In press, NeuroReport
Address all correspondence and reprint requests to : Bruno Rossion,
Université Catholique de Louvain (UCL), Faculté de psychologie, Unité de Neuropsychologie Cognitive (NECO),place du Cardinal Mercier, 10, 1348 Louvain-la-Neuve, Belgium
e-mail: rossion@neco.ucl.ac.be
THE N170 OCCIPITO-TEMPORAL COMPONENT IS DELAYED AND ENHANCED
TO INVERTED FACES BUT NOT TO INVERTED OBJECTS:
AN ELECTROPHYSIOLOGICAL ACCOUNT OF FACE-SPECIFIC PROCESSES IN
THE HUMAN BRAIN.Rossion, B. 1, 2 , Gauthier, I.3 , Tarr, M.J.4, Despland, P. 5, Bruyer, R 1, Linotte, S.1, Crommelinck, M.2
1
Unité de Neuropsychologie Cognitive (NECO), UCL, Louvain-la-Neuve; 2 Laboratoire de Neurophysiologie (NEFY), UCL, Brussels, Belgium; 3 Department of Psychology, Vanderbilt University; 4 Brown University; 5 Unité
d’Exploration du Système Nerveux, CHUV, Switzerland.
ABSTRACT
Behavioral studies have shown that picture-plane inversion impacts face and object recognition differently, thereby suggesting face-specific processing mechanisms in the human brain. Here we used event-related potentials to investigate the time course of this behavioral inversion effect in both faces and novel objects. ERPs were recorded for 14 subjects presented with upright and inverted visual categories, including human faces and novel objects, i.e., Greebles. A N170 was obtained for all categories of stimuli, including Greebles. However, only inverted faces delayed and enhanced N170 (bilaterally). These observations indicate that (1) The N170 is not specific to faces, as it has been previously claimed; (2) The amplitude difference between faces and objects does not reflect face-specific mechanisms since it can be smaller than between non-face object categories; (3) There do exist some early differences in the time-course of categorization for faces and non-faces across inversion – a difference that may be atteibuted either to stimulus category per se (e.g., face-specific mechanisms) or to differences in the level of expertise between these categories.
INTRODUCTION
It is often stated that human face recognition is subserved by specific processes within specific neural structures. Behavioral evidence supporting face-specific mechanisms has been provided by studies showing that stimulus inversion disrupts the processing of faces more than other objects – the face inversion effect1, and that individual feature recognition in faces is particularly dependent upon the relationships between features – the face superiority effect2. The evidence for neuroanatomical specialization has been provided both by neuropsychological patient studies and neuroimaging. Studies of brain injuries have provided cases of impaired of face processing, usually following bilateral occipito-temporal damage3, with intact visual object processing, as well as the opposite deficit, spared face processing with impaired object processing4. More recently, PET and fMRI studies have provided additional evidence that face processing in normal subjects may involve specific neural regions distinct from those regions support general object recognition (e.g. 5). This hypothesis should be evaluated in terms of all factors that play some role in determining visual recognition behavior6. Specifically, stimulus-class membership (face vs. non-face), categorization level (placing objects in basic-level categories such as “chair”,“dog”, and “car” or in subordinate-level categories, such as “Dalmatian”, “beagle”, and “blood hound”), and level of expertise are all critical to understanding some of the observed differences between object and face processing. First, behaviorally, it has been demonstrated that visual expertise with a given dog breed7 or with synthetic nonsense objects – Greebles8, can lead to inversion and/or superiority effects similar to those obtained for faces. Second, recent case descriptions have also shown that prosopagnosics are more affected by manipulations of the level of categorization than normal controls9. This finding suggests that prosopagnosics apparently disproportionate impairment for faces as compared to non-face objects may be partly explained by the fact that faces are usually recognized at the subordinate level, while non-face objects are typically recognized at the basic level. Third, recent fMRI studies suggest that the so-called “face fusiform area (FFA)”5 is more strongly activated when objects are categorized at the subordinate level as compared to the basic level10 and that expertise training with non-face objects (Greebles) may also recruit the FFA to the same degree as faces11.
Of particular interest to the present study, scalp electrophysiological recordings in humans have revealed early face-specific mechanisms which could not have been found using behavioral, neuropsychological, or brain imaging methodologies. Using ERPs, it has been demonstrated that face processing differs from visual object processing at 170 ms following stimulus onset 12-14. This dissociation takes place at the level of the visual N170 component in occipito-temporal regions 13-16 and is characterized by a larger amplitude in the component for faces14 or an absence of N170 for non-faces objects 13,17.
One unresolved issue with these ERP studies is that they generally compared faces to a single mono-oriented object category, e.g., cars13 or houses14. To be able to strongly argue that faces “are “special” one not only has to show specific processes for faces but also that most other object classes are processed the same way, by a generic object recognition18. For example,
differences in level of activity have also been obtained between different non-face object categories in both electrophysiological12 and neuroimaging studies19. It is unlikely that anyone would seriously interpret such results as evidence for separable neural mechanisms between such categories, for instance, chairs and houses. In other words, comparing faces to a single category is not sufficient for claiming a dissociation and one has to be careful to show specific effects for faces that are not observed for the majority of non-face categories. A second weakness of previous ERP studies is that the differential amplitude of the N170 for faces and objects might have been obtained due to potential confounds such as differences in the low-level visual features between faces and objects5, higher visual familiarity for faces (as a class) as compared to objects20, different levels of expertise8.
In light of these concerns, the aim of the present study is to establish a framework in which the modularity of face processing can be better evaluated through ERPs. More precisely, we tested whether the N170 component might really reflect face-specific neural mechanisms occurring early in visual categorization by comparing faces to a variety of non-face mono-oriented stimuli. To avoid the potential confound of differential low-level visual features between faces and objects, these stimuli were not compared directly but were compared relative to their respective picture-plane inverted presentations. While inversion of a face (or an object) preserves the low-level visual features, inverted faces are not believed to involve configural processing2, 4 which is considered to be the hallmark of the face-specific processing as compared to non-face object processing.
We also took advantage of the recent evidence that the N170 is strongly influenced by face inversion21. When faces are presented upside-down, the N170 latency is significantly delayed (around 10 ms) and may also be larger when subjects are engaged in a face discrimination task21. The latency delay, which is fairly robust and has been found in several ERPs studies13, 14, 21 has been proposed to reflect the loss of configural information with inversion 16, 21.
The specificity of the electrophysiological correlate of the inversion effect to faces was tested by presenting subjects with faces and various non-face categories of mono-oriented objects such as houses, chairs, shoes, cars, and Greebles8, in upright and inverted presentations.
According to the face-specific processing hypothesis, which argues for differential neural coding for faces and objects, a latency delay and an increase of activity is expected when faces are inverted but not when inverted non-face objects are presented. The alternative view – a general object-recognition system shared by faces and objects – predicts that the latency delay and the higher activity of the N170 potential reflects generic object rotation effects that should be observed for any mono-oriented objects22.
MATERIALS AND METHODS
Fourteen subjects (7 females; 3 left-handed) with normal or corrected vision participated in this experiment (mean age: 25). Eight different images of six categories of stimuli were used: photographs of faces, full front cars, shoes, chairs, houses, and Greebles. The corresponding inverted stimuli were also presented. All stimuli subtended a visual angle of approximately 2.5 deg. The height of the stimuli was slightly different between categories but identical for upright and inverted versions
of the images. Subjects sat on a comfortable chair in a dimly lit room. They were instructed to visually fixate the center of the screen (distance 1.2 m) during each experimental block. After a short training block of 20 stimuli, subjects received 12 experimental blocks of 120 trials (10 images x 6 categories x 2 orientations). Each stimulus was presented for 500 ms; the inter-stimulus interval was randomized between 1500 and 2000 ms. All of the stimuli were randomized within a block but all subjects viewed the same succession of stimuli. The subject’s task was to press a key if the stimulus was upright and another key if the stimulus was inverted. The upright and inverted orientations of Greebles were shown on the screen to the subjects before the experiment, although without exposure to upright Greebles, they still tend to orient to this particular viewpoint because of their flat base, their bilateral symmetry axis and their rotational axis. All responses were made with the right hand. Subjects were instructed to respond as accurately and as quickly as possible. Responses below 200 ms or above 1250 ms were discarded in the behavioral and ERPs analyses (less than 2% of trials).
EOG was recorded bipolarly from electrodes placed on the outer canthi of the eyes, and in the inferior and superior areas of the orbit. Scalp EEG was recorded from 58 electrodes mounted in an electrode cap (electrocap, Neuroscan Lab) with respect to a left mastoid reference. EEG was amplified with a gain of 30K and bandpass filtered at 0.15 - 70 Hz. Electrode impedance was kept below 5 k?. EEG and EOG were continuously acquired at a rate of 500Hz . After automatic removal of EEG and EOG artifacts, epochs beginning 200 ms prior to stimulus onset and continuing for 800 ms were made. They were referenced off-line to a common average reference. The average waveforms computed for the different categories (6 x 2 averages for each subject) were low pass filtered at 30 Hz. Peak amplitude and latencies of the N170 at occipito-temporal sites (T5 and T6) electrodes were measured relative to a 200 ms pre-stimulus baseline on a computed grand average waveform and in each subject individually, for the 12 types of stimuli used in the experiment. Topographic maps were also computed for the various categories of stimuli presented.
Behavioral (accuracy rates and correct RTs) and electrophysiological (latencies and amplitudes of components) measures on these different electrodes were analyzed by means of repeated-measures analyses of variance (ANOVA) with the Orientation (2 levels), the Visual Category (6 levels) and, when relevant, the Hemispheric Lateralization (2 levels) as factors. Post-hoc paired t-tests were also performed.
RESULTS
Behavioral data: The overall accuracy in the detection task for the categories ranged between 92% (shoes) and 96% (faces and houses); mean reaction times ranged between 556 ms (upright houses) and 615 ms (inverted shoes). The 2 x 6 ANOVA on RTs revealed significant main effects of orientation (F
1,13
= 9.27; p=0.009), the inverted stimuli being slower to
detect (603 ms vs. 579 ms), and of category (F
1,13
=8.13; p<0.001), cars being the fastest stimuli for the orientation decision (576 ms on average) and Greebles being the slowest (613 ms). There was also a significant
interaction (F
1,13
=3.25; p=0.01), mainly due to the absence of any difference between orientation for Greebles and shoes.
ERPs: All subjects elicited an occipito-temporal N170 for all the visual categories tested, including Greebles. The topography of this N170 was very similar for all object categories. Table 1 shows the latencies of the N170 for the various categories tested. Amplitudes are shown on Table 2. On average, the N170 component to faces peaked at 165 ms at T6 and at 163 ms at T5, with similar latencies for objects (see table 1), except for houses which elicited a slightly later N170. Three main results were observed: (1) the N170 was larger for faces than for all other visual categories (Table 1; Fig.1); (2) the N170 was delayed and larger for inverted faces as compared to upright faces at both hemispheres (Fig.1; Tables 1 and 2); (3) and there was no effect of orientation for all the non-face categories tested (Table 1; Fig.2).
Statistical analyses largely confirmed these observations. The ANOVA (Category x Orientation x Hemisphere) on N170 latencies showed a significant
interaction between Category and Orientation (F
5,65
= 10.022; p<0.001). There was also a main effect
of Category (F
5,65
= 6.476, p<0.001), due to the larger latencies for houses, and a main effect for Orientation
(F
1,13
= 8.086; p=0.014). When faces were not taken in the analysis (ANOVA 2 x 2 x 5), there was not a main
effect of Orientation (F
1,13
= 0.263; p=0.617) nor any significant interaction between Orientation and Category
(F
4,52
= 1.070; p=0.381) but still a main effect of
Category (F
4,52
= 9.09, p<0.001). Post-hoc t-tests conducted on each category showed a significant delay for inverted faces only (p<0.0001). For all non-face categories, there was no effect of Orientation (cars: p=0.314; Greebles: p=0.502; Shoes: p=0.869; Chairs: p= 0.364; Houses: p=0.065). The non-significant trend for houses was actually due to an increase in latencies for upright houses. Finally, post-hoc t-tests revealed that the N170 to Greebles and cars peaked earlier than for other categories (p=0.001 and p=0.01, respectively) but did not differ from one another (p=0.13)while N170 to
T6 (right hemisphere)T5 (left hemisphere)
Upright Inverted Upright Inverted Faces164172164170
Greebles166164162166
Cars166167164164
Chairs170170167169
Houses175171171172
Shoes166166165166
Table 1. Mean latencies of the N170 to the differentvisual categories presented, for the two orientations and at the twohemispheres. The latency delay for faces is 8 ms on average at T6 and 6 ms at T5 while there wasn’t anysuch delay for all other objects tested.
shoes and houses was significantly delayed as
compared to all other categories, including faces (p=0.026 and p=0.01, respectively). All other comparisons between Categories (p
s
between 0.194 and 0.462). It is worth noting that all of the 14 subjects tested exhibited a delayed (>6 ms) N170 to inverted faces at least at one hemisphere. 8 subjects showed the effect in both hemispheres.
The ANOVA on N170 amplitudes also revealed a main effect of Visual Category
(F
5,65
= 10.647; p<0.001) and a significant interaction
between Orientation and Category (F
5,65
= 4.942; p<0.001). The first effect reflected the larger amplitude of the N170 to faces (Fig.1) but also differences between object categories (see post-hoc t-tests). The interaction in this ANOVA was due to the specific increase of voltage amplitude for inverted faces as compared to upright faces (Fig.1; table 2).Post-hoc t-tests confirmed the larger amplitude for faces than for all other objects (p=0.003) although the N170 to faces was not significantly larger than for cars (p=0.059). There was also significant differences between categories of objects (e.g. cars vs. others: p=0.025; chairs vs. others: p<0.001; see Table 2) . Post hoc t-tests conducted for each category showed a main effect of Orientation for faces (p=0.014) and the absence of any such effect for all other visual categories (cars: p=0.780; Shoes: p=0.159; Chairs: p= 0.605; Houses: p=0.130), except that the N170 was actually larger for upright than inverted Greebles (see Table 2; p=<0.001).
DISCUSSION
The first observation of our study is that the N170 is not at all specific to faces, as has been claimed in previous reports13, 17. This component is actually observed for completely novel objects such as Greebles. This result demonstrates that long term familiarity with a category is not a prerequisite to trigger a N170 potential.
T6 (right hemisphere)T5 (left hemisphere)
Upright Inverted Upright Inverted Faces-3.22-4.65-4.28-5.48
Greebles-2.31-1.36-2.72-1.89
Cars-3.34-3.02-2.95-2.80
Chairs-2.03-2.27-0.63-1.69
Houses-3.35-2.50-1.99-1.82
Shoes-1.43-1.44-1.55-1.79
Table 2. Mean amplitudes of the N170 to the differentvisual categories presented, for the two orientations and at the twohemispheres.the amplitude of the component is significantly larger for faces tahn forall other categories. The increase for inverted faces is 1.43 mV on averageat T6 and 1.30 mV at T5 while there wasn’t any such increase for allother objects tested
The fact that the N170 amplitude is even larger for Greebles than for some common familiar objects (e.g. chairs and shoes) is not as surprising as it might seem at first glance: Greebles share more physical properties with faces (smooth surfaces, bilateral symmetry, homogeneity of the class, configuration of individual features, organic appearance) than many other objects. A second observation is that the N170 observed for faces zis larger than for the other objects tested here. This confirms previous reports14, 23, eventhough neither these studies nor the present study definitively rule out the possibility that these amplitude differences are due to the low-level visual properties differing between faces and objects. Moreover, when compared only to cars, faces failed to elicit a significantly larger N170. More to the point, our results demonstrate that the amplitude differences in the N170 component can be as large between different categories of non-face objects (e.g. cars and shoes) as between faces and some object categories. In our view, a difference between faces and non-face objects is not sufficient to argue for face-specific processes in the human brain. However, a stronger claim for face-specific processes in early visual categorization can be made on the basis of the main finding of this study, namely the observation of a specific latency shift of the N170, as well as an increase in amplitude, for faces when they are inverted. This result confirms and extends our previous findings21 as none of the other visual categories tested in the present study showed a similar effect when inverted stimuli were presented. To our knowledge, the specificity of this N170 latency delay for inverted faces has not been previously established. The neural correlates of the behavioral face inversion effect have been examined in several recent fMRI studies (e.g.11, 24, 25) but the timing effect observed here are undetectable by these techniques. As we have suggested previously21 the effect observed with inverted faces is compatible with both behavioral studies and neurophysiological findings in monkeys. According to behavioral experiments, face inversion disrupts the configural information - i.e. the spatial relationships between parts - that is used by default and accessed more quickly than individual parts during face recognition2, 26. The loss of configural information through inversion may thus slow down early face processing. Support for the view that the loss of configural information is intimately linked to the latency delay observed for inverted faces is provided by other ERP studies that have observed similar delays for isolated eyes13, or faces with the eyes removed14, 16. A shift in the latency of the N170 has also been observed when subjects are engaged in analytical processing of faces (i.e., focusing on the eyes16).
The latency delay observed for inverted faces is also compatible with neurophysiological recordings in the monkey temporal lobe. The onset of activity for individual face-selective cells is roughly equivalent for normal and inverted faces27. However, differential proportions of cells coding normal and inverted faces cause differential rates of activity accumulation in the cell population as a whole and, thus, may lead to timing differences, that is, delays for uncommon view of faces, in subsequent processing stages27.
The larger N170 observed for inverted faces as compared to upright faces provides further evidence for a face-specific effect in early visual categorization. Two explanations for this increased amplitude for inverted faces have been suggested21. First, inverted faces are more difficult to process than normal faces and this may increase the component’s amplitude due to the superimposition of a long-lasting temporal negativity associated with difficulty15. The second explanation
rests on a recent fMRI study indicating that inverted faces recruit both the regions involved in face and object processing whereas normal faces or normal objects recruit only face-specific substrates or object processing substrates25. The larger amplitude of the N170 observed for inverted faces as compared to upright faces may thus be due to inverted faces recruiting brain areas specifically sensitive to faces and those areas more generally involved in object recognition. At the same time, the fact that general object recognition usually operates by default at the basic-level may account for why an N170 latency delay and increase of activity is not observed in early visual categorization when objects are inverted. Supporting this conjecture, basic-level recognition is only slightly disrupted by inversion.
According to the face-specific processing hypothesis, the result of a specific increase and delay of the N170 to faces is indicative of a process dedicated to face recognition2, 28. An alternative hypothesis emphasizes the recognition processes rather than the object category differences. According to this subordinate-level expertise model, it is the experience of solving the problem of face recognition that tunes our visual recognition system so that it treats faces differently than other object classes. Whereas we recognize most objects at a basic-level level (e.g., chair, car or bird) we recognize faces at the individual level (e.g., Charlie or Lucy). The subordinate-level expertise hypothesis states that when an observer acquires experience discriminating between objects that share a similar configuration of parts (members of a homogeneous object class), the particular recognition processes that are recruited, as well as their corresponding neural substrates, will be the same as those used for face recognition, because the constraints are the same. For instance, previous behavioral studies8 as well as a fMRI studies11 have used the same novel stimuli as used here (Greebles) and found that effects which appeared face-specific for Greeble novices were also obtained for upright but not inverted Greebles for Greeble experts (trained with upright Greebles; see also7, for an inversion effect found in dog experts). Since our present results provide evidence for face-specific differences in the early processing of faces and objects which are not due to low-level visual differences, further studies should build on this finding, testing whether these face-specific effects may be observed in experts for non-face objects belonging to their domain of expertise, e.g. Greebles.
CONCLUSION
Using event-related potentials, we have demonstrated that the N170 potential is not specific to faces per se, as it has been previously claimed13,17. Moreover, our observations show that the N170 difference between non-face categories can be as large as between faces and these non-face categories. Thus such differences are not diagnostics for face-specific mechanisms. In order to claim face-specific mechanisms on the basis of electrophysiological data, one would need to present results revealing other face-specific effects. Our study provides one such result, showing that the N170 is enhanced and delayed only to face stimuli. What is still unclear is whether this robust difference in in the time-course of visual categorization is a consequence of class-specific processing mechanisms or to differences in the level of experience subjects generally have had with faces and control stimuli such as Greebles.
The first author is supported by the Belgian National Fund For Scientific Research (FNRS). We thank R. Zoontjes, B. de Gelder and J.-F Delvenne for use of the stimuli.
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Figure Legends
Figure 1- Above. Grand average waveforms obtained for normal and inverted faces at T6 (right occipito-temporal site) and T5. The N170 is larger and delayed for inverted faces as compared to normal faces at both sites. Below. Grand average waveforms obtained for faces and for all other objects categories. Overall, the N170 is significantly larger for faces than for objects at both occipito-temporal sites.
Figure 2- Comparison of waveforms obtained at T6 (right occipito-temporal electrode site) for normal and inverted stimuli for all the visual categories tested in this study. A larger and delayed N170 to inverted stimuli is observed only for faces.
Faces Inverted faces
Faces Objects
Figure 1
Faces
Cars
Houses Greebles
INVERTED
Figure 2
最新国际质量管理文件 管理体系过程方法的概念和使用指南 1 引言 本文件为理解“过程方法”的概念、意图及其在ISO9000族质量管理体系标准中的应用提供指南。本指南也可用于其他管理体系采用过程方法，不论组织的类型和规模如何。 本指南的目的是推动描述过程的方法的一致性，并使用与过程有关的术语。 过程方法的目的是提高组织在实现规定的目标方面的有效性和效率。 过程方法的好处有： ?对过程进行排列和整合，使策划的结果得以实现； ?能够在过程的有效性和效率上下功夫； ?向顾客和其他相关方提供组织一致性业绩方面的信任； ?组织内运作的透明性； ?通过有效使用资源，降低费用，缩短周期； ?获得不断改进的、一致的和可预料的结果； ?为受关注的和需优先安排的改进活动提供机会； ?鼓励人员参与，并说明其职责。 2 什么是过程？ “过程”可以定义为“将输入转化为输出的一组相互关联、相互作用的活动”。这些活动需要配置资源，如人员和材料。图1所示为通用的过程。
与其他方法相比，过程方法的主要优点是对这些过程间的相互作用和组织的职能层次间的接口进行管理和控制（在第4章中详细说明）。 输入和预期的输出可以是有形的（如设备、材料和元器件）或无形的（如能量或信息）。输出也可能是非预期的，如废料或污染。 每一个过程都有顾客和受过程影响的其他相关方（他们可以是组织内部的，也可以是外部的），他们根据其需求和期望规定所需要的输出。 应通过系统进行收集数据、分析数据，以提供有关过程业绩的信息，并确定纠正措施或改进的需求。 所有过程都应与组织的目标相一致，要规定所有过程都增值，与组织的规模和复杂程度相适应。 过程的有效性和效率可通过内部和外部评审过程予以评审。 3 过程的类型 可规定以下类型的过程 ——组织的管理过程。包括与战略策划、制定方针、建立目标、提供沟通、确保获得所需的资源和管理评审有关的过程。 ——资源管理过程。包括为组织的管理、实现、测量过程提供所需资源的所有过程。 ——实现过程。包括提供组织预期输出的所有过程。 ——测量、分析和改进过程。包括测量和收集业绩分析及提高有效性和效率的数据的那些过程，如测量、监视和审核过程，纠正和预防措施，它们是管理、资源管理和实现过程不可缺少的一部分。 4 过程方法的理解 过程方法是一种对如何使活动为顾客和其他相关方创造价值进行组织和管理的有力方法。
Mathematica函数大全--运算符及特殊符号一、运算符及特殊符号 Line1;执行Line，不显示结果 Line1,line2顺次执行Line1，2，并显示结果 ?name关于系统变量name的信息 ??name关于系统变量name的全部信息 !command执行Dos命令 n! N的阶乘 !!filename显示文件内容
a-b减 a*b或a b 乘 a/b除 a^b 乘方 base^^num以base为进位的数 lhs&&rhs且 lhs||rhs或 !lha非 ++,-- 自加1，自减1 +=,-=,*=,/= 同C语言 >,<,>=,<=,==,!=逻辑判断（同c） lhs=rhs立即赋值 lhs:=rhs建立动态赋值 lhs:>rhs建立替换规则 expr//funname相当于filename[expr] expr/.rule将规则rule应用于expr expr//.rule 将规则rule不断应用于expr知道不变为止param_ 名为param的一个任意表达式（形式变量）param__名为param的任意多个任意表达式（形式变量） 二、系统常数 Pi 3.1415....的无限精度数值 E 2.17828...的无限精度数值 Catalan 0.915966..卡塔兰常数 EulerGamma 0.5772....高斯常数 GoldenRatio 1.61803...黄金分割数 Degree Pi/180角度弧度换算 I复数单位 Infinity无穷大
一．实名登记完整流程 二．半天工程序操作指南 （一）主页（功能简介） 半天工程序了实现工地现场的务工人员实名登记管理、实名考勤管理、工资发放信息公示和其他系统管理功能。 （二）专户信息登记 1.管理工程 左键单击【专户信息登记】→【管理工程】，点击项目名称左边的图标，可查看项目专户信息，该信息不可更改，如需更改请联系半天工客服人员。
2.基本信息 左键单击【专户信息登记】→【基本信息】，可填写项目的基本信息。 注意：请程序操作人员（劳资专管员）正确填写本人身份证号，同时关注半天工微信公众号，在微信号内绑定个人身份证号码即可接收工人登记信息提醒。 3.微信用户管理 左键单击【专户信息登记】→【微信用户管理】，填写人员身份证信息，添加多个微信用户，关注微信公众号并绑定后即可接收工人登记信息提醒。 （三）实名登记管理 注意：务工人员实名制登记时，请严格按照【用工单位登记】→【务工班组登记】→【务工人员实名制进场登记】的顺序来操作。 1. 用工单位管理
左键单击【实名登记管理】→【用工单位管理】，填写用工单位信息。 2. 务工班组登记 左键单击【实名登记管理】→【务工班组登记】，添加务工班组信息（请准确选择务工班组所属企业）。如需删除或修改务工班组信息，请点击班组名称左边图标进行编辑操作。 3. 务工人员实名制进场登记 注意：务工人员实名制登记进场有两种操作方法，一是在【实名登记管理】→【务工人员实名制进场登记】界面进行人员进场登记，二是在【实名登记管理】→【工人花名册】界面进行人员进场登记。这里先介绍第一种方法，第二种方法在下面会介绍。 （1）左键单击【实名登记管理】→【务工人员实名制进场登记】，点击左侧的[全部班组] →[xxx单位] →[xxx班组]，然后点击右侧的[进场登记]，开始进场登记操作。
Math Studio—— Math Studio1M Catalog Catalog Math Studio https://www.doczj.com/doc/7118420231.html,/Manual Manual Wolfram mathematica Det det diff Diff ALGEBRA Apart, Coefficient, Degree , Denominator, Divisors , DivisorSigma, Eval, Expand , Factor, GCD, LCM , PolyDivide, PolyFit, PolyGCD , PolyLCM, PowerExpand, Quotient , Remainder, Sequence, SimplifyPoly , Solve, SolveSystem, Together BASIC Abs, Arg, Conj, Exp, Hyperbolic Functions, Im , Imag, Ln, Log, Re, Real, Trigonometric Functions CALCULUS D, Diff, DSolve , fDiff, FourierCos, FourierSeries , FourierSin, iDiff, iLaplace , Integrate, Laplace, Limit , NIntegrate, pDiff, Product , Series, Sum CAS Append Call,Caps , Char
, Choose, Clear, Command, Date , Delete, Extract , Function, Insert, IsList , IsMatrix, IsNumber , IsPoly, Left, Length , List, Matrix , Part, Replace, Reshape , Reverse, Right , Size, Sort, String , Value Converts a string to a value, Variables DATA Constant, Finance , HRStoHMS , LoadList , LoadMatrix, Table ELEMENTARY Binomial, Ceil, Eulerian, Factorial, Floor , fPart, iPart , Mod, Multinomial, nCr , nPr, nRoot, Pochhammer, Round, Sign, Sqrt GRAPHING clip, FullRectSineWave, HalfRectSineWave( ), SawToothWave(), SquareWave, StaircaseWave, TriangleWave MANUAL Code Files, Commands, Creating Scripts, Entering Expressions, Graphing Equations, Include Folder, Lists, Matrices, Strings , Symbols, Time Graphing MATRIX
小程序使用说明文档 1.登录角色： 本次小程序主要支持的登录角色有：代理商、业务员两种角色 2.功能模块： 本次一期小程序主要实现的功能有三个，第一个商户经营状况查询；第二个商户预警提醒功能；第三个数据罗盘。 (1)商户经营状况 商户经营状况中，可以查看到所登录角色下属的所有活跃商户的交易状况（如果某个商户某一天一条交易记录都没有，那么它不会出现在当天的经营状况列表里面） 商户经营状况可以按照商户名查询某一个商户的经营状况；也可以按照具体某一天，或者按月来查询下属商户的交易状况；当然，这两个条件是可以组合使用的，你可以查询下属商户某一天或者某一个月份的经营情况！ (2)商户预警 预警提醒功能分两个页签：“预警信息”和“等待确认”，都可支持按照商户名进行搜索 预警提醒中会显示登录角色下属的所有昨天交易量相对前天有所下降的商户，并且会按照下降比例从高到低的顺序进行排序。在预警提醒中，可以预警商户进行操作。 对于处于正常波动范围内的商户，点击长按，在弹出框中点击“忽略”，即可从预警信息列表中清楚该数据。
对于下降比例不正常的商户，点击长按，在弹出框中选择“等待确认”，即可把该条记录添加到等待确认列表中（预警列表中的数据每天都会刷新，所以请务必记得把异常商户及时添加到等待确认列表！）。 等待确认列表列表中显示当前登录角色从预警信息列表中添加过来的所有商户数据，在改列表中可以对商户进行处理。 对于不小心误操作过来的商户，可以点击长按，选择“正常”，从该列表中清楚该条数据。 对于无法挽回的商户，点击长按，在弹出框中选择“确认流失”，填写流失原因说明（必填！）后可从该列表中移出该条数据。 对于已经做出处理并挽回的商户，点击长按，在弹出框中选择“确认处理”，填写处理方法（必填）后，可从该列表中移出该记录。 对于所有添加到等待确认列表中的商户，具体的处理方法和处理说明记录，都有在数据库做记录。 (3)数据罗盘 数据罗盘主要是展示当前登录角色下的所有商户的交易情况的一些汇总信息。如：昨日交易总金额、较上周昨日同比增长或下降比例，昨日交易总笔数、较上周昨日同比增长或下降比例，累计开户数、本月新开户数；以及下属商户的星级占比饼图。 昨日交易总金额、较上周昨日同比增长或下降比例：昨日交易金额是指当前登录角色昨天的首款总额；较上周昨日同比增长或下降比例是指，昨天的交易总额和上周的同一天（如昨天是周二，就和上周二进行比较）的交易总额的上浮或下降比例[（昨天交易金额-上周昨日交易金额）/上周昨日交易金额] 昨日交易总笔数、较上周昨日同比增长或下降比例:比较方式与昨日交易总金额一样，只是以笔数为统计单位。 累计开户数和本月新开户数：累计开户数是当前登录角色下属所有的商户个数；本月新开是指进件日期为当前月份的商户个数。 星级排行：即后台的商户星级排行功能以饼图形式的展现，类别“其它”是指暂时没有星级的
(仅供内部使用) 文档作者：_____________________ 日期：___/___/___ 说明书校对：_____________________ 日期：___/___/___ 产品经理：_____________________ 日期：___/___/___ 请在这里输入公司名称 版权所有不得复制
软件使用说明书模板 1引言 1 .1编写目的 编写本使用说明的目的是充分叙述本软件所能实现的功能及其运行环境，以便使用者了解本软件的使用范围和使用方法，并为软件的维护和更新提供必要的信息。 1 .2参考资料 略 1 .3术语和缩写词 略 2 软件概述 2 .1软件用途 本软件的开发是为具有电能质量仪表，可以获取电能数据的技术人员提供一个有利的分析工具。 2 .2软件运行 本软件运行在PC 及其兼容机上，使用WINDOWS 操作系统，在软件安装后，直接点击相应图标，就可以显示出软件的主菜单，进行需要的软件操作。 2 .3系统配置 本软件要求在PC 及其兼容机上运行，要求奔腾II以上CPU，64兆以上内存，10G 以上硬盘。软件需要有WINDOWS 98 操作系统环境。 2 .4软件结构 略 2 .5软件性能 略 2 .6输入、处理、输出 2 .6.1输入 略 2 .6.2处理 略 2 .6.3输出 分析数据为： 略
图表有： 略 3 软件使用过程 3 .1软件安装 直接点击软件的安装软件SETUP.EXE ；然后按照软件的提示进行。 3 .2运行表 略 3 .3运行步骤 略 3 .4运行说明 略 3 .4.1控制输入 按照软件的说明，将测试数据加入到软件中；具体过程如下： 略 3 .4.2管理信息 软件运行过程中的密码键入： 略 3 .4.3输入输出文件 略 3 .4.4输出报告 略 3 .4.5输出报告复制 略 3 .4.6再启动及恢复过程 略 3 .5出错处理 软件运行过程中可能雏形的出物及处理如下： 略 3 .6非常规过程 如果出现不可能处理的问题，可以直接与公司的技术支持人员联系：略
一、运算符及特殊符号 Line1; 执行Line，不显示结果 Line1,line2 顺次执行Line1，2，并显示结果?name 关于系统变量name的信息 ??name 关于系统变量name的全部信息 !command 执行Dos命令 n! N的阶乘 !!filename 显示文件内容 < Expr>> filename 打开文件写 Expr>>>filename 打开文件从文件末写 () 结合率 [] 函数 {} 一个表 <*Math Fun*> 在c语言中使用math的函数(*Note*) 程序的注释 #n 第n个参数 ## 所有参数 rule& 把rule作用于后面的式子 % 前一次的输出 %% 倒数第二次的输出
%n 第n个输出 var::note 变量var的注释 "Astring " 字符串 Context ` 上下文 a+b 加 a-b 减 a*b或a b 乘 a/b 除 a^b 乘方 base^^num 以base为进位的数 lhs&&rhs 且 lhs||rhs 或 !lha 非 ++,-- 自加1，自减1 +=,-=,*=,/= 同C语言 >,<,>=,<=,==,!= 逻辑判断（同c） lhs=rhs 立即赋值 lhs:=rhs 建立动态赋值 lhs:>rhs 建立替换规则 lhs->rhs 建立替换规则 expr//funname 相当于filename[expr] expr/.rule 将规则rule应用于expr expr//.rule 将规则rule不断应用于expr知道不变为止
param_ 名为param的一个任意表达式（形式变量） param__ 名为param的任意多个任意表达式（形式变量） ————————————————————————————————————— 二、系统常数 Pi 3.1415....的无限精度数值 E 2.17828...的无限精度数值 Catalan 0.915966..卡塔兰常数 EulerGamma 0.5772....高斯常数 GoldenRatio 1.61803...黄金分割数 Degree Pi/180角度弧度换算 I 复数单位 Infinity 无穷大 -Infinity 负无穷大 ComplexInfinity 复无穷大 Indeterminate 不定式 ————————————————————————————————————— 三、代数计算 Expand[expr] 展开表达式 Factor[expr] 展开表达式 Simplify[expr] 化简表达式
函数以及其他一些库函数 函数名称: abs 函数原型: int abs(int x); 函数功能: 求整数x的绝对值 函数返回: 计算结果 参数说明: 所属文件: <>,<> 使用范例： #include <> #include <> int main() { int number=-1234; printf("number: %d absolute value: %d",number,abs(number)); return 0; } @函数名称: fabs 函数原型: double fabs(double x); 函数功能: 求x的绝对值. 函数返回: 计算结果 参数说明: 所属文件: <> 使用范例： #include <> #include <> int main() { float number=; printf("number: %f absolute value: %f",number,fabs(number)); return 0; } @函数名称: cabs 函数原型: double cabs(struct complex znum) 函数功能: 求复数的绝对值 函数返回: 复数的绝对值 参数说明: zuum为用结构struct complex表示的复数，定义如下：struct complex{ double m; double n; }
所属文件: <> #include <> #include <> int main() { struct complex z; double val; =; =; val=cabs(z); printf("The absolute value of %.2lfi %.2lfj is %.2lf",,,val); return 0; } @函数名称: ceil 函数原型: double ceil(double num) 函数功能: 得到不小于num的最小整数 函数返回: 用双精度表示的最小整数 参数说明: num-实数 所属文件: <> #include <> #include <> int main() { double number=; double down,up; down=floor(number); up=ceil(number); printf("original number %",number); printf("number rounded down %",down); printf("number rounded up %",up); return 0; } @函数名称: sin 函数原型: double sin(double x); 函数功能: 计算sinx的值.正弦函数 函数返回: 计算结果 参数说明: 单位为弧度 所属文件: <> 使用范例：
Win32程序快速入门指南 1.程序说明 示例程序放在Win32ShapeOrg中 1.1_tWinMain _tWinMain是程序入口。 while (GetMessage(&msg, NULL, 0, 0))开始为消息处理循环。 如果程序运行到此处将进入一个消息响应过程，即如果有消息就会进入消息响应函数 LRESULT CALLBACK WndProc(HWND hWnd, UINT message, WPARAM wParam, LPARAM lParam) 1.2全局初始化 如果要做全局初始化可以在tWinMain函数中while (GetMessage(&msg, NULL, 0, 0))前1.3消息响应机制 win32程序是基于消息响应的，最核心的模块是消息响应函数 LRESULT CALLBACK WndProc(HWND hWnd, UINT message, WPARAM wParam, LPARAM lParam) 消息是依附在某个窗口的。其中hWnd是窗口句柄，windows程序里，每一个窗口都有一个HWND类型的句柄用于标识这个窗口。 message是UINT类型的消息，实质上整数，消息的其它信息包含在wParam和lParam中 1.4绘制函数 WM_PAINT是绘制消息，所有和绘制相关的代码都放在WM_PAINT消息响应部分（具体在hdc = BeginPaint(hWnd, &ps);和EndPaint(hWnd, &ps);之间），win32所有绘制函数都带有一个HDC类型设备上下文句柄的参数。 InvalidateRect(hWnd, NULL, true);语句会发出绘制消息。如果需要更新绘制画面，就可以调用此语句。 Windows绘制机制的基础是图像设备交互(GDI，Graphics Device Interface)。 Brush是用来填充的刷子，绘制的东西是实心的。Pen相当于画笔，用来描述绘制直线曲线时的颜色粗细样式等等。如果不做设置，系统会提供默认的设置。 直线和曲线函数在这里，这是一个直线段的例子。 //显示文字 char cMessage[128]; sprintf(cMessage, "%d, %d", g_xPos, g_yPos); SetBkMode(hdc, TRANSPARENT); //设置背景透明显示模式 TextOut(hdc, g_xPos+10, g_yPos-10, cMessage, strlen(cMessage)); //实心椭圆 int r = 9; Ellipse(hdc, g_xPos-r, g_yPos-r, g_xPos+r, g_yPos+r); //当前位置坐标&位置+直径： //空心椭圆 MoveToEx(hdc, g_xPos+r, g_yPos, 0); AngleArc(hdc, g_xPos, g_yPos, r, 0, 360); //画线
MATLAB 软件使用指南 2009年3月 中国科学院计算机网络信息中心超级计算中心
目录 MATLAB 软件使用指南 (1) 目录 (2) 1. 软件介绍 (3) 2. 软件的安装与测试 (4) 2.1 安装目录及安装信息 (4) 2.2 测试结果 (5) 3. 软件的运行使用方法 (13)
1. 软件介绍 MATLAB 是一种用于算法开发、数据可视化、数据分析以及数值计算的高级技术计算语言和交互式环境。使用 MATLAB，您可以较使用传统的编程语言（如C、C++ 和 Fortran）更快地解决技术计算问题。 MATLAB 的应用范围非常广，包括信号和图像处理、通讯、控制系统设计、测试和测量、财务建模和分析以及计算生物学等众多应用领域。附加的工具箱（单独提供的专用MATLAB 函数集）扩展了 MATLAB 环境，以解决这些应用领域内特定类型的问题。 MATLAB 提供了很多用于记录和分享工作成果的功能。可以将您的 MATLAB 代码与其他语言和应用程序集成，来分发您的 MATLAB 算法和应用。 主要功能 此高级语言可用于技术计算 此开发环境可对代码、文件和数据进行管理 交互式工具可以按迭代的方式探查、设计及求解问题 数学函数可用于线性代数、统计、傅立叶分析、筛选、优化以及数值积分等 二维和三维图形函数可用于可视化数据 各种工具可用于构建自定义的图形用户界面 各种函数可将基于 MATLAB 的算法与外部应用程序和语言（如C、C++、Fortran、Java、COM 以及 Microsoft Excel）集成 更多详细信息，请参考以下网页： https://www.doczj.com/doc/7118420231.html,/products/matlab/description1.html
cmath是c++语言中的库函数,其中的c表示函数是来自c标准库的函数，math为数学常用库函数。cmath 库函数列表: using ::abs; //绝对值 using ::acos; //反余弦 using ::acosf; //反余弦 using ::acosl; //反余弦 using ::asin; //反正弦 using ::asinf; //反正弦 using ::asinl; //反正弦 using ::atan; //反正切 using ::atan2; //y/x的反正切 using ::atan2f; //y/x的反正切 using ::atan2l; //y/x的反正切 using ::atanf; //反正切 using ::atanl; //反正切 using ::ceil; //上取整 using ::ceilf; //上取整 using ::ceill; //上取整 using ::cos; //余弦 using ::cosf; //余弦 using ::cosh; //双曲余弦 using ::coshf; //双曲余弦 using ::coshl; //双曲余弦 using ::cosl; //余弦 using ::exp; //指数值 using ::expf; //指数值 using ::expl; //指数值 using ::fabs; //绝对值 using ::fabsf; //绝对值 using ::fabsl; //绝对值 using ::floor; //下取整 using ::floorf; //下取整 using ::floorl; //下取整 using ::fmod; //求余 using ::fmodf; //求余 using ::fmodl; //求余 using ::frexp; //返回value=x*2n中x的值,n存贮在eptr中 using ::frexpf; //返回value=x*2n中x的值,n存贮在eptr中
公路路面设计程序系统（HPDS2017）使用说明 本说明由十一个部分和四个附件组成，它们是： 一、系统总说明 ----------------------------------------------------------------------- 1 二、系统主菜单窗口使用说明 ----------------------------------------------------------- 5 三、改建路段留用路面结构顶面当量回弹模量计算程序（HOC）使用说明----------------------- 6 四、沥青路面设计与验算程序（HAPDS）使用说明 ------------------------------------------ 8 五、路基验收时路段内实测路基顶面弯沉代表值计算程序（HOCG）使用说明-------------------- 15 六、路面交工验收时路段内实测路表弯沉代表值计算程序（HOCA）使用说明-------------------- 17 七、改建路段原路面当量回弹模量计算程序（HOC1）使用说明-------------------------------- 19 八、新建单层水泥混凝土路面设计程序（HCPD1）使用说明 ---------------------------------- 21 九、新建复合式水泥混凝土路面设计程序（HCPD2）使用说明 -------------------------------- 28 十、旧混凝土路面上加铺层设计程序（HCPD3）使用说明 ------------------------------------ 33 十一、基（垫）层或加铺层及新建路基交工验收弯沉值计算程序（HCPC）使用说明 ------------- 38 附件一、沥青路面材料代码与材料名称对照表 --------------------------------------------- 40 附件二、水泥混凝土路面基（垫）层材料代码与材料名称对照表 ----------------------------- 43 附件三、版权声明 --------------------------------------------------------------------- 44 附件四、作者简介 --------------------------------------------------------------------- 44 现分别叙述如下： 一、系统总说明 1.本系统是根据新发行的《公路沥青路面设计规范》JTG D50-2017和已发行的《公路水泥混凝土路面设计 规范》JTG D40-2011的有关内容编制的，共包括如下九个程序： （1）改建路段留用路面结构顶面当量回弹模量计算程序HOC （2）沥青路面设计与验算程序HAPDS （3）路基验收时路段内实测路基顶面弯沉代表值计算程序HOCG （4）路面交工验收时路段内实测路表弯沉代表值计算程序HOCA （5）改建路段原路面当量回弹模量计算程序HOC1 （6）新建单层水泥混凝土路面设计程序HCPD1 （7）新建复合式水泥混凝土路面设计程序HCPD2 （8）旧混凝土路面上加铺层设计程序HCPD3 （9）基（垫）层或加铺层及新建路基交工验收弯沉值计算程序HCPC 2.系统的特点 （1）采用Visual Basic 6.0 for Windows 语言编程，在Windows系统下运行，有良好的用户界面； （2）功能齐全，凡公路路面设计与计算所需的程序应有尽有； （3）计算速度快，精度高；
/* #include int abs( int num ); double fabs( double arg ); long labs( long num ); 函数返回num的绝对值 #include double acos( double arg ); 函数返回arg的反余弦值，arg的值应该在-1到1之间 #include double asin( double arg ); 函数返回arg的反正弦值，arg的值应该在-1到1之间 #include double atan( double arg ); 函数返回arg的反正切值 #include double atan2( double y, double x ); 函数返回y/x的反正切值，并且它可以通过x，y的符号判断 （x，y）所表示的象限，其返回的也是对应象限的角度值 #include double ceil( double num ); double floor( double arg ); ceil函数返回不小于num的最小整数，如num = 6.04, 则返回7.0 floor函数返回不大于num的最大的数，如num = 6.04, 则返回6.0 #include double cos( double arg ); double sin( double arg ); double tan( double arg ); 函数分别返回arg的余弦，正弦，正切值，arg都是用弧度表示 #include double cosh( double arg ); double sinh( double arg ); double tanh( double arg ); 函数分别返回arg的双曲余弦，双曲正弦，双曲正切，arg都是用弧度表示的 #include double fmod( double x, double y );
GRRM 1.00
Chemistry is a wonderful world with lots of unexplored materials, which is producible from ca. one hundred kinds of chemical elements. More than thirty millions of compounds have already been known, and now two millions of new chemical compounds are produced annually. Invention and discovery of chemical reactions among compounds have extensively been made by chemists. Eighty years ago, when quantum mechanics was discovered, all problems in chemistry seemed to be insolvable. Equations for chemical problems are so complex that many theoreticians had abandoned to solve the problems at that time. However, some theoretical chemists had continually made efforts to improve approximation techniques solving chemical problems until many problems could have been solved effectively by means of electronic computers and computational techniques. By virtue of recent developments, the range of quantum chemical treatments has rapidly been widened so that we are now able to apply them to various chemical problems. A theoretical technique based on quantum chemical calculations has made it possible to determine a stable geometrical structure and its energy in good accuracy for a chemical system without experiments. This is called “structure-optimization”, which can be used by anyone in nowadays. However, it requires an initial guess, which should be made on the basis of our experience or chemical intuition. Since no general method exists to find out suitable initial guesses, one cannot avoid try-and-errors before one finally obtains some valuable conclusions such as new compounds or new chemical reaction pathways. It follows that a global search of isomers and reaction pathways among them has never been accomplished except for very small systems not larger than a four atom system. It has been an unexplored summit to perform a global search of isomers and inter-conversion reaction pathways among them for a chemical system composed of more than four atoms. 1 GRRM 1.00 / Ohno&Maeda
实验五 SRIM计算重离子在材料中的剂量分布 一、实习目的和要求 （一）实习目的： 1、熟悉SRIM程序的基本使用方法，以及在辐射剂量和防护计算中的应用。 2、通过此程序仿真模拟重带电粒子入核的过程，获得离子在材料中的剂量分布。 3、通过进一步自学，利用SRIM程序解决实际工作中的碰到的一些实际问题。 （二）实习要求： 1、掌握SRIM软件的基本组成、操作方法； 2、利用SRIM对离子在不同物质中的射程进行计算分析； 3、对质子在不同固体靶中的径迹及剂量分布进行简单的计算，并对计算结果进行分析并绘图，得出结论。 二、SRIM程序简介 1、SRIM软件介绍 SRIM是模拟计算离子在靶材中能量损失和分布的程序组。它采用Monte Carlo方法，利用计算机模拟跟踪一大批入射粒子的运动。粒子的位置、能量损失以及次级粒子的各种参数都在整个跟踪过程中存储下来，最后得到各种所需物理量的期望值和相应的统计误差。该软件可以选择特定的入射离子及靶材种类，并可设置合适的加速电压。可以算不同粒子，以不同的能量，从不同的位置，以不同的角度入射到靶中的情况。SRIM中包含一个TRIM运算软件。 TRIM（Transport of Ions in Matter）是一个非常复杂的程序。它不仅可以描述离子在物质中的射程，还可以详细计算注入离子在慢化过程中对靶产生损伤等其他信息。它可以使用动画让你看到离子注入到靶中的全过程，并给你展示级联反冲粒子和靶原子混合在一起的情形。为了精确估计每个离子和靶原子间相遇时的物理情形，程序只能一次对一个粒子进行计算。这样的话，计算可能消耗可观的时间——计算每个离子花费的时间从一秒到几分钟不等。而精确度由模拟采用的离子数来决定。典型的情况是，应用1000个离子进行计算将得到好于10%的精确度。 软件特点：
函数
abs(arg)
fabs(arg) ceil（ arg) floor(arg) exp(arg) log(arg) log10(arg) pow(arg1,ar g2)
-camth-的数值函数
说明
返 回 arg 的 绝 对 值 ，其 类 型 与 arg 相 同 ，arg 可 以 是 任 意 浮 点 类 型 。在
-cmath-的三角函数
函数
cos(angle) sin(angle) tan(angle) cosh(angle)
sinh(angle)
tanh(angle)
acos(arg)
asin(arg)
atan(arg)
atan2(arg1,ar g2)
说明
返 回 angle 的 余 弦， angle 变 元 以 弧度 为单位 。 返 回 angle 的 正 弦， angle 变 元 以 弧度 为单位 。 返 回 angle 的 正 切， angle 变 元 以 弧度 为单位 。 返 回 angle 的 双 曲余 弦 { [e^x - e^(-x)]/2 }， angle 变 元 以 弧度 为单位。
返 回 angle 的 双 曲正 弦 { [e^x + e^(-x)]/2 }， angle 变 元 以 弧度 为单位。
返 回 angle 的 双 曲正 切 { [e^x + e^(-x)]/[e^x - e^(-x)] }， angle 变元以弧度为单位。
返 回 arg 的 反 余 弦 。变 元 在 1 到 -1 之 间 。结 果 以 弧 度 为 单 位 ， 其范围是 0 到π 。 返 回 arg 的 反 正 弦 。 变 元 在 1 到 -1 之 间 。 结 果 以 弧 度 为 单 位 ， 其 范 围是 -π /2 到 π /2。 返 回 arg 的 反 正 切 。 结 果 以 弧 度 为 单 位 ， 其 范 围 是 -π /2 到 π /2。
这 个 函 数 需 要 两 个 浮 点 类 型 的 变 元 ，返 回 arg1/arg2 的 反 正 切 。 结果以弧度为单位，范围是 0 到π ，类型与变元相同。
math.h 数学函数库,一些数学计算的公式的具体实现是放在math.h里,具体有： 1、三角函数 double sin (double);正弦 double cos (double);余弦 double tan (double);正切 2 、反三角函数 double asin (double); 结果介于[-PI/2, PI/2] double acos (double); 结果介于[0, PI] double atan (double); 反正切(主值), 结果介于[-PI/2, PI/2] double atan2 (double, double); 反正切(整圆值), 结果介于[-PI, PI] 3 、双曲三角函数 double sinh (double); double cosh (double); double tanh (double); 4 、指数与对数 double exp (double);求取自然数e的幂 double sqrt (double);开平方 double log (double); 以e为底的对数 double log10 (double);以10为底的对数 double pow(double x, double y）;计算以x为底数的y次幂 float powf(float x, float y); 功能与pow一致，只是输入与输出皆为浮点数 5 、取整 double ceil (double); 取上整 double floor (double); 取下整 6 、绝对值 double fabs (double);求绝对值 double cabs(struct complex znum) ;求复数的绝对值 7 、标准化浮点数 double frexp (double f, int *p); 标准化浮点数, f = x * 2^p, 已知f求x, p ( x介于[0.5, 1] ) double ldexp (double x, int p); 与frexp相反, 已知x, p求f 8 、取整与取余