Some comments on DAE theory for IRK methods

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Sci收录机械工程期刊

刊名简称ACTA MECH SINICA-PRC ADV APPL MECHADV VIB ENGAPPL THERM ENG ARCH CIV MECH ENG ASHRAE J ATOMIZATION SPRAY BWK-ENERGIE-FACHMAG CHINA OCEAN ENG CHIN J MECH ENG-EN DRY TECHNOLENG FAIL ANALENG COMPUT-GERMANY EXP HEAT TRANSFER EXP TECHNIQUESEXP THERM FLUID SCI EXP FLUIDSFATIGUE FRACT ENG M FLOW MEAS INSTRUM FORSCH INGENIEURWES HEAT TRANSFER ENG HVAC&R RESIEEE-ASME T MECH IND LUBR TRIBOLINT J AUTO TECH-KOR INT J CRASHWORTHINES INT J ENGINE RESINT J FATIGUEINT J HEAT FLUID FL INT J HEAT MASS TRAN INT J HEAVY VEH SYST INT J IMPACT ENGINT J MACH TOOL MANU INT J MECH SCIP LINN SOC N S WINT J OPTOMECHATRONI INT J PLASTICITYINT J PRECIS ENG MAN INT J PRES VES PIP INT J REFRIGINT J STRUCT STAB DY INT J SURF SCI ENG INT J THERM SCIINT J VEHICLE DESISI BILIM TEK DERGJ ADV MECH DES SYST J COMPUT NONLIN DYN J ELECTRON PACKAGINGJ ENG GAS TURB POWER J ENG MATER-T ASME J ENG MECH-ASCEJ ENG THERMOPHYS-RUS J ENHANC HEAT TRANSF J FLUID STRUCTJ FLUID ENG-T ASME J FRICT WEAR+J HEAT TRANS-T ASME J HYDRAUL ENG-ASCE J JPN SOC TRIBOLOGIS J MANUF SCI E-T ASME J MECH DESIGNJ MECH SCI TECHNOL J MICROELECTROMECH S J OFFSHORE MECH ARCT J POROUS MEDIAJ PRESS VESS-T ASME J SANDW STRUCT MATER J SOL ENERG-T ASME J SOUND VIBJ STRAIN ANAL ENGJ BALK TRIBOL ASSOC J BRAZ SOC MECH SCI J THERM SCIJ THERMOPHYS HEAT TR J TRIBOL-T ASMEJ TURBOMACHJ VIB ACOUSTJ VIB CONTROLJ VIBROENGLAT AM J SOLIDS STRU LUBR SCIMACH SCI TECHNOL MEC INDMECH ENGMECH SYST SIGNAL PR MECH MACH THEORY MECHATRONICS NANOSC MICROSC THERM NONLINEAR DYNAM PROBABILIST ENG MECH P COMBUST INSTP I MECH ENG A-J POW P I MECH ENG B-J ENG P I MECH ENG C-J MEC P I MECH ENG D-J AUT P I MECH ENG E-J PRO P I MECH ENG F-J RAIP I MECH ENG G-J AER P I MECH ENG J-J ENG P I MECH ENG K-J MUL PROF ENGPROG ENERG COMBUST RAPID PROTOTYPING J SCI IRANSHOCK VIBSMART STRUCT SYST SOUND VIB STROJARSTVOSTROJ VESTN-J MECH E STRUCT ENG MECH STRUCT INFRASTRUCT E THEOR APPL FRACT MEC T FAMENAT CAN SOC MECH ENG TRIBOL LUBR TECHNOL TRIBOL INTTRIBOL LETTTRIBOL TVEHICLE SYST DYN WEARWIND ENERGY刊名全称ACTA MECHANICA SINICAADVANCES IN APPLIED MECHANICSAdvances in Vibration EngineeringAPPLIED THERMAL ENGINEERINGArchives of Civil and Mechanical EngineeringASHRAE JOURNALATOMIZATION AND SPRAYSBWKCHINA OCEAN ENGINEERINGChinese Journal of Mechanical EngineeringDRYING TECHNOLOGYENGINEERING FAILURE ANALYSISENGINEERING WITH COMPUTERSEXPERIMENTAL HEAT TRANSFEREXPERIMENTAL TECHNIQUESEXPERIMENTAL THERMAL AND FLUID SCIENCEEXPERIMENTS IN FLUIDSFATIGUE & FRACTURE OF ENGINEERING MATERIALS & STRUCTURESFLOW MEASUREMENT AND INSTRUMENTATIONFORSCHUNG IM INGENIEURWESEN-ENGINEERING RESEARCHHEAT TRANSFER ENGINEERINGHVAC&R RESEARCHIEEE-ASME TRANSACTIONS ON MECHATRONICSINDUSTRIAL LUBRICATION AND TRIBOLOGYINTERNATIONAL JOURNAL OF AUTOMOTIVE TECHNOLOGYINTERNATIONAL JOURNAL OF CRASHWORTHINESSInternational Journal of Engine ResearchINTERNATIONAL JOURNAL OF FATIGUEINTERNATIONAL JOURNAL OF HEAT AND FLUID FLOWINTERNATIONAL JOURNAL OF HEAT AND MASS TRANSFERInternational Journal of Heavy Vehicle SystemsINTERNATIONAL JOURNAL OF IMPACT ENGINEERINGINTERNATIONAL JOURNAL OF MACHINE TOOLS & MANUFACTUREINTERNATIONAL JOURNAL OF MECHANICAL SCIENCESINTERNATIONAL JOURNAL OF OFFSHORE AND POLAR ENGINEERINGInternational Journal of OptomechatronicsINTERNATIONAL JOURNAL OF PLASTICITYInternational Journal of Precision Engineering and Manufacturing INTERNATIONAL JOURNAL OF PRESSURE VESSELS AND PIPINGINTERNATIONAL JOURNAL OF REFRIGERATION-REVUE INTERNATIONALE DU FROID International Journal of Structural Stability and DynamicsInternational Journal of Surface Science and Engineering INTERNATIONAL JOURNAL OF THERMAL SCIENCESINTERNATIONAL JOURNAL OF VEHICLE DESIGNISI Bilimi ve Teknigi Dergisi-Journal of Thermal Science and Technology Journal of Advanced Mechanical Design Systems and ManufacturingJournal of Computational and Nonlinear DynamicsJOURNAL OF ELECTRONIC PACKAGINGJOURNAL OF ENGINEERING FOR GAS TURBINES AND POWER-TRANSACTIONS OF THE A JOURNAL OF ENGINEERING MATERIALS AND TECHNOLOGY-TRANSACTIONS OF THE ASM JOURNAL OF ENGINEERING MECHANICS-ASCEJournal of Engineering ThermophysicsJOURNAL OF ENHANCED HEAT TRANSFERJOURNAL OF FLUIDS AND STRUCTURESJOURNAL OF FLUIDS ENGINEERING-TRANSACTIONS OF THE ASMEJournal of Friction and WearJOURNAL OF HEAT TRANSFER-TRANSACTIONS OF THE ASMEJOURNAL OF HYDRAULIC ENGINEERING-ASCEJOURNAL OF JAPANESE SOCIETY OF TRIBOLOGISTSJOURNAL OF MANUFACTURING SCIENCE AND ENGINEERING-TRANSACTIONS OF THE AS JOURNAL OF MECHANICAL DESIGNJournal of Mechanical Science and TechnologyJOURNAL OF MICROELECTROMECHANICAL SYSTEMSJOURNAL OF OFFSHORE MECHANICS AND ARCTIC ENGINEERING-TRANSACTIONS OF TH JOURNAL OF POROUS MEDIAJOURNAL OF PRESSURE VESSEL TECHNOLOGY-TRANSACTIONS OF THE ASME JOURNAL OF SANDWICH STRUCTURES & MATERIALSJOURNAL OF SOLAR ENERGY ENGINEERING-TRANSACTIONS OF THE ASMEJOURNAL OF SOUND AND VIBRATIONJOURNAL OF STRAIN ANALYSIS FOR ENGINEERING DESIGNJournal of the Balkan Tribological AssociationJournal of the Brazilian Society of Mechanical Sciences and Engineering JOURNAL OF THE CHINESE SOCIETY OF MECHANICAL ENGINEERSJournal of Thermal ScienceJOURNAL OF THERMOPHYSICS AND HEAT TRANSFERJOURNAL OF TRIBOLOGY-TRANSACTIONS OF THE ASMEJOURNAL OF TURBOMACHINERY-TRANSACTIONS OF THE ASMEJOURNAL OF VIBRATION AND ACOUSTICS-TRANSACTIONS OF THE ASMEJOURNAL OF VIBRATION AND CONTROLJournal of VibroengineeringLatin American Journal of Solids and StructuresLubrication ScienceMACHINING SCIENCE AND TECHNOLOGYMecanique & IndustriesMECHANICAL ENGINEERINGMECHANICAL SYSTEMS AND SIGNAL PROCESSINGMECHANISM AND MACHINE THEORYMECHATRONICSNanoscale and Microscale Thermophysical EngineeringNONLINEAR DYNAMICSPROBABILISTIC ENGINEERING MECHANICSPROCEEDINGS OF THE COMBUSTION INSTITUTEPROCEEDINGS OF THE INSTITUTION OF MECHANICAL ENGINEERS PART A-JOURNAL O PROCEEDINGS OF THE INSTITUTION OF MECHANICAL ENGINEERS PART B-JOURNAL O PROCEEDINGS OF THE INSTITUTION OF MECHANICAL ENGINEERS PART C-JOURNAL O PROCEEDINGS OF THE INSTITUTION OF MECHANICAL ENGINEERS PART D-JOURNAL O PROCEEDINGS OF THE INSTITUTION OF MECHANICAL ENGINEERS PART E-JOURNAL O PROCEEDINGS OF THE INSTITUTION OF MECHANICAL ENGINEERS PART F-JOURNAL OPROCEEDINGS OF THE INSTITUTION OF MECHANICAL ENGINEERS PART G-JOURNAL O PROCEEDINGS OF THE INSTITUTION OF MECHANICAL ENGINEERS PART J-JOURNAL O PROCEEDINGS OF THE INSTITUTION OF MECHANICAL ENGINEERS PART K-JOURNAL O PROFESSIONAL ENGINEERINGPROGRESS IN ENERGY AND COMBUSTION SCIENCERAPID PROTOTYPING JOURNALScientia IranicaSHOCK AND VIBRATIONSmart Structures and SystemsSOUND AND VIBRATIONSTROJARSTVOSTROJNISKI VESTNIK-JOURNAL OF MECHANICAL ENGINEERINGSTRUCTURAL ENGINEERING AND MECHANICSStructure and Infrastructure EngineeringTHEORETICAL AND APPLIED FRACTURE MECHANICSTransactions of FAMENATRANSACTIONS OF THE CANADIAN SOCIETY FOR MECHANICAL ENGINEERING TRIBOLOGY & LUBRICATION TECHNOLOGYTRIBOLOGY INTERNATIONALTRIBOLOGY LETTERSTRIBOLOGY TRANSACTIONSVEHICLE SYSTEM DYNAMICSWEARWIND ENERGY刊名中文翻译ISSN大类名称力学学报0567-7718物理应用力学进展0065-2156工程技术先进震动工程0972-5768工程技术实用热力工程1359-4311工程技术1644-9665工程——土木、机械、美国供暖、制冷与空调工程师学会志0001-2491工程技术-化学、机械、雾化与喷射1044-5110工程技术燃料、热能与动力1618-193X工程技术中国海洋工程0890-5487工程技术机械工程学报英文版1000-9345工程干燥技术0737-3937工程技术工程事故分析1350-6307工程技术计算机在工程中的应用0177-0667工程技术实验传热0891-6152工程技术实验技术0732-8818工程技术实验热力和流体科学0894-1777工程技术流体力学实验0723-4864工程技术工程材料及结构的疲劳与断裂8756-758X工程技术流动测量和仪表设备0955-5986工程技术工程研究0015-7899工程技术传热工程0145-7632工程技术国际供暖、通风、空调、制冷研究杂志1078-9669工程技术IEEE-ASME机械电子学汇刊1083-4435工程技术工业润滑与摩擦学0036-8792工程技术国际汽车技术杂志1229-9138工程技术国际防撞性能杂志1358-8265工程技术国际发动机研究杂志1468-0874工程技术国际疲劳杂志0142-1123工程技术国际热与流体流杂志0142-727X工程技术国际传热与传质杂志0017-9310工程技术国际重型机动车系统杂志1744-232X工程技术国际冲击工程杂志0734-743X工程技术国际机床与制造杂志0890-6955工程技术国际机械科学杂志0020-7403工程技术国际近海和极地工程师杂志0370-047X环境科学国际机电光杂志1559-9612工程技术国际塑性力学杂志0749-6419工程技术国际精密工程与制造杂志1229-8557工程技术国际压力容器与管道杂志0308-0161工程技术国际制冷杂志0140-7007工程技术国际结构稳定性与动力学杂志0219-4554工程技术国际表面科学与工程杂志1749-785X工程技术国际热科学杂志1290-0729工程技术国际机动车设计杂志0143-3369工程技术d Technology 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& SONS LTD；英国Quarterly 3是否为TOP期刊SCI/SCIE/EI目前内容2011年影响因子N Engineering, Computing & Technology0.86Y5N0.328N 2.064N0.855N0.392N0.526N SCI/SCIE Engineering, Computing & Technology0.056N SCIE Engineering, Computing & Technology0.468N SCIE0.208N SCI/SCIE Engineering, Computing & Technology 2.084N SCIE Engineering, Computing & Technology 1.086N SCIE0.739N SCIE0.537N SCIE0.257N SCIE 1.414N SCI/SCIE 1.735N SCI/SCIE0.847N SCIE0.8N SCIE0.268N SCIE0.892N SCIE0.683N SCI/SCIE 2.865N SCIE0.35N SCIE0.606N SCIE0.789N SCIE0.969N SCI/SCIE 1.546N SCI/SCIE 1.927Y SCI/SCIE 2.407N SCIE0.2N SCI/SCIE 1.701N SCI/SCIE 2.169N SCI/SCIE 1.231 SCIEN SCIE0.556Y SCI/SCIE 4.603N SCI/SCIEN SCIE0.989N SCI/SCIE 1.817N SCIE0.45N SCIE0.414N SCIE 2.142N SCIE0.457N SCIE0.229N SCIE0.205N SCIE0.827N SCIE0.694N SCI/SCIE0.679 N SCI/SCIE0.954 N SCI/SCIE0.99 N SCIE0.259 N SCIE0.275 N SCI/SCIE 1.567 N SCI/SCIE0.747 N SCIE0.409 N SCI/SCIE 1.83 N SCI/SCIE 1.429 N SCIE0.009 N SCIE0.727 N SCIE 1.017 N SCIE0.448 N SCI/SCIE 2.098 N SCIE0.427 N SCI/SCIE0.516 N SCI/SCIE0.516 N SCIE0.561 N SCI/SCIE0.846 N SCI/SCIE 1.588 N SCI/SCIE 1.085 N SCIE0.158 N SCIE0.2 SCIEN SCIE0.31 N SCI/SCIE0.739 N SCI/SCIE 1.196 N SCIE0.542 N SCIE 1.022 N SCIE1N SCIE0.346 N SCIE0.4 N SCIE0.53 N SCIE0.804 N SCIE0.221 N SCIE0.209 N SCIE 1.824 N SCI/SCIE 1.366 N SCI/SCIE 1.255 N SCI/SCIE 1.032 N SCI/SCIE 1.247 N SCI/SCIE 1.245 N SCIE 3.633 N SCIE0.7 N SCI/SCIE0.725 N SCI/SCIE0.473 N SCI/SCIE0.636 N SCI/SCIE0.393 N SCIE0.436N SCIE0.488 N SCIE0.733 N SCIE0.566 N SCIE0.000 0 Y SCI/SCIE14.22 N SCIE 1.023 N SCIE0.348 N SCIE0.26N SCIE 1.231 N SCIEN SCIEN SCIE0.398 N SCIE0.863 N SCIE0.966 N SCIE0.99N SCIE0.103 N SCIE0.188 N SCIE0.114 N SCI/SCIE 1.553 N SCI/SCIE 1.582 N SCI/SCIE0.854 N SCIE0.722 N SCI/SCIE 1.872 N SCIE2010年影响因子2009年影响因子2008年影响因子5年平均影响因子0.7490.8650.9390.8653.000 5.500 5.000 6.3570.1250.0000.0001.826 1.922 1.3492.3890.3830.5150.0000.4040.1880.2520.3630.9280.7540.4940.5780.0310.0990.0690.0310.3020.2600.4300.4030.1940.0000.0001.662 1.048 1.3932.1510.7700.9450.441 1.0540.6250.704 1.155 1.0000.4500.4880.5350.7980.5050.5000.2680.4341.267 1.234 1.037 1.6011.599 1.817 1.8542.0100.8940.8350.9340.9180.8080.7210.819 1.1770.3480.3910.1700.3080.9370.8410.792 1.1120.658 1.2240.637 1.0342.577 2.331 1.614 2.8520.3970.1780.2310.3620.5110.5170.5150.6150.6070.5530.4120.8520.9470.9030.0001.806 1.602 1.556 1.8221.802 1.498 1.3352.2471.899 1.947 1.8942.9130.2130.2220.2090.4001.522 1.301 1.398 1.7451.919 1.956 1.5762.5641.266 1.288 1.077 1.4950.6820.3540.0000.6245.082 4.791 3.875 4.7381.060 1.2070.0001.518 1.2560.862 1.2931.439 1.537 1.4582.0890.6440.7210.5370.6220.4140.5000.0000.3511.667 1.770 1.6832.3900.3580.4800.3890.5170.1790.2080.0000.2620.2100.0000.2590.5710.5570.000 1.1710.5820.7810.8270.8390.4820.6350.7350.8420.6950.8150.938 1.1070.9560.9800.792 1.1440.2090.0000.0000.4000.3140.4380.5481.482 1.256 1.380 1.7450.4400.4520.6280.9070.2040.0000.0000.9420.959 1.421 2.2861.227 1.478 1.272 1.8620.0330.0480.0610.0210.5670.5990.740 1.1170.6170.869 1.532 1.4830.4120.3740.2580.4012.157 1.922 2.226 2.5120.2770.2890.4470.5070.7070.6840.6120.6470.2930.3510.4490.4760.7730.5910.6460.7900.6440.8260.662 1.1641.334 1.414 1.364 1.7320.8970.7480.6260.9920.1610.1030.0000.2560.1210.0000.2120.1650.0000.2390.8230.6870.6470.7500.4490.7020.722 1.2640.3450.391 1.2970.9400.3900.7080.7280.9780.8630.8960.656 1.205 0.3230.3570.0000.5790.1880.0000.5880.0000.0000.4590.4330.6840.9090.1030.0650.0000.2500.2860.2770.2801.7622.075 1.984 2.4711.210 1.407 1.437 1.5400.944 1.198 1.434 1.4961.903 1.900 1.000 1.0101.741 1.658 1.295 1.4381.252 1.221 1.105 1.2451.797 3.256 1.906 3.5330.7990.6550.6090.7930.6990.4120.2810.7900.4510.4160.3170.5600.4410.4020.3420.7020.5200.4320.4630.4880.3890.4340.4070.6540.4800.7730.3660.5790.7210.6300.4900.8220.3460.3700.2790.7430.0300.0450.0710.00310.36211.0248.00017.1330.7200.662 1.086 1.285 0.2540.1230.0000.2600.0950.4650.4161.316 1.064 1.137 1.2480.1590.1890.3210.2220.0480.0000.4660.5330.2350.3740.4290.4380.5000.8120.5920.847 1.1910.9761.0730.771 1.011 1.0870.1430.2080.0000.2810.1950.1330.2280.1100.0690.1450.0981.560 1.690 1.423 1.8691.574 1.664 1.385 1.732 0.6690.3440.5780.7580.7520.6590.724 1.0191.635 1.771 1.5092.2601.716 1.342 1.271备注。

CORRESPONDENCE ADDRESS

DISCRIMINATIVE COMMON VECTORS FOR FACE RECOGNITION Hakan Cevikalp1, Marian Neamtu2, Mitch Wilkes1, and Atalay Barkana31Department of Electrical Engineering and Computer Science, Vanderbilt University, Nashville, Tennessee, USA.2Center for Constructive Approximation, Department of Mathematics, Vanderbilt University, Nashville, Tennessee, USA.3Department of Electrical and Electronics Engineering, Osmangazi University, Eskisehir, Turkey.CORRESPONDENCE ADDRESS:Prof. Mitch WilkesDepartment of Electrical Engineering and Computer Science,Vanderbilt University, Nashville, Tennessee, USATel: (615) 343-6016Fax: (615) 322-7062e-mail: mitch.wilkes@AbstractIn face recognition tasks, the dimension of the sample space is typically larger than the number of the samples in the training set. As a consequence, the within-class scatter matrix is singular and the Linear Discriminant Analysis (LDA) method cannot be applied directly. This problem is known as the “small sample size” problem. In this paper, we propose a new face recognition method called the Discriminative Common Vector method based on a variation of Fisher’s Linear Discriminant Analysis for the small sample size case. Two different algorithms are given to extract the discriminative common vectors representing each person in the training set of the face database. One algorithm uses the within-class scatter matrix of the samples in the training set while the other uses the subspace methods and the Gram-Schmidt orthogonalization procedure to obtain the discriminative common vectors. Then the discriminative common vectors are used for classification of new faces. The proposed method yields an optimal solution for maximizing the modified Fisher’s Linear Discriminant criterion given in the paper. Our test results show that the Discriminative Common Vector method is superior to other methods in terms of recognition accuracy, efficiency, and numerical stability.Index Terms: Common Vectors, Discriminative Common Vectors, Face Recognition, Fisher’s Linear Discriminant Analysis, Principal Component Analysis, Small Sample Size, Subspace Methods.I. INTRODUCTIONRecently, due to military, commercial, and law enforcement applications, there has been much interest in automatically recognizing faces in still and video images. This research spans several disciplines such as image processing, pattern recognition, computer vision and neural networks. The data come from a wide variety of sources. One group of sources is the relatively controlled format images such as passports, credit cards, photo ID’s, driver’s licenses, and mug shots. A more challenging class of application imagery includes real-time detection and recognition of faces in surveillance video images, which present additional constraints in terms of speed and processing requirements [1].Face recognition can be defined as the identification of individuals from images of their faces by using a stored database of faces labeled with people’s identities. This task is complex and can be decomposed into the smaller steps of detection of faces in a cluttered background, localization of these faces followed by extraction of features from the face regions, and finally recognition and verification [2]. It is a difficult problem as there are numerous factors such as 3-D pose, facial expression, hair style, make up, and so on, which affect the appearance of an individual’s facial features.In addition to these varying factors, lighting, background, and scale changes make this task even more challenging. Additional problematic conditions include noise, occlusion, and many other possible factors.Many methods have been proposed for face recognition within the last two decades [1], [3]. Among these methods, appearance-based approaches operate directly on images or appearances of face objects, and process the images as two-dimensional (2-D) holistic patterns. In these approaches, a two-dimensional image of size w by h pixels is represented by a vector in a wh-dimensional space. Therefore, each facial image corresponds to a point in this space. This spaceis called the sample space or the image space, and its dimension typically is very high [4]. However, since face images have similar structure, the image vectors are correlated, and any image in the sample space can be represented in a lower-dimensional subspace without losing a significant amount of information. The Eigenface method has been proposed for finding such a lower-dimensional subspace [5]. The key idea behind the Eigenface method, which uses Principal Component Analysis (PCA), is to find the best set of projection directions in the sample space that will maximize the total scatter across all images such that ||max arg )(W S W W J T T Wopt PCA = is maximized. Here T S is the total scatter matrix of the trainingset samples, and W is the matrix whose columns are the orthonormal projection vectors. The projection directions are also called the eigenfaces. Any face image in the sample space can be approximated by a linear combination of the significant eigenfaces. The sum of the eigenvalues that correspond to the eigenfaces not used in reconstruction gives the mean square error of reconstruction. This method is an unsupervised technique, since it does not consider the classes within the training set data. In choosing a criterion that maximizes the total scatter, this approach tends to model unwanted within-class variations such as those resulting from differences in lighting, facial expression, and other factors [6], [7]. Additionally, since the criterion does not attempt to minimize within-class variation, the resulting classes may tend to have more overlap than other approaches. Thus, the projection vectors chosen for optimal reconstruction may obscure the existence of the separate classes.The Linear Discriminant Analysis (LDA) method is proposed in [6] and [7]. This method overcomes the limitations of the Eigenface method by applying the Fisher’s Linear Discriminant criterion. This criterion tries to maximize the ratio ||||maxarg )(W S W W S W W J W T B T W opt FLD =, where B S isthe between-class scatter matrix, and W S is the within-class scatter matrix. Thus, by applying this method, we find the projection directions that on one hand maximize the Euclidean distance between the face images of different classes and on the other minimize the distance between the face images of the same class. This ratio is maximized when the column vectors of the projection matrix W are the eigenvectors of B W S S 1−. In face recognition tasks, this method cannot be applied directly since the dimension of the sample space is typically larger than the number of samples in the training set. As a consequence, W S is singular in this case. This problem is also known as the “small sample size problem” [8].In the last decade numerous methods have been proposed to solve this problem. Tian et al . [9] used the Pseudo-Inverse method by replacing 1−W S with its pseudo-inverse. The Perturbation method is used in [2] and [10], where a small perturbation matrix ∆ is added to W S in order to make it nonsingular. Cheng et al . [11] proposed the Rank Decomposition method based on successive eigen-decompositions of the total scatter matrix T S and the between-class scatter matrix B S . However, the above methods are typically computationally expensive since the scatter matrices are very large (e.g., images of size 256 by 256 yield scatter matrices of size 65,536 by 65,536). Swets and Weng [7] proposed a two stage PCA+LDA method, also known as the Fisherface method, in which PCA is first used for dimension reduction so as to make W S nonsingular before the application of LDA. In this method the final optimal projection vector matrix becomes FLD PCA opt W W W =, where ||max arg W S W W T T W PCA =, and||||max arg W W S W W W W S W W W PCA W T PCA T PCA B T PCA T WFLD =. However, in order to make W S nonsingular, some directions corresponding to the small eigenvalues of T S are thrown away in the PCA step. Thus,applying PCA for dimensionality reduction has the potential to remove dimensions that contain discriminative information [12]-[16]. Chen et al . [17] proposed the Null Space method based on the modified Fisher’s Linear Discriminant criterion, W S W W S W W J T T B T W opt MFLD maxarg )(=. Thismethod was proposed to be used when the dimension of the sample space is larger than the rank of the within-class scatter matrix, W S . It has been shown that the original Fisher’s Linear Discriminant criterion can be replaced by the modified Fisher’s Linear Discriminant criterion in the course of solving the discriminant vectors of the optimal set in [18]. In this method, all image samples are first projected onto the null space of W S , resulting in a new within-class scatter that is a zero matrix. Then, PCA is applied to the projected samples to obtain the optimal projection vectors. Chen et al . also proved that by applying this method, the modified Fisher’s Linear Discriminant criterion attains its maximum. However, they did not propose an efficient algorithm for applying this method in the original sample space. Instead, a pixel grouping method is applied to extract geometric features and reduce the dimension of the sample space. Then they applied the Null Space method in this new reduced space. In our experiments, we observed that the performance of the Null Space method depends on the dimension of the null space of W S in the sense that larger dimension provides better performance. Thus, any kind of pre-processing that reduces the original sample space should be avoided.Another novel method, the PCA+Null Space method was proposed by Huang et al . in [15] for dealing with the small sample size problem. In this method, at first, PCA is applied to remove the null space of T S , which contains the intersection of the null spaces of B S and W S . Then, the optimal projection vectors are found in the remaining lower-dimensional space by using the Null Space method. The difference between the Fisherface method and the PCA+Null Space methodis that for the latter, the within-class scatter matrix in the reduced space is typically singular. This occurs because all eigenvectors corresponding to the nonzero eigenvalues of T S are used for dimension reduction. Yang et al . applied a variation of this method in [16]. After dimensionreduction, they split the new within-class scatter matrix, PCA W T PCAW P S P S =~ (where PCA P is the matrix whose columns are the orthonormal eigenvectors corresponding to the nonzeroeigenvalues of T S ), into its null space },...,{)~(1t r W span S N ξξ+= and orthogonal complement(i.e., range space) },...,{)~(1r W span S R ξξ= (where r is the rank of W S , and )(T S rank t = is thedimension of the reduced space). Then, all the projection vectors that maximize the between-class scatter in the null space are chosen. If, according to some criterion, more projection vectors are needed, the remaining projection vectors are obtained from the range space. Although the PCA+Null Space method and the variation proposed by Yang et al ., use the original sample space, applying PCA and using all eigenvectors corresponding to the nonzero eigenvalues make these methods impractical for face recognition applications when the training set size is large. This is due to the fact that the computational expense of training becomes very large.Lastly, the Direct-LDA method is proposed in [12]. This method uses the simultaneous diagonalization method [8]. First, the null space of B S is removed, and then the projection vectors that minimize the within-class scatter in the transformed space are selected from the range space of B S . However, removing the null space of B S by dimensionality reduction will also remove part of the null space of W S and may result in the loss of important discriminative information [13], [15], [16]. Furthermore, B S is whitened as a part of this method. This whitening process can be shown to be redundant and therefore should be skipped.In this paper, a new method is proposed which addresses the limitations of other methods that use the null space ofS to find the optimal projection vectors. Thus, the proposed method canWbe only used when the dimension of the sample space is larger than the rank ofS. TheW remainder of the paper is organized as follows. In Section II, the Discriminative Common Vector approach is introduced. In Section III, we describe the data sets and experimental results. Finally, we formulate our conclusions in Section IV.II. DISCRIMINATIVE COMMON VECTOR APPROACHThe idea of common vectors was originally introduced for isolated word recognition problems in the case where the number of samples in each class was less than or equal to the dimensionality of the sample space [19], [20]. These approaches extract the common properties of classes in the training set by eliminating the differences of the samples in each class. A common vector for each individual class is obtained by removing all the features that are in the direction of the eigenvectors corresponding to the nonzero eigenvalues of the scatter matrix of its own class. The common vectors are then used for recognition. In our case instead of using a given class’s own scatter matrix, we use the within-class scatter matrix of all classes to obtain the common vectors. We also give an alternative algorithm based on the subspace methods and the Gram-Schmidt orthogonalization procedure to obtain the common vectors. Then, a new set of vectors, called the discriminative common vectors, which will be used for classification are obtained from the common vectors. We introduce algorithms for obtaining the common vectors and the discriminative common vectors below.A. Obtaining the Discriminative Common Vectors by Using the Null Space of W S Let the training set be composed of C classes, where each class contains N samples, and let i mx be a d -dimensional column vector which denotes the m-th sample from the i-th class. There will be a total of M=NC samples in the training set. Suppose that d>M-C. In this case, W S , B S , and T S are defined as,T i i m i C i N m i m W x x S ))((11µµ−−=∑∑==, (1)T i C i i B N S )()(1µµµµ−−=∑=, (2)andB W T i mC i Nm i m T S S x x S +=−−∑∑===))((11µµ, (3) where µ is the mean of all samples, and i µ is the mean of samples in the i -th class.In the special case where 0=w S w W T and 0≠w S w B T , for all }0{\d R w ∈, the modified Fisher’s Linear Discriminant criterion attains a maximum. However, a projection vector w , satisfying the above conditions, does not necessarily maximize the between-class scatter. In this case, a better criterion is given in [6] and [13], namely||max arg ||max arg )(0||0||W S W W S W W J T T W S W B T W S W opt W T W T ====. (4)To find the optimal projection vectors w in the null space of W S , we project the face samples onto the null space of W S and then obtain the projection vectors by performing PCA. To do so, vectors that span the null space of W S must first be computed. However, this task is computationally intractable since the dimension of this null space can be very large. A moreefficient way to accomplish this task is by using the orthogonal complement of the null space of W S , which typically is a significantly lower-dimensional space.Let d R be the original sample space, V be the range space of W S , and ⊥V be the null space of W S . Equivalently,},...,1,0|{r k S span V k W k =≠=αα (5)and},...,1,0|{d r k S span V k W k +===⊥αα, 6)where d r < is the rank of W S , },....,{1d αα is an orthonormal set, and },....,{1r αα is the set of orthonormal eigen vectors corresponding to the nonzero eigenvalues of W S .Consider the matrices ]....[1r Q αα= and ]....[1d r Q αα+=. Since⊥⊕=V V R d , every face image d i m R x ∈ has a unique decomposition of the formi m i m i m z y x +=, (7)where V x QQ Px y i m T i m i m ∈==, ⊥∈==V x Q Q x P z i m T i m i m , and P and P are the orthogonal projection operators onto V and ⊥V , respectively. Our goal is to computei m i m i m i m i m Px x y x z −=−=. (8)To do this, we need to find a basis for V , which can be accomplished by an eigen-analysis of W S . In particular, the normalized eigenvectors k α corresponding to the nonzero eigenvalues of W S will be an orthonormal basis for V. The eigenvectors can be obtained by calculating the eigenvectors of the smaller M by M matrix, A A T , defined such that T W AA S =, where A is a d by M matrix of the form]........[22111111C C N N x x x x A µµµµ−−−−=. (9)Let k λand k v be the k -th nonzero eigenvalue and the corresponding eigenvector of A A T , where C M k −≤. Then k k Av =α will be the eigenvector that corresponds to the k -th nonzero eigenvalue of W S . The sought-for projection onto ⊥V is achieved by using (8). In this way, it turns out, we obtain the same unique vector for all samples of the same class,i m T i m T i m i com x Q Q x QQ x x =−=, m=1,…,N , i=1,...,C , (10)i.e., the vector on the right-hand side of (10) is independent of the sample index m . We refer tothe vectors i com x as the common vectors. The above fact is proved in the following theorem.Theorem 1: Suppose Q is a matrix whose column vectors are the orthonormal vectors thatspan the null space ⊥V of W S . Then, the projections of the samples i mx of the class i onto ⊥V produce a unique common vector i com x such thati m T i com x Q Q x =, m =1,…,N , i=1,…,C . (11)Proof : By definition, a vector d R ∈α is in ⊥V if 0=αW S . Let i µ be the mean vector of the i -th class, G be the N by N matrix whose entries are all 1−N , and i X be the d by N matrix whosem -th column is the sample i m x . Thus, multiplying both sides of identity 0=αW S by T α andwriting∑==Ci i W S S 1, (12)withT i i i i T i i m i Nm i m i G X X G X X x x S ))(())((1−−=−−=∑=µµ, (13)immediately leads to 211||))((||)())((0ααα∑∑==−=−−=Ci T i T i T i C i T X G I X G I G I X , (14)where ||.|| denotes the Euclidean norm. Thus, (14) holds if 0))((=−k T i X G I α, or k T i k T i X G X αα)()(=. From this relation we can see that,d r k C i N m x k T i k T i m ,...,1,,...,1,, (1))()(+====αµα. (15) Thus, the projection of i m x onto ⊥V ,k k d r k i k k d r k i m i com x x ααµαα〉〈=〉〈=∑∑+=+=,,11, (16)is independent of m , which proves the theorem.The theorem states that it is enough to project a single sample from each class. This will greatly reduce the computational burden of the calculations. This computational savings has not been previously reported in the literature.After obtaining the common vectors i com x , optimal projection vectors will be those thatmaximize the total scatter of the common vectors,||max arg ||max arg ||max arg )(0||0||W S W W S W W S W W J com T WT T W S W B T W S W opt W T W T =====, (17)where W is a matrix whose columns are the orthonormal optimal projection vectors k w , and com S is the scatter matrix of the common vectors,T com i com com Ci i com com x x S ))((1µµ−−=∑=, i =1,…,C , (18) where com µ is the mean of all common vectors, ∑==C i i com com x C 11µ.In this case optimal projection vectors k w can be found by an eigen-analysis of com S . In particular, all eigenvectors corresponding to the nonzero eigenvalues of com S will be the optimal projection vectors. com S is typically a large d by d matrix and thus we can use the smaller matrix, com T com A A , of size C by C, to find nonzero eigenvalues and the corresponding eigenvectors ofT com com com A A S =, where com A is the d by C matrix of the form]....[1com C com com com com x x A µµ−−=. (19)There will be C -1 optimal projection vectors since the rank of com S is C -1 if all common vectors are linearly independent. If two common vectors are identical, then the two classes which are represented by this vector cannot be distinguished. Since the optimal projection vectors k wbelong to the null space of W S , it follows that when the image samples i m x of the i -th class areprojected onto the linear span of the projection vectors k w , the feature vectorT C i m i m i w x w x ],....,[11><><=Ω− of the projection coefficients ><k i m w x , will also be independent of the sample index m . Thus, we havei m T i x W =Ω, m=1,…,N, i=1,…,C. (20)We call the feature vectors i Ω discriminative common vectors , and they will be used for classification of face images. The fact that i Ω does not depend on the index m in (20) guarantees 100% accuracy in the recognition of the samples in the training set. This guarantee has not been reported in connection with other methods [15], [17].To recognize a test image test x , the feature vector of this test image is found bytest T test x W =Ω, (21)which is then compared with the discriminative common vector i Ω of each class using the Euclidean distance. The discriminative common vector found to be the closest to test Ω is used to identify the test image.Since test Ω is only compared to a single vector for each class, the recognition is very efficientfor real-time face recognition tasks. In the Eigenface, the Fisherface, and the Direct-LDA methods, the test sample feature vector test Ω is typically compared to all feature vectors ofsamples in the training set, making these methods impractical for real-time applications for large training sets.The above method can be summarized as follows:Step 1: Compute the nonzero eigenvalues and corresponding eigenvectors of W S by using the matrix A A T , where T W AA S = and A is given by (9). Set ]....[1r Q αα=, where r is therank of W S . Step 2: Choose any sample from each class and project it onto the null space of W S to obtain the common vectorsi m T i m i com x QQ x x −=, N m ,...,1=, C i ,...,1=. (22)Step 3: Compute the eigenvectors k w of com S , corresponding to the nonzero eigenvalues, byusing the matrix com T comA A , where T com com com A A S = and com A is given in (19). There are at most C -1 eigenvectors that correspond to the nonzero eigenvalues. Use these eigenvectors to form the projection matrix ]....[11−=C w w W , which will be used to obtain feature vectors in (20) and (21).B. Obtaining the Discriminative Common Vectors by Using Difference Subspaces and the Gram-Schmidt Orthogonalization ProcedureTo find an orthonormal basis for the range of W S , the algorithm described above uses the eigenvectors corresponding to the nonzero eigenvalues of the M by M matrix A A T , whereTW AA S =. Assuming that C M S rank W −=)(, then dC C M dM M M M l +−+−+)(2)234(233floating point operations (flops) are required to obtain an orthonormal basis set spanning the range of W S by using this approach. Here l represents the number of iterations required for convergence of the eigen-decomposition algorithm. However, the computations may become expensive and numerically unstable for large values of M . Since we do not need to find the eigenvalues (i.e., an explicit symmetric Schur decomposition) of W S , the following algorithm can be used for finding the common vectors efficiently. It requires only ))()(2(2C M d C M d −+− flops to find an orthonormal basis for the range of W S and is based on the subspace methods and the Gram-Schmidt orthogonalization procedure.Suppose that d >M-C . In this case, the subspace methods can be applied to obtain the commonvectors i com x for each class i . To do this, we choose any one of the image vectors from the i -th class as the subtrahend vector and then obtain the difference vectors i k b of the so-called difference subspace of the i -th class [20]. Thus, assuming that the first sample of each class istaken as the subtrahend vector, we have i i k i k x x b 11−=+, 1,...,1−=N k .The difference subspace i B of the i -th class is defined as },....,{11i N i i b b span B −=. Thesesubspaces can be summed up to form the complete difference subspace},....,,,....,{....12111111C N N C b b b b span B B B −−=++=. (23)The number of independent difference vectors i k b will be equal to the rank of W S . Forsimplicity, suppose there are M -C independent difference vectors. Since by Theorem 3, B and the range space V of W S , are the same spaces, the projection matrix onto B is the same as the matrix P (projection matrix onto the range space of W S ) defined previously in Section II-A. This matrix can be computed asT T D D D D P 1)(−=, (24)where ]........[1211111C N N b b b b D −−= is a d by M-C matrix [21]. This involvesfinding the inverse of an M -C by M -C nonsingular, positive definite symmetric matrix D D T . A computationally efficient method of applying the projection uses an orthonormal basis for B . Inparticular, the difference vectors i k b can be orthonormalized by using the Gram-Schmidtorthogonalization procedure to obtain orthonormal basis vectors C M −ββ,....,1. The complement of B is the indifference subspace ⊥B such that]....[1C M U −=ββ, T UU P =, (25) ]....[1d C M U ββ+−=, T U U P =, (26)where P and P are the orthogonal projection operators onto B and ⊥B , respectively. Thus matrices P and P are symmetric and idempotent, and satisfy I P P =+. Any sample from each class can now be projected onto the indifference subspace ⊥B to obtain the corresponding common vectors of the classes,.,...,1,,...,1,C i N m x UU x x U U Px x x P x i m T i m im T i mi m i m i com ==−==−== (27)The common vectors do not depend on the choice of the subtrahend vectors and they are identical to the common vectors obtained by using the null space of W S . This follows from Theorem 3 below, which uses the results of Lemma 1 and Theorem 2.Theorem 2: Let ⊥i V be the null space of the scatter matrix i S , and ⊥i B be the orthogonal complement of the difference subspace i B . Then ⊥⊥=i i B V and i i B V =.Proof : See [20].Lemma 1: Suppose that C S S ,....,1 are positive semi-definite scatter matrices. ThenI Ci i C S N S S N 11)()....(==++, (28) where N ( ) denotes the null space.Proof : The null space on the left-hand side of the above identity contains elements α such that0)....(1=++αC S S (29)or0....)....(11=++=++ααααααC T T C T S S S S , (30)by the positive semi-definiteness of C S S ++....1. Thus, again by the positive semi-definiteness, )....(1C S S N ++∈α if and only if0=ααi T S , i=1,...,C , (31)or, equivalently, I Ci i S N 1)(=∈α. Theorem 3: Let C S S ,....,1 be positive semi-definite scatter matrices. ThenC C C W B B S R S R S S R S R B ++=++=++==....)(....)()....()(111, (32)where R denotes the range.Proof : Since it is well known that the null space and the range of a matrix are complementary spaces, using the previous Lemma 1, we have,....)(....)())((....))(())(())....(()....(111111C C C C i i C C B B S R S R S N S N S N S S N S S R ++=++=++==++=++⊥⊥⊥=⊥I (33)where the last equality is a consequence of Theorem 2.After calculating the common vectors, the optimal projection vectors can be found by performing PCA as described previously in Section II-A. The eigenvectors corresponding to the nonzero eigenvalues of com S will be the optimal projection vectors. However, optimal projection vectors can also be obtained more efficiently by computing the basis of the difference subspace com B of the common vectors, since we are only interested in finding an orthonormal basis for the range of com S .The algorithm based on the Gram-Schmidt orthogonalization can be summarized as follows.Step 1: Find the linearly independent vectors i k b that span the difference subspace B and set},....,,,....,{1211111C N N b b b b span B −−=. There are totally r linearly independent vectors, where r is at most M -C .Step 2: Apply the Gram-Schmidt orthogonalization procedure to obtain an orthonormal basis r ββ,....,1 for B and set ]....[1r U ββ=.Step 3: Choose any sample from each class and project it onto B to obtain common vectors by using (27).Step 4: Find the difference vectors that span com B as11com k com k com x x b −=+, 1,...,1−=C k , (34)。

Generalized WDVV equations for B_r and C_r pure N=2 Super-Yang-Mills theory

a r X i v :h e p -t h /0102190v 1 27 F eb 2001Generalized WDVV equations for B r and C r pure N=2Super-Yang-Mills theoryL.K.Hoevenaars,R.MartiniAbstractA proof that the prepotential for pure N=2Super-Yang-Mills theory associated with Lie algebrasB r andC r satisﬁes the generalized WDVV (Witten-Dijkgraaf-Verlinde-Verlinde)system was given by Marshakov,Mironov and Morozov.Among other things,they use an associative algebra of holomorphic diﬀter Ito and Yang used a diﬀerent approach to try to accomplish the same result,but they encountered objects of which it is unclear whether they form structure constants of an associative algebra.We show by explicit calculation that these objects are none other than the structure constants of the algebra of holomorphic diﬀerentials.1IntroductionIn 1994,Seiberg and Witten [1]solved the low energy behaviour of pure N=2Super-Yang-Mills theory by giving the solution of the prepotential F .The essential ingredients in their construction are a family of Riemann surfaces Σ,a meromorphic diﬀerential λSW on it and the deﬁnition of the prepotential in terms of period integrals of λSWa i =A iλSW ∂F∂a i ∂a j ∂a k .Moreover,it was shown that the full prepotential for simple Lie algebras of type A,B,C,D [8]andtype E [9]and F [10]satisﬁes this generalized WDVV system 1.The approach used by Ito and Yang in [9]diﬀers from the other two,due to the type of associative algebra that is being used:they use the Landau-Ginzburg chiral ring while the others use an algebra of holomorphic diﬀerentials.For the A,D,E cases this diﬀerence in approach is negligible since the two diﬀerent types of algebras are isomorphic.For the Lie algebras of B,C type this is not the case and this leads to some problems.The present article deals with these problems and shows that the proper algebra to use is the onesuggested in[8].A survey of these matters,as well as the results of the present paper can be found in the internal publication[11].This paper is outlined as follows:in theﬁrst section we will review Ito and Yang’s method for the A,D,E Lie algebras.In the second section their approach to B,C Lie algebras is discussed. Finally in section three we show that Ito and Yang’s construction naturally leads to the algebra of holomorphic diﬀerentials used in[8].2A review of the simply laced caseIn this section,we will describe the proof in[9]that the prepotential of4-dimensional pure N=2 SYM theory with Lie algebra of simply laced(ADE)type satisﬁes the generalized WDVV system. The Seiberg-Witten data[1],[12],[13]consists of:•a family of Riemann surfacesΣof genus g given byz+µz(2.2)and has the property that∂λSW∂a i is symmetric.This implies that F j can be thought of as agradient,which leads to the followingDeﬁnition1The prepotential is a function F(a1,...,a r)such thatF j=∂FDeﬁnition2Let f:C r→C,then the generalized WDVV system[4],[5]for f isf i K−1f j=f j K−1f i∀i,j∈{1,...,r}(2.5) where the f i are matrices with entries∂3f(a1,...,a r)(f i)jk=The rest of the proof deals with a discussion of the conditions1-3.It is well-known[14]that the right hand side of(2.1)equals the Landau-Ginzburg superpotential associated with the cor-∂W responding Lie ing this connection,we can deﬁne the primaryﬁeldsφi(u):=−∂x (2.10)Instead of using the u i as coordinates on the part of the moduli space we’re interested in,we want to use the a i .For the chiral ring this implies that in the new coordinates(−∂W∂a j)=∂u x∂a jC z xy (u )∂a k∂a k )mod(∂W∂x)(2.11)which again is an associative algebra,but with diﬀerent structure constants C k ij (a )=C k ij(u ).This is the algebra we will use in the rest of the proof.For the relation(2.7)weturn to another aspect of Landau-Ginzburg theory:the Picard-Fuchs equations (see e.g [15]and references therein).These form a coupled set of ﬁrst order partial diﬀerential equations which express how the integrals of holomorphic diﬀerentials over homology cycles of a Riemann surface in a family depend on the moduli.Deﬁnition 6Flat coordinates of the Landau-Ginzburg theory are a set of coordinates {t i }on mod-uli space such that∂2W∂x(2.12)where Q ij is given byφi (t )φj (t )=C kij (t )φk (t )+Q ij∂W∂t iΓ∂λsw∂t kΓ∂λsw∂a iΓ∂λsw∂a lΓ∂λsw∂t r(2.15)Taking Γ=B k we getF ijk =C lij (a )K kl(2.16)which is the intended relation (2.7).The only thing that is left to do,is to prove that K kl =∂a mIn conclusion,the most important ingredients in the proof are the chiral ring and the Picard-Fuchs equations.In the following sections we will show that in the case of B r ,C r Lie algebras,the Picard-Fuchs equations can still play an important role,but the chiral ring should be replaced by the algebra of holomorphic diﬀerentials considered by the authors of [8].These algebras are isomorphic to the chiral rings in the ADE cases,but not for Lie algebras B r ,C r .3Ito&Yang’s approach to B r and C rIn this section,we discuss the attempt made in[9]to generalizethe contentsof the previoussection to the Lie algebras B r,C r.We will discuss only B r since the situation for C r is completely analogous.The Riemann surfaces are given byz+µx(3.1)where W BC is the Landau-Ginzburg superpotential associated with the theory of type BC.From the superpotential we again construct the chiral ring inﬂat coordinates whereφi(t):=−∂W BC∂x (3.2)However,the fact that the right-hand side of(3.1)does not equal the superpotential is reﬂected by the Picard-Fuchs equations,which no longer relate the third order derivatives of F with the structure constants C k ij(a).Instead,they readF ijk=˜C l ij(a)K kl(3.3) where K kl=∂a m2r−1˜C knl(t).(3.4)The D l ij are deﬁned byQ ij=xD l ijφl(3.5)and we switched from˜C k ij(a)to˜C k ij(t)in order to compare these with the structure constants C k ij(t). At this point,it is unknown2whether the˜C k ij(t)(and therefore the˜C k ij(a))are structure constants of an associative algebra.This issue will be resolved in the next section.4The identiﬁcation of the structure constantsThe method of proof that is being used in[8]for the B r,C r case also involves an associative algebra. However,theirs is an algebra of holomorphic diﬀerentials which is isomorphic toφi(t)φj(t)=γk ij(t)φk(t)mod(x∂W BC2Except for rank3and4,for which explicit calculations of˜C kij(t)were made in[9]we will rewrite it in such a way that it becomes of the formφi(t)φj(t)=rk=1 C k ij(t)φk(t)+P ij[x∂x W BC−W BC](4.3)As aﬁrst step,we use(3.4):φiφj= Ci·−→φ+D i·−→φx∂x W BC j= C i−D i·r n=12nt n2r−1 C n·−→φ+D i·−→φx∂x W BCj(4.4)The notation −→φstands for the vector with componentsφk and we used a matrix notation for thestructure constants.The proof becomes somewhat technical,so let usﬁrst give a general outline of it.The strategy will be to get rid of the second term of(4.4)by cancelling it with part of the third term,since we want an algebra in which theﬁrst term gives the structure constants.For this cancelling we’ll use equation(3.4)in combination with the following relation which expresses the fact that W BC is a graded functionx ∂W BC∂t n=2rW BC(4.5)Cancelling is possible at the expense of introducing yet another term which then has to be canceled etcetera.This recursive process does come to an end however,and by performing it we automatically calculate modulo x∂x W BC−W BC instead of x∂x W BC.We rewrite(4.4)by splitting up the third term and rewriting one part of it using(4.5):D i·−→φx∂x W BC j= −12r−1 D i·−→φx∂x W BC j= −D i2r−1·−→φx∂x W BC j(4.6) Now we use(4.2)to work out the productφkφn and the result is:φiφj= C i·−→φ−D i2r−1·r n=12nt n D n·−→φx∂x W BC j +2rD i2r−1·rn=12nt n −D n·r m=12mt m2r−1[x∂x W BC−W BC]j(4.8)Note that by cancelling the one term,we automatically calculate modulo x∂x W BC −W BC .The expression between brackets in the ﬁrst line seems to spoil our achievement but it doesn’t:until now we rewrote−D i ·r n =12nt n 2r −1C m ·−→φ+D n ·−→φx∂x W BCj(4.10)This is a recursive process.If it stops at some point,then we get a multiplication structureφi φj =r k =1C k ij φk +P ij (x∂x W BC −W BC )(4.11)for some polynomial P ij and the theorem is proven.To see that the process indeed stops,we referto the lemma below.xby φk ,we have shown that D i is nilpotent sinceit is strictly upper triangular.Sincedeg (φk )=2r −2k(4.13)we ﬁnd that indeed for j ≥k the degree of φk is bigger than the degree ofQ ij5Conclusions and outlookIn this letter we have shown that the unknown quantities ˜C k ijof[9]are none other than the structure constants of the algebra of holomorphic diﬀerentials introduced in [8].Therefore this is the algebra that should be used,and not the Landau-Ginzburg chiral ring.However,the connection with Landau-Ginzburg can still be very useful since the Picard-Fuchs equations may serve as an alternative to the residue formulas considered in [8].References[1]N.Seiberg and E.Witten,Nucl.Phys.B426,19(1994),hep-th/9407087.[2]E.Witten,Two-dimensional gravity and intersection theory on moduli space,in Surveysin diﬀerential geometry(Cambridge,MA,1990),pp.243–310,Lehigh Univ.,Bethlehem,PA, 1991.[3]R.Dijkgraaf,H.Verlinde,and E.Verlinde,Nucl.Phys.B352,59(1991).[4]G.Bonelli and M.Matone,Phys.Rev.Lett.77,4712(1996),hep-th/9605090.[5]A.Marshakov,A.Mironov,and A.Morozov,Phys.Lett.B389,43(1996),hep-th/9607109.[6]R.Martini and P.K.H.Gragert,J.Nonlinear Math.Phys.6,1(1999).[7]A.P.Veselov,Phys.Lett.A261,297(1999),hep-th/9902142.[8]A.Marshakov,A.Mironov,and A.Morozov,Int.J.Mod.Phys.A15,1157(2000),hep-th/9701123.[9]K.Ito and S.-K.Yang,Phys.Lett.B433,56(1998),hep-th/9803126.[10]L.K.Hoevenaars,P.H.M.Kersten,and R.Martini,(2000),hep-th/0012133.[11]L.K.Hoevenaars and R.Martini,(2000),int.publ.1529,www.math.utwente.nl/publications.[12]A.Gorsky,I.Krichever,A.Marshakov,A.Mironov,and A.Morozov,Phys.Lett.B355,466(1995),hep-th/9505035.[13]E.Martinec and N.Warner,Nucl.Phys.B459,97(1996),hep-th/9509161.[14]A.Klemm,W.Lerche,S.Yankielowicz,and S.Theisen,Phys.Lett.B344,169(1995),hep-th/9411048.[15]W.Lerche,D.J.Smit,and N.P.Warner,Nucl.Phys.B372,87(1992),hep-th/9108013.[16]K.Ito and S.-K.Yang,Phys.Lett.B415,45(1997),hep-th/9708017.。

热应力专题序

第52卷第5期力学学报V ol. 52, N o. 5 2020年9月C h in e s e J o u r n a l o f T h e o r e tic a l a n d A p p lie d M e c h a n ic s S e p t., 2020热应力专题热应力专题序王骥u 杜建科(宁波大学工学院，浙江宁波315211)在全国从事热应力研究的同行们热情支持下，第三届全国热应力大会于2019年12月6—8日在宁波举行，近200位代表参会，提交了约70篇论文.会议日程紧凑，参会代表来自工程技术企业、科研院所和高等院校，参会人数和会议报告稳定，充分表明热应力问题日益受到工程技术人员的高度重视，全国热应力大会也逐渐成为大家探讨学术、启发思考、交流经验的重要平台.全国热应力大会由南京航空航天大学高存法教授发起，第一届会议于2015年在南京召开，是国内第一个热应力专题的学术会议，顺应了科学技术发展过程中大家对热应力及其相关问题的高度关注，以及对研究和应用中所涉及的挑战和难题的关心.当然，这一会议的发起也和国际热应力领域的活跃和中国学者们的逐渐融入有关.高存法教授作为主编之一参加了《Encyclopedia of T h e r m a l Stresses》的编写与出版，后来也和陈伟球教授加入了《J o u r n a l o f T h e r m a l S t r e s s e s》编委会，并于2013年组织在南京召开了国际热应力大会，使中国的热应力领域的研究人员融入了国际交流.这方面的努力取得了很好的成效，激发了大家加强国内同行交流的热情.把握这一大好时机，浙江大学陈伟球教授在2017年在杭州举办了第二届全国热应力大会，同时决定第三届会议由王骥教授负责在宁波举办.由于大家有着长期密切合作，与国内外学术界有着紧密联系并熟悉学术交流的形式和组织工作，因此我们共同推动热应力大会成为又一项重要学术活动.下…届会议将于2021年由重庆大学承办，预祝会议取得更大成功.热应力大会此前没有出版会议论文集，但每届都会向国内的主要力学刊物推荐优秀论文.今年承蒙《力学学报》慷慨承诺，为我们提供专辑安排.经会议组委会和学报评审，最终有7篇论文汇集成了热应力大会论文专辑.这些论文涉及和热应力相关的广泛工程领域，如我们熟知的机械、土木、电子和航空航天等，展现出热应力问题的普遍性和重要性.李妍等的《超短激光脉冲加热薄板的广义热弹扩散问题》研究了目前广泛使用的超短波长激光脉冲辐照所产生的热扩散和热迟滞等参数，为激光热加工技术发展提供理论指导；李吉伟和何天虎在《考虑应变率的广义压电热弹理论及其应用》中考虑了应力松弛对压电弹性体热效应分析的影响，提出了一个更精准的瞬态效应分析方法；伏培林等在《考虑材料温度相关性的二维轮轨弹塑性滑动接触温升分析》中利用经典弹性接触理论分析了轮轨接触模式与温度变化的关系，对改善高速轮轨体系的性能和可靠性有重要的参考作用；王现辉等的《板中热弹波传播:一种改进的勒让德多项式方法》在长期持续工作的基础上，改进了基于勒让德多项式解的计算方法，高效计算了分数阶弹性板热传播特征；李杨等的论文《一维准晶功能梯度层合圆柱壳热电弹性精确解》考虑了复杂材料结构壳的热电弹性耦合解；胡克强2020~09-04 收稿.1) E-mail: **************.cn引用格式：王骥，杜建科.热应力专题序.力学学报，2020, 52(5): 1233-1234Wang Ji, Du Jianke. Preface of theme articles on thermal stresses. Chinese Journal of Theoretical and Applied Mechanics, 2020, 52(5):________________________________________________________________________________________________1234 力学学报2020年第52卷等在《磁-电-弹性半空间在轴对称热载荷作用下的三维问题研究》中继续分析多场耦合中的热效应及其特性，对智能结构和器件的精准设计来说非常重要；程若然和张春利在《多个局部温度载荷下压电半导体纤维杆的压电电子学行为分析》中也分研究了多场耦合问题，而着眼点则放在电子器件应用方面.总的来说，电磁热弹多场耦合问题是一个研究热点，和智能结构与器件的开发密切相关.考虑材料的热性能对应力和变形的影响在许多工程领域都是需要的，也表现了热应力问题的重要性和无处不在的特色.从这些角度出发，我们希望工程技术领域的研究机构和工程师们能够更多地参与未来的热应力大会，丰富目前以高校为主的学术交流，拓展领域，深化问题.这次热应力大会的组织工作得到宁波大学力学与工程科学系的许多年轻老师的全力支持，我们再次表示衷心感谢.这一专辑的论文的收集和评审工作主要由张爱兵博士负责，先后多次同作者、审稿人沟通，卓有成效地完成了这项非常繁重和细致的工作.doi:10.6052/0459-1879-20-314。

材料类期刊投稿指南

材料类期刊投稿指南SCI(EI)收录的科技期刊（2003）1 NATURE NATURE 自然0028-0836 27.955/2 SCIENCE SCIENCE 科学0036-8075 23.329/3 SURF SCI REP SURFACE SCIENCE REPORTS 表面科学报告0167-5729 14.091/science/journal/016757294 PROG MATER SCI PROGRESS IN MATERIALS SCIENCE 材料科学进展0079-6425 14http//www.elsevier.nl/inca/publications/store/4/1/4/5 PROG SURF SCI PROGRESS IN SURFACE SCIENCE 表面科学进展0079-6816 7.96/science/journal/007968166 PHYS REV LETT PHYSICAL REVIEW LETTERS 物理评论快报0031-9007 6.668 /7 MAT SCI ENG R MATERIALS SCIENCE & ENGINEERING R-REPORTS 材料科学与工程报告0927-796X 6.143/science/journal/0927796X8 ADV POLYM SCI ADVANCES IN POLYMER SCIENCE 聚合物科学发展0065-3195 6.053/science/journal/007967009 ADV MATER ADVANCED MATERIALS 先进材料0935-9648 5.579http://www.wiley-vch.de/publish/en/journals/alphabeticIndex/2089/10 ANNU REV MATER SCI ANNUAL REVIEW OF MATERIALS SCIENCE 材料科学年度评论0084-6600 5.405/loi/matsci?cookieSet=111 APPL PHYS LETT APPLIED PHYSICS LETTERS 应用物理快报0003-6951 3.849 /aplo/12 PROG POLYM SCI PROGRESS IN POLYMER SCIENCE 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原子层沉积的蒙特卡洛模拟及其空间位阻效应的研究

2018届研究生硕士学位论文分类号：学校代码：10269密级：学号：51151213023East China Normal University硕士学位论文MASTER’S DISSERTATION原子层沉积的蒙特卡洛模拟及其空间位阻效应的研究院系：信息科学技术学院电子工程系专业：微电子学与固体电子学研究方向：原子层沉积模拟研究指导教师：李亚巍副教授学位申请人：金凌鹏2018年5月4日A dissertation for Master’s Degree in 2018School code: 10269Student ID: 51151213023 East China Normal University Simulation of Atomic Layer Deposition and the Research of Steric HindranceEffectSchool：Department of Electronic Engineering,School of Information Science and Technology Major:Microelectronics and Solid ElectronicsSpeciality：Simulation of Atomic Layer DepositionSupervisors：Associate Professor Li YaweiAuthor：Jin LingpengMay 2018金凌鹏硕士学位论文答辩委员会成员名单摘要原子层沉积（ALD）的饱和沉积特性使得该技术被广泛应用于对材料厚度和纯度要求极高的电子行业，进而引起人们的广泛关注。

材料生长机制是材料科学研究的重要内容，对ALD过程进行计算机仿真，能够帮助人们深入了解实验手段无法获得的沉积机理，进而对改进和优化薄膜生长工艺，提高薄膜质量和性质具有极为重要的意义。

为了研究难以通过实验方法了解的ALD纳米薄膜沉积的形貌演化过程，本文引入了一个以随机沉积和扩散限制聚集为基础的蒙特卡罗模型。

地学SCI-国内EI及核心

序号全称简称中文刊名语种1Annual review of earth and ANNU REV EART地球与行星科学年评英语2Reviews of geophysics REV GEOPHYS地球物理学评论3Global Ecology and Biogeogr GLOBAL ECOL B全球生态学与生物地理学英语4Global biogeochemical cycle GLOBAL BIOGEO全球生物地球化学循环英语5Earth-science reviews EARTH-SCI REV地学评论英语6Quaternary science reviews QUATERNARY SC第四纪科学评论英语7Earth and Planetary Science Letters地球与行星科学通讯德语，法语，8Journal of petrology J PETROL岩石英语9Geology地质学英语10Geochimica et cosmochimica GEOCHIM COSMO地球化学与宇宙化学学报英语11Journal of Biogeography J BIOGEOGR生物地理学杂志12Paleoceanography古海洋学美国地质学会通报英语13The Geological Society of America bulle14Precambrian research前寒武纪研究15Chemical geology CHEM GEOL化学地质学英语16Contributions to mineralogy CONTRIB MINER矿物学与岩石学论文集英语17Geobiology地球生物学英语18Geotextiles and Geomembrane GEOTEXT GEOME土工用纺织物与土工用膜19Astrobiology天体生物学英语地质标准与地质分析研究德语，法语，20Geostandards and geoanalytical research英语21Journal of geophysical research:space p地球物理学研究杂志-空间物理地球物理学研究杂志-大地英语21Journal of geophysical research:solid e地球物理学研究杂志-行星21Journal of geophysical research:planets21Journal of geophysical research:oceans地球物理学研究杂志-海洋英语21Journal of geophysical research:atmosph地球物理学研究杂志-大气英语地球物理学研究杂志-地表英语21Journal of geophysical research:earth s22Lithos国际矿物学、岩石学与地球化英语23Biogeosciences生物地球科学英语24Journal of metamorphic geology变质地质学杂志英语25Geophysical research letters地球物理学研究快报英语26The journal of geology地质学杂志英语27Biogeochemistry生物地球化学英语28Journal of quaternary scien J QUA TERNARY第四纪科学杂志英语29Turkish Journal of Earth Sc TURKJ EATTH S土耳其地球学杂志英语30Tectonics 构造地质学英语31Progress in Physical Geogra PROG PHYS GEO物理地理学进展英语32American journal of science AM J SCI美国科学杂志33Meteoritics&Planetary Scien METEORIT PLAN陨星学与行星科学34Geochemistry,geophysics,geosystems地球化学，地球物理学，地球英语35Marine and petroleum geology海洋与石油地质学英语36IEEE transactions on geoscience and remIEEE地学与遥感汇刊英语37The Holocene全新世英语38Global and planetary change地球和行星的变化英语39Basin research盆地研究英语40Journal of the Geological society地质学会志英语41Journal of the paleolimnology古湖沼学杂志英语42Hydrology and earth system sciences水文学与地球系统科学英语43Elements元素英语44Quaternary research第四纪研究英语45American mineralogist美国矿物学家46Reviews in mineralogy and geochemistry矿物学和地球化学评论英语古地理学、古气候学、古生态学47Palaeogeography,palaeoclimatology,palae48Journal of hydrology水文学杂志英语49Organic geochemistry有机地球化学50Geophysical journal international国际地球物理学杂志英语51Bulletin of volcanology火山学通报英语52Terra nova地球新星英语53Boreas国际第四纪研究杂志德语，法语，54Landscape Ecology景观生态学德语，法语，地球与行星内部物理学英语55Physics of the earth and planetary inte56Marine geology海洋地质学英语57Palaios古代58Earth surface processes and landforms地球表面变化过程与地形英语59Seismological Research Letters 地震学研究快报英语60Journal of sedimentary research沉积研究杂志英语62Applied clay science应用粘土科学63Geomorphology地貌学英语64Journal of contaminant Hydr J CONTAM HYDR污染物水文学杂志英语65Geochemical transactions地球化学汇刊英语国际地理信息科学杂志英语66International journal of geographical i67Earth Interactions EARTH IN TERA地球相互作用英语68Acta geologica Sinica中国地质学报69Economic geology and the bulletin of th经济地质学与经济地质学家学英语70Journal of marine systems海洋系统杂志英语71Sedimentary geology沉积地质学英语72Applied geochemistry应用地球化学英语美国地震学会通报英语73Bulletin of the seismological society o火山学与地热研究杂志德语，法语，74Journal of volcanology and geothermal r75Tectonophysics地壳构造物理学德语，法语，国际地球科学杂志德语，法语，76International journal of earth sciences77Journal of Asian earth sciences亚洲地学杂志英语78Sedimentology沉积学英语79Journal of structural geology结构地质学杂志英语80Geomicrobiology journal地质微生物学杂志英语81Journal of Nuclear Material J NUCL MATER核材料杂志英语82Journal of Geodesy J GEODESY大地测量学杂志英语83Landscape and Urban Plannin LANDSCAPE URB园林与城市规划英语84Permafrost and periglacial processes永久冻土与冰缘过程85International journal of coal geology国际煤炭地质学杂志英语86Geological magazine地质学杂志英语87Journal of Atmosphere and S J ATMOS SOL-T大气与日地物理学杂志英语88International geology review国际地质学评论英语89Geodinamica acta地球动力学报法语90Geografiska Annaler: Series GEOGR ANNA法语91Quaternary International国际第四纪研究法语92Ground water地下水英语93Journal of Archaeological S J ARCHAEOL SC考古科学杂志英语美国水资源协会志英语94 Journal of the American Water Resource95Annales Geophysicae ANN GEOPHYS-G地球物理层编年史英语96Aquatic geochemistry水地球化学英语97GeoArabia中东石油地球科学杂志英语98Clays and clay minerals粘土与粘土矿物英语99Gondwana research冈瓦纳研究英语100CATENA专业丛书英语101Geofluids地热流体英语103Nonlinear Processes in Geop NONL IN EAR P地球物理学的非线性进程英语104Dynamics of Atmospheres and DYNAM ATMOS O大气与海洋动力学英语105AAPG bulletin美国石油地质学家学会通报英语106Mineralogical magazine矿物学杂志英语107Antarctic science南极科学英语108Zeitschrift fur Geomorphologie地球形态学杂志英语109Mineralium deposita矿床英语110Physics and chemistry of minerals矿物物理学与矿物化学111Gems & gemology宝石与宝石学英语112Australian journal of earth sciences澳大利亚地球科学杂志英语113European journal of mineralogy欧洲矿物学杂志德语，法语114Geologica Acta地质学报德语，法语115Earth, planets and space地球、行星与太空英语116Chemie der Erde地球化学德语，英语117Quaternary geochronology第四纪地质年代118Surveys in geophysics地球物理学综论英语119Geophysics地球物理英语120Radiocarbon放射性碳英语121Geosynthetics International国际土工合成材料学英语122IEEE Xplore: Geoscience and IEEE GEOSCI R IEEE地学与遥感汇刊英语123Bulletin of Earthquake Engineering地震工程通报英语124Hydrogeology journal水文地质学杂志法语，英语国际摄影测量和遥感学会志英语125ISPRS journal of photogrammetry and rem摄影测量工程与遥感126Photogrammetric engineering and remote127Journal of petroleum geology石油地质学杂志128Journal of African Earth Sc J AFR EARTH S非洲地学杂志英语128Journal of African earth sciences非洲地学杂志法语，英语矿物,金属材料科学学会杂志法语，英语129Journal of the Minerals Metals & Materi131Petrology+岩石学法语，英语132SpaceWeather空间气象法语，英语133The Canadian mineralogist加拉大矿物学者法语，英语134 The Canadian mineralogist加拉大矿物学者法语，英语自然灾害与地球系统科学英语135Natural hazards and earth system scienc136Geophysical and Astrophysic GEOPHYS ASTRO地球物理与天体物理流体动力英语137Geological journal地质学杂志英语138Natural Hazards NAT HAZARDS自然风险139Journal of caves and karst studies洞穴与岩溶研究杂志英语140Mineral Processing and Extr MINER PROCESS矿物处理和提取冶金英语141Facies相英语142Ore geology reviews矿物地质学评论143Landslides滑坡144Near Surface Geophysics近地表地球物理145Radio science无线电科学国际矿物处理杂志英语146International journal of mineral proces147Arctic, antarctic, and alpine research北极、南极与高山研究148Canadian journal of earth sciences加拿大地球科学杂志法语，英语149Engineering geology工程地质学德语，法语，150Norwegian Journal of Geology挪威地质学杂志151Geomagnetism and Aeronomy国际地磁学与高空科学协会152Minerals Engineering矿物工程154Geothermics地热155Mineralogy and petrology矿物学和岩石学156Computers & geosciences计算机与地学英语157The photogrammetric record摄影测绘记录英语158Archaeometry考古定年学英语159Cold Regions Science and Technology寒冷地区科学160Comptes Rendus Geosciences C R GEOSCI161Arctic北极南美地学杂志英语162Journal of South American earth science矿物与岩石学杂志英语163Journal of mineralogical and petrologic164Episodes幕英语165Geoinformatica地学信息英语166Pure and applied geophysics理论与应用地球物理学英语167Proceedings of the geologists associati地质学家协会会报英语168Petroleum geoscience石油地质科学169The island arc岛弧170Netherlands Journal of Geosciences荷兰地学杂志171Russian Geology and Geophysics俄罗斯地质学与地球物理学172Geo-marine letters地质海洋快报英语173Resource geology地质资源174New Zealand Journal of Geology and Geop新西兰地质学与地球物理学杂志爱丁堡地学皇家学会事物175Transactions of the Royal Society of Ed176Acta geologica Polonica波兰地质学报177Polar research极地研究178Advances in Space Research空间研究进展179Journal of seismology地震学杂志英语180Mathematical Geology数学地质英语181Journal of Geophysics and Engineering地球物理学与工程学英语法国地质学会通报法语，英语182Bulletin de la Societe Geologique de Fr183Journal of Environmental Engineering Ge环境与工程地球物理学杂志土工技术与地质环境工程杂志英语184Journal of Geotechnical & Geoenvironmen185Computational geosciences计算地球科学英语国际岩石力学与采矿科学杂志英语186International journal of rock mechanics187Studia geophysica et geodaetica地球物理学与大地测量学研究德语，法语，188Geophysical prospecting地球物理勘探德语，法语，189Environmental geology环境地质学英语190Scottish journal of geology苏格兰地质学杂志英语191Geoarchaeology地质考古学英语英语中国科学D辑：地球科学（英文191Science in China. series D earth scienc192Geological quarterly地质学季刊英语193Ofioliti蛇绿岩195Clay minerals粘土矿物英语石油科学和石油工程杂志英语196Journal of petroleum science & engineer197Marine geophysical researches海洋地球物理研究英语工程地质学季刊英语198Quarterly journal of engineering geolog地层学及地质学的相互关系199Stratigraphy and Geological Correlation200Geochemical journal地球化学杂志英语201Eclogae Geologicae Helvetiae英语202Geochronometria英语203Chinese Journal of Geophysi CHINESE J GEO地球物理学报204Archaeological Prospection考古学展望205Physics and chemistry of the earth地球物理学与地球化学土壤动力学与地震工程206Soil Dynamics and Earthquake Engineerin207Journal of Cultural Heritage文化遗产杂志208South African journal of geology南非地质学杂志英语209Journal of geochemical exploration地球化学勘探杂志英语210Revista mexicana de ciencias geologicas211Geotechnique土工英语矿物学新年鉴. 论文辑德语，法语，212Neues Jahrbuch fur Mineralogie, Abhandl213Computers and geotechnics计算机与土工学214Revista geologica de Chile智利地质杂志西班牙语，英215Geosciences journal地球科学杂志英语丹麦地质学会通报德语，法语216Bulletin of the geological society of D217Journal of Earthquake Engineering地震工程杂志218Soils and Foundations地基及基础219Earth, Moon, and Planets地球，月球及行星220Geotectonics大地构造地质学221Bollettino della Societa Geologica Ital意大利地质学通报英语222Canadian geotechnical journal加拿大土工杂志英语223Astronomy and Geophysics天文学与地球物理学英语224Journal of Cold Regions Engineering寒冷地区工程杂志英语225Geologica Carpathica喀尔巴阡山地质学英语226Geologiska Foreningens i St GFF瑞典地质学会汇刊德语，法语，227Geochemistry International国际地球化学德语，法语，228Rock mechanics and rock engineering岩石力学与岩石工程英语229Rivista Italiana di Paleontologia e Stratigrafia英语英语230Journal of earth system science印度科学院会刊：地球与行星231Geotechnical Testing Journal土工试验杂志英语工程地质学与环境通报法语，英语232Bulletin of engineering geology and the233Izvestiya Physics of the Solid Earth法语，英语234Doklady Earth Sciences法语，英语235Physical Geography自然地理法语，英语陆地，大气，海洋科学法语，英语236Terrestrial, atmospheric and oceanic sc237Geology of Ore Deposits矿床地质学法语，英语238Geologica Belgica比利时地质239Journal of coastal research海岸研究杂志英语240Lithology and mineral resources岩相学与矿物资源英语印度地质学会志英语241Journal of the geological society of In242SPE reservoir evaluation & engineering石油工程师协会油藏评估与工英语243Mountain Research and Development山区研究与开发英语244Nuovo Cimento Societa Intaliana di Fisi诺沃西门托会刊意大利迪记事英语北京科技大学学报：矿物冶金英语245Journal of University of Science and Te246Annals of geophysics地球物理学纪事英语247Minerals & Metallurgical Processing矿产及冶金加工英语248Coal Preparation备煤英语249Journal of Mining Science采矿科学杂志英语250Environmental & engineering geoscience环境与工程地质科学英语251Newsletters on stratigraphy地层学通讯地球化学：探索，环境，分析252Geochemistry: Exploration, Environment,253Journal of seismic exploration地震探测杂志英语254Petrophysics岩石物理学英语约克郡地质学会会报英语255Proceedings of the Yorkshire geological256Carbonates and evaporites碳酸盐与蒸发岩英语英国太阳系内杂志英语257Journal of the British Interplanetary S258Marine georesources and geotechnology海洋地资源与土工学英语259Proceedings of the Institution of Civil知名土木工程师议程：地球应用技术工程260Geotimes地质时代英语261Survey review测量评论英语丹麦和格陵兰岛地质勘测期刊英语262Geological Survey of Denmark and Greenl南非采矿与冶金学会志英语263Journal of the South African Institute264Canadian mining journal加拿大采矿杂志英语265Engineering and mining journal工程与采矿杂志英语266Advances in Geophysics地球物理学发展英语267Swiss Journal of Geoscience瑞士地学杂志类别ISSN影响因子年发文数自然科学0084-65977.73224自然科学8755-1209 6.925自然科学1466-822X 4.43577自然科学0886-6236 4.33596地质学0012-8252 4.3141地质学0277-3791 4.11205自然科学0012-821X 3.873503自然科学0022-3530 3.80686地质学0091-7613 3.754285地质学0016-7037 3.665395自然科学0305-0270 3.5391590883-8305 3.391760016-7606 3.35495地质学0301-9268 3.24796地质学0009-2541 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1.91960地质学0197-9337 1.917148地质学0895-0695 1.91646地质学1527-1404 1.8971地质学0169-1317 1.861112地质学0169-555X 1.854244地质学0169-7722 1.85298地质学1467-4866 1.842121365-8816 1.82259测量普查勘探1087-3562 1.80822测量普查勘探地质学1000-9515 1.7811090361-0128 1.77665矿床与矿产资地质学0924-7963 1.772128地质学0037-0738 1.757201自然科学0883-2927 1.744200地质学0037-1106 1.743181地质学0377-0273 1.742154地质学0040-1951 1.729161地质学1437-3254 1.71970地质学1367-9120 1.714151地质学0037-0746 1.66762地质学0191-8141 1.66130地质学0149-0451 1.65553地质学0022-3115 1.643730地质学0949-7714 1.63657地质学0169-2046 1.6331571045-6740 1.633280166-5162 1.625100矿床与矿产资地质学0016-7568 1.58463地质学1364-6826 1.566195地质学0020-6814 1.5560地质学0985-3111 1.52922地质学0435-3676 1.524地质学1040-6182 1.468143地质学0017-467X 1.44188地质学0305-4403 1.439169地质学1093-474X 1.436121地质学0992-7689 1.427214地质学1380-6165 1.412181025-6059 1.38123矿床与矿产资0009-8604 1.37556矿床与矿产资地质学1342-937X 1.35766地质学0341-8162 1.3461201468-8115 1.33332矿床与矿产资1023-5809 1.31471矿床与矿产资0377-0265 1.315矿床与矿产资0149-1423 1.27375矿床与矿产资0026-461X 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井间ERT电极阵列优化及监测实验系统设计与开发

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关键词井间电阻层析成像$监测实验系统$电极阵列优化$有限元$二氧化碳$地质封存
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非参数GKNN估计的高效独立成分分析算法

独立成分分析（／盲源分离（）的主要目的是从观测到的混合信号中恢复出统计独立的源信号．ＩＣＡ）ＢＳＳ “ 盲” 就意味着源信号和混合过程都未知．通信信号处理、语ＩＣＡ算法由于可广泛应用于生物医学信号处理、系统辨识、数据分析和压缩等方面，在信号处理（和神经网络（领域引起了很大的兴趣，音信号处理、ＳＰ）ＮＮ）
］１～３目前已成为这些领域的研究热点［．［］３］自从从统计的观点研究Ｉ已出现了许多有效算法，如：自然梯度算法［，ＣＡ问题以来，Ｉｎｆｏｍａｘ算法２，
收稿日期：２００７０７１０基金项目：国家自然科学基金资助（）；河南工业大学校青年科研基金资助（）６０６７２０４７０６ＸＪＣ０３２作者简介：王法松（），男，工程师，：１９７９Ｅｍａｉｌｆａｓｏｎｗａｎ２６．ｃｏｍ．＠１ｇｇ
第４期王法松等：非参数ＧＫＮＮ估计的高效独立成分分析算法
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［］］［］５］６分阶段学习算法［，灵活评价函数算法［，ＪＡＤＥ算法４，ＦａｓｔＩＣＡ算法７等．Ｉｎｆｏｍａｘ算法在分离源信号是超高斯混合情形时效果比较理想，当混合信号中包含亚高斯信号时，分离效果就不令人满意．为了解决上面［］遇到的问题，提出了一类扩展的基于参数模型的Ｉ这类算法在一定程度上改善了传统参数ＣＡ分离算法６，

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