomarker studies have provided a new direction for CRC screening, among which the SDC2 (Syndecan-2) gene methylation assay has attracted much attention due to its high sensitivity and specificity. Existing studies have shown that SDC2 methylation has been aberrant in early CRC lesions (e.g., adenomas and precancerous polyps) and can be efficiently screened by non-invasive fecal DNA testing. Still, the establishment of a standardized testing process and the optimization of cost-effectiveness need to be further explored. This article reviews the methylation mechanism of the SDC2 gene in CRC development, the optimization of detection technology, and its clinical application progress. In the future, SDC2 methylation detection is expected to become an essential tool for early screening of CRC.
基金State Key Research Development Program of China,Grant/Award Number:2021YFC3001301。
文摘As the first gold mine discovered at the sea in China and the only coastal gold mine currently mined there,Sanshandao Gold Mine faces unique challenges.The mine's safety is under continual threat from its faulted structure coupled with the overlying water.As the mining proceeds deeper,the risk of water inrush increases.The mine's maximum water yield reaches 15000 m3/day,which is attributable to water channels present in fault zones.Predominantly composed of soil–rock mixtures(SRM),these fault zones'seepage characteristics significantly impact water inrush risk.Consequently,investigating the seepage characteristics of SRM is of paramount importance.However,the existing literature mostly concentrates on a single stress state.Therefore,this study examined the characteristics of the permeability coefficient under three distinct stress states:osmotic,osmotic–uniaxial,and osmotic–triaxial pressure.The SRM samples utilized in this study were extracted from in situ fault zones and then reshaped in the laboratory.In addition,the micromechanical properties of the SRM samples were analyzed using computed tomography scanning.The findings reveal that the permeability coefficient is the highest under osmotic pressure and lowest under osmotic–triaxial pressure.The sensitivity coefficient shows a higher value when the rock block percentage ranges between 30%and 40%,but it falls below 1.0 when this percentage exceeds 50%under no confining pressure.Notably,rock block percentages of 40%and 60%represent the two peak points of the sensitivity coefficient under osmotic–triaxial pressure.However,SRM samples with a 40%rock block percentage consistently show the lowest permeability coefficient under all stress states.This study establishes that a power function can model the relationship between the permeability coefficient and osmotic pressure,while its relationship with axial pressure can be described using an exponential function.These insights are invaluable for developing water inrush prevention and control strategies in mining environments.
文摘In order to solve the problem of the variable coefficient ordinary differen-tial equation on the bounded domain,the Lagrange interpolation method is used to approximate the exact solution of the equation,and the error between the numerical solution and the exact solution is obtained,and then compared with the error formed by the difference method,it is concluded that the Lagrange interpolation method is more effective in solving the variable coefficient ordinary differential equation.
基金Supported by the Shaanxi Provincial Key Research and Development Plan Project,No.2020ZDLSF01-02.
文摘BACKGROUND At present,the conventional methods for diagnosing cerebral edema in clinical practice are computed tomography(CT)and magnetic resonance imaging(MRI),which can evaluate the location and degree of peripheral cerebral edema,but cannot realize quantification.When patients have symptoms of diffuse cerebral edema or high cranial pressure,CT or MRI often suggests that cerebral edema is lagging and cannot be dynamically monitored in real time.Intracranial pressure monitoring is the gold standard,but it is an invasive operation with high cost and complications.For clinical purposes,the ideal cerebral edema monitoring should be non-invasive,real-time,bedside,and continuous dynamic monitoring.The dis-turbance coefficient(DC)was used in this study to dynamically monitor the occu-rrence,development,and evolution of cerebral edema in patients with cerebral hemorrhage in real time,and review head CT or MRI to evaluate the development of the disease and guide further treatment,so as to improve the prognosis of patients with cerebral hemorrhage.AIM To offer a promising new approach for non-invasive adjuvant therapy in cerebral edema treatment.METHODS A total of 160 patients with hypertensive cerebral hemorrhage admitted to the Department of Neurosurgery,Second Affiliated Hospital of Xi’an Medical University from September 2018 to September 2019 were recruited.The patients were randomly divided into a control group(n=80)and an experimental group(n=80).Patients in the control group received conventional empirical treatment,while those in the experimental group were treated with mannitol dehydration under the guidance of DC.Subsequently,we compared the two groups with regards to the total dosage of mannitol,the total course of treatment,the incidence of complications,and prognosis.RESULTS The mean daily consumption of mannitol,the total course of treatment,and the mean hospitalization days were 362.7±117.7 mL,14.8±5.2 days,and 29.4±7.9 in the control group and 283.1±93.6 mL,11.8±4.2 days,and 23.9±8.3 in the experimental group(P<0.05).In the control group,there were 20 patients with pulmonary infection(25%),30 with electrolyte disturbance(37.5%),20 with renal impairment(25%),and 16 with stress ulcer(20%).In the experimental group,pulmonary infection occurred in 18 patients(22.5%),electrolyte disturbance in 6(7.5%),renal impairment in 2(2.5%),and stress ulcers in 15(18.8%)(P<0.05).According to the Glasgow coma scale score 6 months after discharge,the prognosis of the control group was good in 20 patients(25%),fair in 26(32.5%),and poor in 34(42.5%);the prognosis of the experimental group was good in 32(40%),fair in 36(45%),and poor in 12(15%)(P<0.05).CONCLUSION Using DC for non-invasive dynamic monitoring of cerebral edema demonstrates considerable clinical potential.It reduces mannitol dosage,treatment duration,complication rates,and hospital stays,ultimately lowering hospital-ization costs.Additionally,it improves overall patient prognosis,offering a promising new approach for non-invasive adjuvant therapy in cerebral edema treatment.
文摘BACKGROUND Various stone factors can affect the net results of shock wave lithotripsy(SWL).Recently a new factor called variation coefficient of stone density(VCSD)is being considered to have an impact on stone free rates.AIM To assess the role of VCSD in determining success of SWL in urinary calculi.METHODS Charts review was utilized for collection of data variables.The patients were subjected to SWL,using an electromagnetic lithotripter.Mean stone density(MSD),stone heterogeneity index(SHI),and VCSD were calculated by generating regions of interest on computed tomography(CT)images.Role of these factors were determined by applying the relevant statistical tests for continuous and categorical variables and a P value of<0.05 was gauged to be statistically significant.RESULTS There were a total of 407 patients included in the analysis.The mean age of the subjects in this study was 38.89±14.61 years.In total,165 out of the 407 patients could not achieve stone free status.The successful group had a significantly lower stone volume as compared to the unsuccessful group(P<0.0001).Skin to stone distance was not dissimilar among the two groups(P=0.47).MSD was significantly lower in the successful group(P<0.0001).SHI and VCSD were both significantly higher in the successful group(P<0.0001).CONCLUSION VCSD,a useful CT based parameter,can be utilized to gauge stone fragility and hence the prediction of SWL outcomes.
基金funding from the National Natural Science Foundation of China (Grant No.42277175)the pilot project of cooperation between the Ministry of Natural Resources and Hunan Province“Research and demonstration of key technologies for comprehensive remote sensing identification of geological hazards in typical regions of Hunan Province” (Grant No.2023ZRBSHZ056)the National Key Research and Development Program of China-2023 Key Special Project (Grant No.2023YFC2907400).
文摘Joint roughness coefficient(JRC)is the most commonly used parameter for quantifying surface roughness of rock discontinuities in practice.The system composed of multiple roughness statistical parameters to measure JRC is a nonlinear system with a lot of overlapping information.In this paper,a dataset of eight roughness statistical parameters covering 112 digital joints is established.Then,the principal component analysis method is introduced to extract the significant information,which solves the information overlap problem of roughness characterization.Based on the two principal components of extracted features,the white shark optimizer algorithm was introduced to optimize the extreme gradient boosting model,and a new machine learning(ML)prediction model was established.The prediction accuracy of the new model and the other 17 models was measured using statistical metrics.The results show that the prediction result of the new model is more consistent with the real JRC value,with higher recognition accuracy and generalization ability.
文摘Let f be a primitive holomorphic cusp form with even integral weight k≥2 for the full modular groupΓ=SL(2,Z)andλ_(sym^(j)f)(n)be the n-th coefficient of Dirichlet series of j-th symmetric L-function L(s,sym^(j)f)attached to f.In this paper,we study the mean value distribution over a specific sparse sequence of positive integers of the following sum∑(a^(2)+b^(2)+c^(2)+d^(2)≤x(a,b,c,d)∈Z^(4))λ_(sym^(j))^(i)f(a^(2)+b^(2)+c^(2)+d^(2))where j≥2 is a given positive integer,i=2,3,4 andαis sufficiently large.We utilize Python programming to design algorithms for higher power conditions,combining Perron's formula,latest results of representations of natural integers as sums of squares,as well as analytic properties and subconvexity and convexity bounds of automorphic L-functions,to ensure the accuracy and verifiability of asymptotic formulas.The conclusion we obtained improves previous results and extends them to a more general settings.
