On the mean values of Dedekind sums and Hardy sums

Huaning Liu 대한수학회 2009 대한수학회지 Vol.46 No.1

For a positive integer $k$ and an arbitrary integer $h$, the classical Dedekind sums $s(h, k)$ is defined by <수식> where <수식>J. B. Conrey et al proved that <수식> For m≥2, C. Jia reduced the error terms to O(k^{2m-1}). While for m=1, W. Zhang showed <수식> In this paper we give some formulae on the mean value of the Dedekind sums and and Hardy sums, and generalize the above results. For a positive integer $k$ and an arbitrary integer $h$, the classical Dedekind sums $s(h, k)$ is defined by <수식> where <수식>J. B. Conrey et al proved that <수식> For m≥2, C. Jia reduced the error terms to O(k^{2m-1}). While for m=1, W. Zhang showed <수식> In this paper we give some formulae on the mean value of the Dedekind sums and and Hardy sums, and generalize the above results.

On the mean values of the homogeneous Dedekind sums and Cochrane sums in short intervals

Huaning Liu 대한수학회 2007 대한수학회지 Vol.44 No.6

In this paper, we study the mean values of the homogeneousDedekind sums and Cochrane sums in short intervals1;p3and1;p4,and give some asymptotic formulae by using the mean values of the Dirich-let L-functions.x 1. IntroductionFor integersa and q > 0, the classical Dedekind sum is dened byS(a;q) =qXr=1rqarq;where((x)) =x [x]12; ifx is not an integer;0; ifx is an integer:The sum S(a;q) plays an important role in the transformation theory of theDedekind function (see [7] and Chapter 3 of [1] for details).J. B. Conrey, E. Fransen, R. Klein and C. Scott [3] studied the mean valuesof Dedekind sums and proved the following:Proposition 1.1.Suppose thatm is a given positive integer andk is anysuciently large integer. ThenkX 0h=1S2m(h;k) = fm(k)k122m+ Ok9=5 + k2m 1+1 =(m+1)log3 k;Received November 4, 2005; Revised September 10, 2007.2000 Mathematics Subject Classication.11F20.Key words and phrases. Dedekind sums, Cochrane sums, homogeneous, mean values.Research partially supported by the National Natural Science Foundation of China un-der Grant No.60472068 and No.10671155; Natural Science Foundation of Shaanxi provinceof China under Grant No.2006A04; and the Natural Science Foundation of the Education

Hybrid mean value of generalized Bernoulli numbers, general Kloosterman sums and Gauss sums

Huaning Liu,Wenpeng Zhang 대한수학회 2007 대한수학회지 Vol.44 No.1

The main purpose of this paper is to use the properties ofprimitive characters, Gaus ums and Ramanujan’s um to stu dy thehybrid mean value of generalized Bernouli numbers, general Klostermansums and Gaus ums, and give two asymptotic formulae.

ON THE MEAN VALUES OF DEDEKIND SUMS AND HARDY SUMS

Liu, Huaning Korean Mathematical Society 2009 대한수학회지 Vol.46 No.1

ON THE MEAN VALUES OF THE HOMOGENEOUS DEDEKIND SUMS AND COCHRANE SUMS IN SHORT INTERVALS

Liu, Huaning Korean Mathematical Society 2007 대한수학회지 Vol.44 No.6

In this paper, we study the mean values of the homogeneous Dedekind sums and Cochrane sums in short intervals $[1,\;\frac{p}{3}]\;and\;[1,\;\frac{p}{4}]$, and give some asymptotic formulae by using the mean values of the Dirichlet L-functions.

HYBRID MEAN VALUE OF GENERALIZED BERNOULLI NUMBERS, GENERAL KLOOSTERMAN SUMS AND GAUSS SUMS

Liu, Huaning,Zhang, Wenpeng Korean Mathematical Society 2007 대한수학회지 Vol.44 No.1

The main purpose of this paper is to use the properties of primitive characters, Gauss sums and Ramanujan's sum to study the hybrid mean value of generalized Bernoulli numbers, general Kloosterman sums and Gauss sums, and give two asymptotic formulae.

Chlorophyll Fluorescence Characteristics and Rapid Light Response Curves of Alpine Rhododendron

Yuan-Huan Liu,Fang-Li Liu,Bo Long,Xiong-Li Zhou,Xue Zhang,Yue Zhang,Wen-Li Wang,Shi-Kang Shen 한국원예학회 2019 원예과학기술지 Vol.37 No.4

The aim of this study was to determine the photosynthetic adaptability of Rhododendron species toalpine environments. The chlorophyll fluorescence parameters and rapid light response curves ofeight Rhododendron species were determined under field conditions across elevation gradients (atelevations of 2,950, 3,560, 3,660, 3,770, and 4,030 m) in the Jiaozi Mountain National NaturalReserve, Yunnan Province, southwestern China. The effect of different elevations, species, and theirinteractions significantly affected most of the chlorophyll fluorescence and rapid light response curveparameters. The variable to maximum fluorescence ratio (Fv/Fm) ranged from 0.78 to 0.81 at the fiveelevation gradients. This result indicated that the studied species were well grown and adapted to thecurrent environment. The correlation analysis indicated that the elevation was positively significantlycorrelated with the photochemical efficiency of photosystem II, electron transport rate, maximumelectron transport rate, light saturation coefficient (Ek), and chlorophyll relative content (SPAD: leafchlorophyll content index) and was negatively significantly correlated with photochemicalquenching, nonphotochemical quenching, and linear initial slope values. Although no significantrelationship was observed between the elevations and Fv/Fm, the apparent difference in Fv/Fm both atelevation gradients and elevation × species levels indicated that the Rhododendron speciesdemonstrated species-specific adaptation to the environment at different elevations. Our resultsprovided evidence that Rhododendron species exhibit variations in photosynthetic activities in analpine environment at different elevations. These differences may improve the understanding of thephysiological adaptation variations of Rhododendron species across elevation gradients in associationwith climate change in the mountains of southwestern China.

Feasibility of a Clinical-Radiomics Model to Predict the Outcomes of Acute Ischemic Stroke

Zhou Yiran,Wu Di,Yan Su,Xie Yan,Zhang Shun,Lv Wenzhi,Qin Yuanyuan,Liu Yufei,Liu Chengxia,Lu Jun,Li Jia,Zhu Hongquan,Liu Weiyin Vivian,Liu Huan,Zhang Guiling,Zhu Wenzhen 대한영상의학회 2022 Korean Journal of Radiology Vol.23 No.8

Objective: To develop a model incorporating radiomic features and clinical factors to accurately predict acute ischemic stroke (AIS) outcomes. Materials and Methods: Data from 522 AIS patients (382 male [73.2%]; mean age ± standard deviation, 58.9 ± 11.5 years) were randomly divided into the training (n = 311) and validation cohorts (n = 211). According to the modified Rankin Scale (mRS) at 6 months after hospital discharge, prognosis was dichotomized into good (mRS ≤ 2) and poor (mRS > 2); 1310 radiomics features were extracted from diffusion-weighted imaging and apparent diffusion coefficient maps. The minimum redundancy maximum relevance algorithm and the least absolute shrinkage and selection operator logistic regression method were implemented to select the features and establish a radiomics model. Univariable and multivariable logistic regression analyses were performed to identify the clinical factors and construct a clinical model. Ultimately, a multivariable logistic regression analysis incorporating independent clinical factors and radiomics score was implemented to establish the final combined prediction model using a backward step-down selection procedure, and a clinical-radiomics nomogram was developed. The models were evaluated using calibration, receiver operating characteristic (ROC), and decision curve analyses. Results: Age, sex, stroke history, diabetes, baseline mRS, baseline National Institutes of Health Stroke Scale score, and radiomics score were independent predictors of AIS outcomes. The area under the ROC curve of the clinical-radiomics model was 0.868 (95% confidence interval, 0.825–0.910) in the training cohort and 0.890 (0.844–0.936) in the validation cohort, which was significantly larger than that of the clinical or radiomics models. The clinical radiomics nomogram was well calibrated (p > 0.05). The decision curve analysis indicated its clinical usefulness. Conclusion: The clinical-radiomics model outperformed individual clinical or radiomics models and achieved satisfactory performance in predicting AIS outcome

A Miniaturized Catadioptric Laser-Irradiation-Precision Test System

Huan Liu,Hao Sun,Chunyan Wang 한국광학회 2021 Current Optics and Photonics Vol.5 No.2

In this paper a catadioptric laser-irradiation-precision test system is designed, to achieve a high-precision laser-irradiation-accuracy test. In this system, we adopt the method of imaging the entire target surface at a certain distance to realize the measurement of laser-irradiation precision. The method possesses the advantages of convenient operation, high sensitivity, and good stability. To meet the test accuracy requirement of 100 mm/km (0.01%), the coma, field curvature, and distortion over the entire field of view should be eliminated from the optical system’s design. Taking into account the whole length of the tube and the influence of stray light on the structure type, a catadioptric system with a hood added near the primary imaging surface is designed. After optimization using the ZEMAX software, the modulation transfer function (MTF) of the designed optical system is 0.6 at 30 lp/mm, the full-field-of-view distor-tion is better than 0.18%, and the energy concentration in the 10-μm-radius surrounding circle reachesabout 90%. The illumination-accuracy test results show that the measurement accuracy of the radiation hit rate is better than 50 mm when the test distance is 1 km, which is better than the requirement of 100 mm/km for the laser-irradiation-accuracy test.

PREPARATION AND CHARACTERIZATION OF SILVER NANOPARTICLES/CARBON NANOFIBERS VIA ELECTROSPINNING WITH RESEARCH ON THEIR CATALYTIC PROPERTIES

Huan Liu,Jie Bai,Chunping Li,Shan Wang,Yarong Huang,QI WANG,WEIYAN SUN 성균관대학교(자연과학캠퍼스) 성균나노과학기술원 2014 NANO Vol.9 No.3

Ag nanoparticles (Ag NPs) embedded carbon nano¯bers (CNFs) were prepared by a new routewhich included chemical reaction process, electrospinning and calcination technique. The mor-phology and structure of the composite nano¯bers were investigated by scanning electron mi-croscopy, energy dispersive X-ray spectroscopy, transmission electron microscopy and X-raydi®raction. It indicated that Ag NPs were uniformly distributed in the CNFs. This e®ectivesynthesis method can be used to prepare other composite nano¯bers with functionality. The AgNPs/CNFs that served as supported catalysts were used in the styrene epoxidation by TBHP. The Ag NPs/CNFs catalyst showed its highly catalytic activity for the epoxidation of styrene(conversion: 40.6%, SO selectivity: 35.9%). This kind of composite nano¯ber membrane wasproved e®ectively catalytic activity and recycled easily in the styrene epoxidation.