text:
Motivation and emotion/Textbook/Emotion/Emotional expression

The History of Emotional Expression
The way in which individuals can, and do, express themselves emotionally is a phenomenon that has deep historical roots (Russell, Bachorowski, & Fernandez-Dols, 2003) with the importance of emotional expression being well documented as a result (Scherer & Ellgring, 2007). One of the first traditional assumptions of emotional expression was the suggestion that expressions of emotion were a God-given and universal language that was developed to reveal an individual’s passion, virtues, and vices (Russell, Bachorowski, & Fernandez-Dols). Having received a consensus of support, this hypothesis of emotional expression has since been incorporated into many philosophical, religious, and artistic theories from ancient times through to the nineteenth century (Montagu, 1994, cited in Russell, Bachorowski, & Fernandez-Dols). Further, this traditional assumption of emotional expression has also appeared in later works that were conducted by anatomists, physiologists, and other scientists, including the work of Charles Darwin (1872, cited in Russell, Bachorowski, & Fernandez-Dols). With focus on Darwin’s work, Darwin argued that, although relying on the traditional assumptions of emotional expression, emotional expression was the result of natural selection, not God-given (Russell, Bachorowski, & Fernandez-Dols). Further, in an attempt to support his argument for the substitution, Darwin conducted important observational studies that examined cross-species and cross-cultural similarities in emotional expression (Russell, Bachorowski, & Fernandez-Dols). It has been suggested that results of these studies did support the substitution (Russell, Bachorowski, & Fernandez-Dols). Further, in regard to facial expression analysis, Darwin demonstrated the universality off facial expressions and their continuity in both humans and animals. He also claimed, amongst other things, that individuals have specific inborn emotions, which originated in serviceable associated habits (Fasel & Luettin, 2003).

It was not until 1962 that an additional theory of emotional expression was proposed by Sylvan Tomkins and advances were made to the traditional assumptions of emotional expression (Russell, Bachorowski, & Fernandez-Dols, 2003). Similar to Darwin (1872), Tomkins (1962) theory of emotional expression perpetuated many of the traditional assumptions about emotional expression (Russell, Bachorowski, & Fernandez-Dols). However, Tomkins further suggested that individuals have a fixed number of discrete emotions that can and will vary in levels of intensity (Russell, Bachorowski, & Fernandez-Dols). Further, according to Tomkins theory, each discrete emotion consists of a single brain process that produces all the various manifestations of emotion, including facial and vocal expression (Russell, Bachorowski, & Fernandez-Dols).

Furthermore, in 1971, Ekman and Friesen (1971, cited in Fasel & Luettin, 2003) developed the hypothesis that individuals have six primary emotions, each that were said to possess a distinctive content together with a unique facial expression. Further, it was also stated that these six primary emotions and the emotional displays that come with them, are universal across all human ethnicities and cultures and comprise happiness, sadness, fear, disgust, surprise, and anger (Fasel & Luettin). Moreover, modern evolutionary theories of emotional expression have been developed that place emphasis on the idea of natural selection, the interests of the individual, adaption and function (Dawkins & Krebs 1978; Fridlund 1994; Owren &Rendall 2001, cited in Russell, Bachorowski, & Fernandez-Dols, 2003). Moreover, as stated by Russell, Bachorowski and Fernandez-Dols, modern theories have suggested that emotional expression is a multicomponent dynamic processes that is laced with cognition (Scherer, 2001; Smith & Kirby, 2001) that provides a looser, more flexible and context-dependent relation among the components of emotional expression (Bradley & Lang, 2000b).

Further to this, a fundamental recognition that has been made in these modern theories of emotional expression is the idea that, rather than Individuals broadcasting their emotional expressions to all other individuals, individuals will express their emotions in a way that will be directed at an intended receiver in an attempt to influence that receiver in ways that are beneficial to them (Russell, Bachorowski, & Fernandez-Dols, 2003).

A second fundamental recognition that has been made in regard to emotional expression is the suggestion that a receiver does not simply decode the message of a sender in a reflex like manner (Russell, Bachorowski, & Fernandez-Dols, 2003). Rather, it is suggested that a variety of effects can occur (Russell, Bachorowski, & Fernandez-Dols). To elaborate, Russell, Bachorowski, and Fernandez-Dols suggest that vocal stimuli has the ability to capture the receivers attention and alter their affective state without any emotion being encoded or decoded (Owren et al, 2002, cited in Russell, Bachorowski, & Fernandez-Dols).

Emotional Expression- A Social Cognition
Social cognition can be defined as an individual’s ability to interpret and predict the behaviour of other individuals, as well as interact in complex social environments and relationships (Baron-Cohen et al., 2000, cited in Keightley, Chiew, Winocur, & Grady, 2007). Further, an individual’s ability to comprehend and respond to the emotional content and cues that are present in the environment as well as their ability to remember this emotional content are stated as integral parts of social cognition ( Grady & Keightley, 2002; Adolphs, 2003, cited in Keightley et al.). With focus on the emotional content and cues that are present in an individual’s environment, it is stated that individuals will express their emotions through signs in their face, voice, gestures, and bodily posture (Scherer & Ellgring, 2007). With particular focus on an individual’s face and voice, it is stated that the changes that occur in an individual’s face and voice are essential aspects of human social interaction (Russell, Bachorowski, & Fernandez-Dols, 2003). Further, it is often suggested that individuals who lack such aspects in their social interactions could be considered to be suffering from a mental illness, as this is one of several indicators (Russell, Bachorowski, & Fernandez-Dols).

Furthermore, it has been stated that facial expressions are important as a part of non-verbal communication that is used in everyday life (Batty & Taylor, 2003). Facial expressions are complex stimuli that have the ability to convey an individual’s identity, as well as their emotional state (4). They provide individuals with a means of communication that are more rapid than language, that give people the ability to quickly infer the state of mind of their companions (Batty & Taylor). Further, it is suggested that a person’s ability to perceive and differentiate between another person’s facial expressions is vital for successful social communication, as well as survival (Hadj-Bouziane, Bell, Knusten, Ungerleider, & Tootell, 2008; Jiang, & He, 2006). Further to this, changes in the emotional expressions of individuals are believed to be informative as individuals can use this information to judge others’ attitudes, regulate others’ emotions and to guide social interaction (Salovey & Mayer, 1990, cited in Niedenthal, Brauer, Halberstadt, Innes-Ker, 2001). Moreover, it is also suggested that facial expressions allow an individual to easily understand the opinions and attitudes of others, thus providing a powerful tool in social coordination (Batty & Taylor).

Facial Expressions and the Brain
As previously mentioned, Ekman and Friesen (1971) proposed that individuals have six primary, or basic, emotions, each that possess a distinctive content together with a unique facial expression (Fasel & Luettin, 2003). These basic emotions consist of happiness, sadness, fear, disgust, surprise, and anger (Fasel & Luettin; Batty & Taylor, 2003; Keightley et al., 2007). Moreover, several research studies have been conducted to assess if there are different brain processes involved in the processing of these six basic emotions and the way they are expressed by individuals. Further, it has been hypothesised that different neural mechanisms may be important depending on the particular facial expression that is being displayed and processed (Phelps, 2006; Jiang & He, 2006). Although, it is has also been suggested that these brain regions should not be thought to be specialized for recognizing a specific facial expression, but rather should be thought to have a more general role in the processing of different emotions. To elaborate, due to the results of several brain imaging studies, it has firstly been suggested that some regions of the fusiform gyrus are specialized for the recognition of facial identity ( Kanwisher et al., 1997, cited in Phelps). Further, it has also been suggested that the amygdala is a critical component of a network of regions that are involved in social cognition, particularly for the processing of emotions in facial expressions ( Gobbini & Haxby, 2007, cited in Keightley et al., 2007) and lesions to the amygdale will commonly disrupt this ability in effected individuals (Adolphs et al., 1994, 1999; Anderson & Phelps, 2000, cited in Keightley et al). Moreover, several studies have concluded that an individual’s amygdala will become highly active when negative facial expressions are viewed, particularly fear (Breiter et al., 1999; Morris et al., 1996; Whalen et al., 1998b; Blair et al., 1999, Pessoa et al., 2002; Anderson et al., 2003, cited in Keightley et al). This suggestion is supported by research that has been conducted on patients suffering from amygdale damage. Results of these studies demonstrated that individuals with amygdale damage commonly experience impairments in their ability to identify the intensity of fear in any facial expressions they viewed (Adolph et al., 1999, cited in Phelps). However, these individuals are still able to generate normal facial expressions of fear (Anderson & Phelps, 2000) but consistently rate expressions of fear in others as less fearful than do participating individuals with no amygdale damage (Phelps).

Furthermore, several studies have also suggested that the amygdala is involved in the processing of sad faces (Blair, Morris, Frith, Perrett, Dolan, 1999; Schneider, Habel, Kessler, Salloum, & Posse, cited in Batty & Taylor, 2003). Additionally, several studies have also demonstrated that positive facial expressions, such as happy facial expressions, activate the cingulated sulcus in the individual’s brain (Kesler-West, Andersen, Smith, Avison, & Mapp, 2000; Davis, R.J. Kryscio et al, 39, cited in Batty & Taylor), and negative facial expressions, such as angry faces, are processes by the orbital frontal regions of the brain (Blair et al., cited in Batty & Taylor), with the basal ganglia also playing a role in the recognition of anger (Phelps). Lastly, it appears that facial expressions that portray disgust activate the basal ganglia and insular cortex area of the brain when present in social and non-social stimuli (Cohen, Minor, Najolia, & Lee Hong, 2009; Phelps).

The Recognition of Facial Expressions
Given the evidence that suggests that different neural mechanisms may underlie the recognition of facial expressions, it is almost predictable that it would be suggest that different types of information are necessary to make judgements about these facial expressions (Phelps, 2006). For example, two studies that have investigated the amygdalas role in an individual’s ability to identify fearful facial expressions have revealed that an individual’s ability to recognize fear in facial stimuli depends critically on the specific configuration of the different facial features of the individuals face (Phelps). To elaborate, it has been suggested that any subtle changes that occur in the configuration of an individual’s face, such as altering the distance between the eyes and mouth, can significantly impair an individual’s ability to recognize the correct facial expression (Phelps). Additionally, results of other studies have revealed that an individual’s ability to recognize fear from facial stimuli seems to depend critically on a single facial feature, that is, the eyes (Phelps).

Furthermore, Adolphs et al. (2005a) used a technique that helped to identify which aspects of a face are most important when recognizing facial expressions (Phelps, 2006). Consistent with previous imaging results (Whalen et al., 2004) an individual’s ability to identify fear in facial expression depends critically on the eyes (Phelps). Moreover, their research examined the eye movement that was involved when an individual was asked to recognise fear in a facial expression (Phelps). Participants consisted of individuals with bilateral damage to the amygdale, with the control group consisting of individuals with no bilateral damage to their amygdale. Results revealed that individuals with no bilateral damage predominately fixated on the eyes (Phelps). Further to this, participants with bilateral damage to the amygdale showed a different pattern of eye movements, which suggests that they rely on different aspects of the face to identify fear (Phelps). However, when these participants were instructed to remain fixated on the eyes, their ability to recognise fear in facial expressions improved to the levels demonstrated by individuals with no bilateral damage (Phelps). However, they failed to adopt this strategy when not given the instructions to do so (Phelps). These results demonstrate that the amygdale not only responds preferentially to the eyes in facial stimuli, but may also be involved in generating behaviours that aid in the identification of facial expressions of fear (Phelps).

Facial Feedback Hypothesis
The facial feedback hypothesis (FFH) is the assumption that the feedback an individual receives from another person’s facial behaviour, otherwise known as facial expressions, that is then transformed into conscious awareness, constitutes the experience of emotion (Liard, 1974, Tomkins, 1962, 1963, cited in Reeve, 2009). Further to this, there are two testable versions of the FFH, that is, the strong version and the weak version (McIntosh, 1996; Rutledge & Hupke, 1985, cited in Reeve). As these two testable versions exist, research on the validity of the FFH has used two different methodologies (Reeve). To elaborate, the strong version of the FFH proposes that if an individual manipulates their facial musculature into a pattern that corresponds to an emotional display, the emotion that is displayed will activate that emotional experience in the individual (Reeve). For example, Reeve suggests that if an individual is instructed to manipulate their facial musculature so that their eyebrows become raised and pulled together, their upper eyelids also become raised and their lips are stretched horizontally back towards your ears, and were then instructed to complete a questionnaire that assessed their emotional state, it would be suspected that fear would become activated in their emotional experience and their responses would support this.

Moreover, the weaker and more conservative version of the FFH, that has also received a consensus of support, proposes that an individual’s facial feedback modifies the intensity of the emotion, not causes it (Reeve, 2009). Further, this version suggests that an individual’s particular facial musculature display will help to exaggerate an emotional experience (Reeve). However, it will not necessarily activate the emotional experience as suggested by the stronger version of the FFH (Reeve). To elaborate, if an individual intentionally produces a smile when they are already experiencing feelings of happiness, then they will feel more intense happiness (Reeve). This is supported by a study that was conducted to assess the validity of this version of the FFH (Reeve). In this study, participating individuals were required to either exaggerate or suppress the spontaneous facial expressions they produce while watching a video, which depicted either a pleasant, neutral, or unpleasant scenario (Zuckerman, Klorman, Larrance, & Spiegal, 1981, cited in Reeve). Results of this study supported this suggestion as participants demonstrated an exaggerated emotional and psychological experience when instructed to exaggerate their naturally occurring facial expressions (Reeve). Alternatively, participants demonstrated a softened emotional and psychological experience when instructed to suppress their naturally occurring facial expressions (Lanzetta, Cartwright-Smith, & Kleck, 1976, Reeve).

Cultural Differences in Facial Emotion
Multiple research studies have been conducted in an attempt identify if there is a proposition that human beings display similar facial expressions regardless of any cultural differences that may be present (Ekman, 1972, 1994b; Izard, 1994, cited in Reeves, 2009). In each of these studies, participants that were diverse in their nationalities reviewed three photographs, each which displayed a different facial expression from the last (Ekman, 1972, 1993; Ekman & Friesen, 1971: Ekman, Sorenson, & Friesen, 1969, Izard, 1971, 1980, 1994, cited in Reeves). Once all photographs had been viewed, all participants were then instructed to select the photograph that they believed expressed a particular emotion (Reeve). This was done through a multiple choice format (Reeve). For example, participants reviewed three photographs of three different faces, one that expressed anger, one that expressed joy, and one that expressed fear (Reeve). Participants were instructed to select the photograph that they believed best demonstrated what a face would look like when they encountered a specified emotion, such as anger (Reeve). Results of this study revealed that individuals matched the same facial expressions with the same emotions regardless of their cultures, thus providing evidence that facial emotion is cross-culturally universal (Ekman, 1994b; Ekman & Friesen, 1971; Izard, 1971, cited in Reeve). This is also evidence that emotional related facial behaviour has an innate unlearned component (Reeve).

Age Related Differences in Perceiving Emotion
The idea that age can affect an individual’s social cognition has also received considerable attention in recent years, particularly the effect of age on perceiving emotions in faces (Keightley et al., 2007). As a result, many studies have been conducted to assess if any age related differences exist in regard to the recognition of facial expressions (Keightley). For example, Keightley et al. conducted a study that used Functional Magnetic Resonance Imaging (fMRI) to explore the brain activity of younger and older adults while they reviewed and labelled faces that expressed different emotions as well as neutral expressions. Similar to the results of a study conducted by McDowell et al. (1994), results demonstrated that older adults experienced significantly greater difficulty in the task of identifying negative and neutral facial expressions such as those of sadness, anger, and disgust compared to younger adults (Keightley). Further, there were no significant findings that indicated age related differences in regard to identifying expressions of happiness (Keightley). Additionally, the functional neuroimaging data that was obtained in this study revealed that both younger and older adults recruit a different pattern of activity when distinguishing certain facial expressions (Keightley). To elaborate, results demonstrated that older adults have increased activity in their ventromedial prefrontal cortex, lingual gyrus and premotor cortex when distinguishing happy expressions, whereas younger adults recruited a more widely distributed set of regions which included the amygdala, ventromedial prefrontal cortex, lateral prefrontal regions and bilateral inferior and superior temporal areas (Keightley). Conversely, younger adults demonstrated an increase in activity in their dorsal anterior cingulated when distinguishing other types of facial expressions, whereas older adults had more activity in their dorsal cingulated, as well as their middle and inferior frontal gyri, somatosensory cortex, insula, and middle temporal regions (Keightley). These results indicate that there are possible age related differences in the cognitive strategies that are employed by individuals of different ages during the identification of facial expressions (Keightley).

Emotional Suppression- The Consequences
Few studies have been conducted that have assessed the impact of suppression on participants’ emotional experience and several consequences have been suggested (Niedenthal, Krauth-Gruber, & Ric, 2006). Firstly, McCanne and Anderson (1987) conducted a study that demonstrated that individuals who suppress their emotional experience can impair their capacity to feel any corresponding emotions (Niedenthal, Krauth-Gruber, & Ric). Further, based on the research results of Gross and Levenson (1997), it has also been suggested that, although suppression has been found to reduce facial expressions of positive and negative emotions, suppression does not completely inhibit it (Niedenthal, Krauth-Gruber, & Ric). Moreover, research results seem to indicate that suppression of emotional experience can reduce the experience of positive emotions in an individual but will not reduce the experience of negative emotions (Niedenthal, Krauth-Gruber, & Ric).

Vocal Emotional Expression
A significant amount of theoretical and empirical research has also been conducted concerning the production of and response of vocal expressions of emotion in the past decade (Cohen, Minor, Najolia, & Lee Hong, 2009). Further, it is suggested that vocal expressions provide an attractive medium for understanding emotional expression as it also contains a wealth of information (Russell, Bachorowski, & Fernandez-Dols, 2003). Moreover, it is suggested that this information can be conveyed across multiple domains, including those in prosodic and content channels (Cohen et al.). To elaborate, prosodic channels involve the nonverbal aspects of spoken communication, such as intensity, speech rate, and fundamental frequency and can be measured through acoustic analysis of the communications physical properties (Schirmer, Striano, & Friederici, 2005). Further to this, a specific profile of these parameters is believed to be indicative of a specific emotion (Schirmer, Striano, & Friederici). For example, it is suggested that anger can be characterised by a relatively high intensity, speech rate and fundamental frequency whereas sadness is characterised by low intensity, speech rate and fundamental frequency (Banse R, Scherer, 1996, cited in Schirmer, Striano, & Friederici). Conversely, content channels involve the semantic aspects of communication, which are assessed using a variety of content- analytic procedures (Russell, Bachorowski, & Fernandez-Dols, 2003).

Key terms

 * Social Cognition
 * Facial Expressions
 * Fusiform Gyrus
 * Amygdala
 * Cingulated Sulcus
 * Orbital Frontal Regions
 * Basal Ganglia
 * Insular Cortex
 * Bilateral Damage
 * Facial Feedback Hypothesis (FFH)
 * Facial Musculature
 * Functional Magnetic Resonance Imaging (fMRI)
 * Ventromedial Prefrontal Cortex
 * Lingual Gyrus
 * Premotor Cortex
 * Lateral Prefrontal Regions
 * Bilateral Inferior Area
 * Superior Temporal Area
 * Dorsal Anterior Cingulated
 * Somatosensory Cortex
 * Middle Temporal Regions
 * Prosodic Channel
 * Content Channel
 * Intensity
 * Speech Rate
 * Fundamental Frequency

Test your Knowledge: A Quick Quiz
{The first traditional assumptions of emotional expression was the suggestion that expressions of emotion? - Were a result of natural selection. - Were a result of an individuals hormones. + A God-given and universal language. - All of the above.
 * type=""}

{True or False? The amygdala is a critical component of a network of regions that are involved in social cognition, particularly for the processing of emotions in facial expressions. - False. + True.
 * type=""}

{Subtle changes that occur in the configuration of an individual’s face can significantly impair an individual’s ability to? - Identify the person. + Recognize the correct facial expressions. - Recognize happy facial expressions. - All of the above.
 * type=""}

{True of False? The strong version of the FFH proposes that an individual’s facial feedback modifies the intensity of the emotion, not causes it: + False - True
 * type=""}

{True or False? An individual’s ability to identify fear in facial expression depends critically on the eyes. + True - False
 * type=""}


license: Creative Commons - Attribution Share-Alike - https://creativecommons.org/licenses/by-sa/4.0/

title: Template:95%done


text: Template:Multiple image

title: Template:Multiple image


text:
Template:Multiple image/doc

This template creates a box with two to ten images arranged vertically or horizontally with captions for the entire box and/or for each images.

Usage
Important: If the image is under a free license like the GFDL or a CC license, you must not use this parameter as the terms of those licenses require the license, or a link to it, to be reproduced with the image. The image must, therefore, link to its image page. Public domain and (theoretically) fair use images are not subject to this restriction.
 * align defaults to "right"
 * direction defaults to "horizontal"
 * header_align defaults to "center"
 * footer_align defaults to "left"
 * image1 is name of first image. Similarly for image2, etc.
 * width1 is width of first image. Similarly for width2, etc.
 * width overrides width1, width2, width3, etc. If you want the images to be different sizes, do not provide the width parameter.
 * alt1 is alt text of first image (see WP:ALT). Similarly for alt2, etc.
 * caption1 is caption of first image. Similarly for caption2, etc.
 * link1 is where the first page links to. Similarly for link2, etc. If not specified, image links to the image description page as usual. If specified to the empty value, the image does not link anywhere; this should be used for purely decorative images as per WP:ALT.
 * background color defines the background color between the border of the box and the images.
 * header, footer, background color and captions are optional.

Horizontal placement: matching image heights

 * This trick is not required if both images have the same aspect ratio, where width suffices

 * Decide on a common height for the images, say 100 pixels.
 * Examine the individual image pages to obtain the full resolution. For example:
 * File:Donkey in Clovelly, North Devon, England.jpg shows "Donkey in Clovelly, North Devon, England.jpg (750 × 536 pixels, file size: 125 KB, MIME type: image/jpeg)"
 * File:Rainbow trout.png shows "Rainbow_trout.png (300 × 300 pixels, file size: 158 KB, MIME type: image/png)"
 * For non-square images, use these figures to construct an expression for the widthn parameters, of the form:
 * where
 * dh is the desired height in pixels (100)
 * ow is the original width in pixels (750 or 300)
 * oh is the original height in pixels (536 or 300)
 * For square images, simply set the appropriate widthn to be the desired height. If all are square, only width needs to be set.
 * For square images, simply set the appropriate widthn to be the desired height. If all are square, only width needs to be set.

title: Template:Multiple image/doc


text: Template:Multiple image/numImgs

title: Template:Multiple image/numImgs


text:
Numerical Analysis/Romberg's method

Romberg's method approximates a definite integral by applying Richardson extrapolation to the results of either the trapezoid rule or the midpoint rule.

The initial approximations $$R_{k,0}$$ are obtained by applying either the trapezoid or midpoint rule with $$2^k + 1$$ points. In the case of the trapezoid rule on $$\left[a,b\right]$$,

$$R_{k,0} = h_k \left(\frac{f(a) + f(b)}{2} + \sum_{i=1}^{2^k - 1} f(a + h_k i)\right)$$

where

$$h_{k} = \frac{b-a}{2^k}$$

For $$k > 0$$, we can reduce the number of places the function is evaluated by using our previously obtained approximations instead of re-sampling. For the trapezoid rule, this improvement gives

$$R_{0,0} = \left(\frac{b-a}{2}\right)[f(a)+f(b)]$$

and

$$R(k,0) = \frac{1}{2} R(k-1,0) + h_k \sum_{i=1}^{2^{k-1}} f(a + (2i-1)h_k)$$

Richardson's extrapolation is then applied recursively, giving

$$R_{k,j} = \frac{4^j R_{k,j-1}-R_{k-1,j-1}}{4^{j}-1}$$

Each successive level of improvement increases the order of error term from $$ O(h^{2j})$$ to $$O(h^{2j+2})$$ at the expense of doubling the number of places the function is evaluated.

title: Numerical Analysis/Romberg's method


text:
The periodic table/Iron

Discovery
It's harder to trace Iron (Latin: ferrum) than Gold (Au) and Silver (Ag) because Iron corrodes (meaning it's destroyed by chemical reactions) easily.

Iron has been used since the ancient times, from meteorites (which was probably the first time human beings dealt with iron) to Iraq [Mesopotamia] to Egypt (there has been evidence found that they were people in Iraq and Egypt smelting iron since 5000 BC). Iron was used commonly back in the Roman days (as Pliny the Elder said, "It is by the aid of iron that we construct houses, cleave rocks, and perform so many other useful offices in life").

The adding of carbon to iron was most likely an accident at first, but the incident, that happened in 1000BC, has been a way of improving iron, making steel.

Iron is used today in the modern world.

Quick Facts
Name: Iron

Symbol: Fe

Mass: 55.845

Classification: Transition Metals

Protons: 26

Electrons: 26

Neutrons: 29

Color: silvery

Discovered in: Unknown

Uses

 * Iron is used in steel, in which steel is used in civil engineering and manufacturing industries.
 * Iron is a necessary nutrient for all life forms.

title: The periodic table/Iron


text:
Numerical Analysis/Romberg Example

Use Romberg Integration to compute $$ R_{3,3} $$ for the following integral $$ \int_{0}^{\frac{\pi}{2}}cos x \,dx

$$

Solution:

$$

R_{1,1} = \frac{\pi}{4}[cos(0)+cos(\frac{\pi}{2})]

$$

$$ R_{1,1} = \frac{\pi}{4} $$

$$

R_{2,1} = \left(\frac{1}{2}\right)[R_{1,1}+h_{1}f(a+h_{2})]

$$

$$ R_{2,1} = \left(\frac{1}{2}\right)[\frac{\pi}{4}+\frac{\pi}{2}cos\left(\frac{\pi}{4}\right)] $$ $$ R_{2,1} = 1.178023457 $$

$$ R_{3,1} = \left(\frac{1}{2}\right)[R_{2,1}+h_{2}(f(a+h_{3})+f(a+3h_{3}))] $$ $$ R_{3,1} = \left(\frac{1}{2}\right)[1.178023457+\frac{\pi}{4}(cos(\frac{\pi}{8})+cos(\frac{3\pi}{8})] $$ $$ R_{3,1} = 1.374317658 $$

$$ R_{2,2} = R_{2,1}+ \frac{R_{2,1}-R_{1,1}}{4-1} $$ $$ R_{2,2} = 1.178023457 + \frac{.3926252936}{3} $$ $$ R_{2,2} = 1.308898555 $$

$$ R_{3,2} = R_{3,1} + \frac{R_{3,1}-R_{2,1}}{4-1} $$ $$ R_{3,2} = 1.374317658 + \frac{.196294201}{3} $$ $$ R_{3,2} = 1.439749058 $$

$$ R_{3,3} = R_{3,2} + \frac{R_{3,2}-R_{2,2}}{16-1} $$ $$ R_{3,3} = 1.439749058 + \frac{.1308505033}{15} $$ $$ R_{3,3} = 1.448472425 $$

title: Numerical Analysis/Romberg Example


text:
Numerical Analysis/Romberg Excercise

Use Romberg Integration to compute $$R_{3,3} $$ for $$ \int_0^1 x^2 e^{-x} \,dx$$ Solution: $$ R_{1,1} = \frac{h_{1}}{2}[f(0)+f(1)] $$ $$ R_{1,1} = \frac{1}{2}[0+\frac{1}{e}] $$ $$ R_{1,1} = .1839397206 $$

$$ R_{2,1} = \left(\frac{1}{2}\right)[R_{1,1}+h_{1}f(a+h_{2})] $$ $$ R_{2,1} = .1379547904 $$

$$ R_{3,1} = \left(\frac{1}{2}\right)[R_{2,1}+h_{2}(f(a+h_{3})+f(a+3h_{3}))] $$ $$ R_{3,1} = .1475727039 $$

$$ R_{2,2} = R_{2,1}+ \frac{R_{2,1}-R_{1,1}}{4-1} $$ $$ R_{2,2} = .1226264803 $$

$$ R_{3,2} = R_{3,1} + \frac{R_{3,1}-R_{2,1}}{4-1} $$ $$ R_{3,2} = .1507786751 $$

$$ R_{3,3} = R_{3,2} + \frac{R_{3,2}-R_{2,2}}{16-1} $$ $$ R_{3,3} = .1526554881 $$

title: Numerical Analysis/Romberg Excercise


text:
Numerical Analysis/Vandermonde example

We'll find the interpolating polynomial passing through the three points $$ (1,-6)$$, $$(2,2)$$, $$(4,12)$$, using the Vandermonde matrix.

For our polynomial, we'll take $$(1,-6) = (x_{0},y_{0})$$, $$(2,2) = (x_{1},y_{1})$$, and $$ (4,12) = (x_{2},y_{2})$$.

Since we have 3 points, we can expect degree 2 polynomial.

So define our interpolating polynomial as:

$$p(x) = a_{2}x^{2} + a_{1}x + a_{0}$$.

So, to find the coefficients of our polynomial, we solve the system $$ p(x_{i}) = y_{i}$$, $$i\in \{0,1,2\}$$.
 * $$ \left( \begin{array}{ccc}

x_{0}^{2} & x_{0} & 1 \\ x_{1}^{2} & x_{1}& 1 \\ x_{2}^{2} & x_{2} & 1 \\ \end{array} \right) *\left( \begin{array}{c} a_{2} \\ a_{1} \\ a_{0} \end{array} \right)=\left( \begin{array}{c} y_{0} \\ y_{1} \\ y_{2} \end{array} \right)$$

In order to solve the system, we will use an augmented matrix based on the Vandermonde matrix, and solve for the coefficients using Gaussian elimination. Substituting in our $$x$$ and $$y$$ values, our augmented matrix is:

$$ \left( \begin{array}{ccc|c} 1^{2} & 1 & 1 & -6 \\ 2^{2} & 2 & 1 & 2 \\ 4^{2} & 4 & 1 & 12 \end{array} \right) $$

Then, using Gaussian elimination,

$$ \left( \begin{array}{ccc|c} 1 & 1 & 1 & -6 \\ 4 & 2 & 1 & 2 \\ 16 & 4 & 1 & 12 \end{array} \right) \Rightarrow \left( \begin{array}{ccc|c} 1 & 1 & 1 & -6 \\ 0 & -2 & -3 & 26 \\ 0 & -12 & -15 & 108 \end{array} \right) \Rightarrow \left( \begin{array}{ccc|c} 1 & 1 & 1 & -6 \\ 0 & -2 & -3 & 26 \\ 0 & 0 & 3 & -48 \end{array} \right) \Rightarrow \left( \begin{array}{ccc|c} 1 & 1 & 0 & 10 \\ 0 & 2 & 0 & -22 \\ 0 & 0 & 1 & -16 \end{array} \right) \Rightarrow \left( \begin{array}{ccc|c} 1 & 0 & 0 & -1 \\ 0 & 1 & 0 & 11 \\ 0 & 0 & 1 & -16 \end{array} \right) $$

Our coefficients are $$ a_{2} = -1$$, $$ a_{1} = 11$$, and $$ a_{0} = -16$$. So, the interpolating polynomial is

$$ p(x) = -x^{2} +11x -16 $$.

Adding a point
Now we add a point, $$(3,-10) = (x_{3},y_{3})$$, to our data set and find a new interpolation polynomial with this method.

Since we have 4 points, we will have degree 3 polynomial.

Thus our polynomial is $$p(x) = a_{3}x^{3} + a_{2}x^{2} + a_{1}x + a_{0}$$,

and we get the coefficients by solving the system $$p(x_{i}) = y_{i}$$.

Constructing our augmented matrix as before and using Gaussian elimination, we get:

$$ \left( \begin{array}{cccc|c} 1^{3} & 1^{2} & 1 & 1 & -6 \\ 2^{3} & 2^{2} & 2 & 1 & 2 \\ 4^{3} & 4^{2} & 4 & 1 & 12 \\ 3^{3} & 3^{2} & 3 & 1 & -10 \end{array} \right) \Rightarrow \left( \begin{array}{cccc|c} 1 & 1 & 1 & 1 & -6 \\ 0 & -4 & -6 & -7 & 50 \\ 0 & -48 & -60 & -63 & 396 \\ 0 & -18 & -24 & -26 & 152 \end{array} \right) \Rightarrow \left( \begin{array}{cccc|c} 1 & 1 & 1 & 1 & -6 \\ 0 & -4 & -6 & -7 & 50 \\ 0 & 0 & 12 & 21 & -204 \\ 0 & 0 & 3 & \frac{11}{12} & -73 \end{array} \right) $$

$$\Rightarrow \left( \begin{array}{cccc|c} 1 & 1 & 1 & 1 & -6 \\ 0 & -4 & -6 & -7 & 50 \\ 0 & 0 & 12 & 21 & -204 \\ 0 & 0 & 0 & \frac{1}{4} & -22 \end{array} \right) \Rightarrow \left( \begin{array}{cccc|c} 1 & 1 & 1 & 0 & 82 \\ 0 & -4 & -6 & 0 & -566 \\ 0 & 0 & 12 & 0 & 1644 \\ 0 & 0 & 0 & 1 & -88 \end{array} \right) \Rightarrow \left( \begin{array}{cccc|c} 1 & 1 & 0 & 0 & -55 \\ 0 & -4 & 0 & 0 & 256 \\ 0 & 0 & 1 & 0 & 137 \\ 0 & 0 & 0 & 1 & -88 \end{array} \right) $$

$$\Rightarrow \left( \begin{array}{cccc|c} 1 & 0 & 0 & 0 & 9 \\ 0 & 1 & 0 & 0 & -64 \\ 0 & 0 & 1 & 0 & 137 \\ 0 & 0 & 0 & 1 & -88 \end{array} \right) $$

Therefore, our polynomial is:

$$p(x) = 9x^{3} -64x^{2} + 137x -88 $$.

title: Numerical Analysis/Vandermonde example


text:
Ideas in Geometry/Logic

Lesson 10 logic
Logic In order to understand logic, there are certain key words. These key words consist of: not, and, or, if-then, and if-and-only-if. Suppose I have a statement P= I love tea. The only way this statement can be false, is if P is false. In other words, since P is I love tea, then I must love tea in order for this statement to be true. When we use the symbol ¬, this represents not. In order to deny the truth of the above statement for P, we just put ¬ in front of the P: ¬P=It is not the case that I love tea. The only way ¬P can be true is when P is false. Truth table :

By using the truth table, we can see that not P is simply the opposite of P. It swaps true for false, and false for true. When we use the symbol ∧, it represents and. Let’s take a look at the statement: I love tea and I’m happy, let P=I love tea, and Q=Im happy. This statement is only true when P and Q are true. For example, I need to love tea and be happy in order for this statement to be true. Truth Table:

Looking at the truth table, we see that if P is true, but Q is false, then the statement is false. When P and Q are both false, the statement is still false.

When we look at the statement (¬P)∧(¬Q), it is saying that it is not P and it is not Q. This is the same thing as saying its not P or Q.

We can relate this concept back to venn diagrams. The symbol ∧ can also be the same as the symbol ∩. The symbol ∩ means intersection. In a venn diagram, X ∩ Y are elements that overlap X and Y. When we use the statement P ∧ Q, we are saying both P and Q need to be true in order for the whole statement to be true.

¬(A∧B) and ¬(A ∩ B) both represent “not (A intersects B).. In a venn diagram, this means that this is everything outside of the overlap of A and B.

We can also relate this concept to probability. If two events E1 and E2 are independent, meaning that the occurrence of one has no effect on the occurrence of the other, then the probabiloity that E1 and E2 will happen is the product of the probability that E1 happens with the probability that E2 happens. Symbolically, if P(Ei) is the probability that event Ei happens, then: P(E1 ∧ E2)=P(E1)*P(E2). When we ask “what is the probability that a die will roll a five and a six, we multiply 1/6*1/6 which equal 1/36 chance of rolling a five and a six. Similarly, for the statement Both P ∧ Q, the statement can hold true only when both P and Q are true.

When we use the symbol V, this represents or. Let’s look at the statement: I love tea or I am happy. P=I love tea. Q=I am happy. We can see that the only way this statement can hold true either P or Q is true. For example, if I love tea, but I am not happy, this statement is still true. When I hate tea, but I am happy, the statement is still held true. The only way this statement can be false is when both P and Q are false. It is one or the other, or even both. Truth Table:

Looking at the truth table, we can see that the statement not (p or Q) has completely the opposite results from (P or Q). The only way the statement (not P) or (not Q) can be false is when both are not P and not Q. In other words, the statement is false when P and Q both hold true. We see that not (P or Q) have the same results as (not P) or (not Q).

When we look at the statement ¬(P V Q), the results for this statement is different from (¬P)V(¬)Q. It is saying different things. (¬P)V(¬)Q means its not P or its not Q. In other words, its either not one or not the other. Whereas, the statement ¬(P V Q), it is saying its not P or Q. We can also say its not P and its not Q.

Here is a link to an example of this concept: Section 2.2 this example goes back to using "not" and relating with statements with "or" and "and"

We can relate this concept to probability. The probability (P) that E2 or E2 will happen is the sum of the probability that E1 happens with the probability that E2 happens. Symbolically, this means that if P(Ei) is the probability that event Ei happens, then: P(E1 V E2)=P(E1)+P(E2). This means that the probability the event 1 will happen or event 2 will happen. We are finding either one or the other that can happen, and we add them all up. Similarly, in the statement PVQ, we know the statement will hold true when either P or Q are true.

When we use the symbol =>, the symbol represents if-then statements. Let’s look at the statement: If I am under 21, then I can’t drink. P=I am under 21, Q=I can’t drink. Both P and Q need to relate. If this, then that. The statement will be false, only when P is true but Q is false. For example, If I am under 21, then I can’t drink, this statement is false because I cannot drink if I am under 21. On the other hand, If I am not under 21, then I can’t drink, this can hold true. It can hold true because even if I am not under 21, I don’t have to drink for different reasons. Maybe one person just cannot drink because they need to drive, or they get sick when they drink. Truth Table:

Looking at the truth table, we see that when both statements are false, it still holds true. For example, If I am not under 21 then I can drink, this statement holds true because since I am not under 21, I can drink. We can simply say that when P=>Q, this means P implies Q.

Here are links for a helpful example: If-Then Truth Tables, Section 2.2

Lastly, if you see if-and-only-if, it is denoted by the symbol<=>. this is the same thing as (p=>Q) ∧ (Q=>P). In other words they are both relational. If anlly only hold true (If P then its Q), and (if Q then its P). Smoo1244 00:17, 7 November 2010 (UTC)

title: Ideas in Geometry/Logic


text: File:If p then Q.pdf

title: File:If p then Q.pdf


text:
User:Avellar

Plena nomo: Evandro Avellar

Mi estas brazilano, kaj naskigxis en 1975 en Valença.

Mi studis matematikon kaj kriptografion. Nun mi logxas en Brasilia (Brazil).

title: User:Avellar


text:
Numerical Analysis/Vandermonde exercise

Using a Vandermonde matrix, find the interpolating polynomial that passes through the points $$(1,9)$$, $$(2,9)$$, $$(3,6)$$, $$(7,3)$$. Give both the polynomial, and the augmented matrix you used. Solution: The augmented matrix depends on your choice of index for the given points. One possibility is:

$$ \left( \begin{array}{ccccc} 1 & 1 & 1 & 1 & 9 \\ 8 & 4 & 2 & 1 & 9 \\ 27 & 9 & 3 & 1 & 6 \\ 343 & 49 & 7 & 1 & 3 \end{array} \right) $$

The polynomial is:

$$ p(x) = \frac{13}{40}x^{3} - \frac{69}{20}x^{2} + \frac{323}{40}x + \frac{81}{20}$$.

title: Numerical Analysis/Vandermonde exercise


text:
Numerical Analysis/Bisection Method MATLAB Code

The following is taken from the Ohio University Math 344 Course Page. The program mybisect.m finds roots using the Bisection Method.


text:
User:Haytham abulela

Haytham Abulela

Freelance certified translator

book worm :)

title: User:Haytham abulela


text:
Numerical Analysis/Newton form example

We'll find the interpolating polynomial passing through the points $$ (1,-6) = (x_{0},y_{0})$$, $$(2,2) = (x_{1},y_{1})$$, $$(4,12) = (x_{2},y_{2})$$, using the Newton form of the interpolation polynomial.

The Newton form is given by the formula $$ p(x) = \sum_{j=0}^{k}a_{j}n_{j}(x)$$, where $$ a_{j} = [y_{0},\ldots,y_{j}]$$ and $$ n_{j}(x) = \prod_{i = 0}^{j-1}(x-x_{i})$$, with $$n_{0}(x) = 1$$. We start by finding each $$n_{j}(x)$$.

$$n_{0}(x) = 1$$

$$n_{1}(x) = x-1$$

$$n_{2}(x) = (x-1)(x-2) = x^{2} -3x +2$$

Next, we find the necessary divided differences. First, $$[y_{0}] = -6$$, $$[y_{1}] = 2$$, and $$[y_{2}] = 12$$. For the next level, we have:

$$[y_{0},y_{1}] = \frac{2+6}{2-1} = 8$$

$$[y_{1},y_{2}] = \frac{12-2}{4-2} = 5$$

Finally, we can find:

$$[y_{0},y_{1},y_{2}] = \frac{5-8}{4-1} = -1$$.

Now, we can find the coefficients $$a_j$$.

$$a_{0} = [y_{0}] = -6$$

$$a_{1} = [y_{0},y_{1}] = 8$$

$$a_{2} = [y_{0},y_{1},y_{2}] = -1$$

Substituting and simplifying, we get our interpolating polynomial:

$$p(x) = -6 +8(x-1) - (x^{2} -3x +2) = -x^{2} + 11x -16$$.

Adding a point
Now let's add the point $$(3,-10) = (x_{3},y_{3})$$ to our data set and find the new polynomial using the same method. Due to the formula for the Newton form, we only have to add the term $$ a_{3}n_{3}(x)$$ to our previous interpolating polynomial.

First, we have

$$n_{3}(x) = (x-1)(x-2)(x-4) = x^{3} - 7x^{2} +14x -8$$.

Now to find $$a_{3}$$ we calculate some more divided differences.

$$[y_{3}] = -10$$

$$[y_{2},y_{3}] = \frac{-10-12}{3-4} = 22$$

$$[y_{1},y_{2},y_{3}] = \frac{22-5}{3-2} = 17$$

$$a_{3} = [y_{0},\ldots,y_{3}] = \frac{17+1}{3-1} = 9$$

So, our new interpolating polynomial is:

$$p(x) = -x^{2} + 11x -16 + 9(x^{3}-7x^{2}+14x-8) = 9x^{3}-64x^{2}+137x-88$$.

title: Numerical Analysis/Newton form example


text:
Numerical Analysis/Bisection Method Worked Example

X^3=20 ,f(X),   X4=4, X1=1 take four iterations.

Example
Find an approximation of $$\sqrt {3}$$ correct to within 10-4 by using the bisection method on $$ f(x) = x^2 - 3 \ $$. starting on [1, 2].

Analysis of the Problem
The number of iterations we will use, n, must satisfy the following formula: $$ n > \log_2\frac{b-a}{\epsilon} = \log_2\frac{2-1}{10^{-4}} = \log_2\frac{1}{10^{-4}} = \log_2{10^4} = \frac{\log{10^4}}{\log{2}} = \frac{4}{.3010} = 13.29$$. Thus, we will use 14 iterations of the bisection method.

Iteration #1
First, we find the midpoint of the interval [1, 2]: $$ c_1 = \frac{1 + 2}{2} = 1.5 \ $$  Then we check if $$ f(c_1) \ $$ is positive or negative:  $$ f(c_1) = 1.5^2 - 3 = -0.75 \ $$  The value of $$ f(c_1) \ $$ is negative, which means that $$ c_1 \ $$ is less than $$ \sqrt{3} $$. We therefore use $$ c_1 \ $$ as the left endpoint of our new interval and keep 2 as the right endpoint. yes

Iteration #2
Repeat the process from Iteration #1 to do Iteration #2: Solution:

We continue by finding the midpoint of our new interval, [1.5, 2]: $$ c_2 = \frac{1.5 + 2}{2} = 1.75 \ $$  Then we check if $$ f(c_2) \ $$ is positive or negative:  $$ f(c_2) = 1.75^2 - 3 = 0.0625 \ $$  The value of $$ f(c_2) \ $$ is positive, which means that $$ c_2 \ $$ is greater than $$ \sqrt{3} $$. We therefore keep 1.5 as the left endpoint of our new interval and use $$ c_2 \ $$ as the right endpoint.

Iteration #3 - Iteration #14
Complete Iteration #3 - Iteration #14: Solution: the final answer is follow or describe below

Conclusion
Thus, we have found that Solution: 1.73201  is an approximation of $$\sqrt {3}$$ correct to within 10-4.


text:
Numerical Analysis/Matlab Sample Code

Matlab code given below can also be used for excuting Romberg Integration method.

Sample Matlab code:

Resource: Numerical Methods using Matlab by Mathews

title: Numerical Analysis/Matlab Sample Code


text: File:Cylinder.jpg

title: File:Cylinder.jpg


text:
Numerical Analysis/Romberg Quiz

{$$\int_0^3 \frac{sin(2x)}{1+x^2}\,dx $$ - -1.2345435 + 0.4761463020 - 0.2345632784 - 0.567894567
 * type=""}

{ Romberg Integration is an extrapolation formula of the trapezoidal rule for integration. +TRUE. -FALSE.
 * type=""}

{True error in a multiple segment Trapezoidal rule with n segments for an integral $$ I = \int_a^b f(x) \,dx $$ is + $$ E_t = \frac{(b-a)^3}{12n^2}\frac{\sum_{i=1}^nf''(\xi_i)}{n} $$ - $$ E_t = \frac{(b)^3}{11n^2}\frac{\sum_{i=1}^nf''(\xi_i)}{n} $$ - $$ E_t = \frac{(b-a)^3}{12n^2}$$ - $$ E_t = \frac{\sum_{i=1}^nf''(\xi_i)}{n}$$
 * type=""}

{$$\int_0^3 sin(4x)e^{-2{x}}\,dx$$ +0.1997146621 -0.1345234567 -0.2345234234 -1.0000000000
 * type=""}

{$$\int_{0.04}^1 \frac{1}{\sqrt{x}}\,dx $$ -2.6 +1.6 -4.6 -5.6
 * type=""}

title: Numerical Analysis/Romberg Quiz


text:
Numerical Analysis/Newton form exercise

1. Using the Newton form, find the interpolating polynomial passing through the points $$(x_{0},y_{0}) = (9,9)$$, $$(x_{1},y_{1}) = (5,7)$$, $$(x_{2},y_{2}) = (3,7)$$, and $$(x_{3},y_{3}) = (7,1)$$.

Solution:

To find the four coefficients, we can use the divided differences.

 * $$\begin{matrix}

x_i&y_i& [y_0,y_1]                 & [y_0,y_1,y_2]                          & [y_0,y_1,y_2,y_3] \\ 9 & 9 &                            & \\       &   & {7-9\over 5-9} = {1\over 2} & \\ 5 & 7 &                            & {{0 - 1\over 2}\over {3-9}} = {1\over 12} \\ &  & {7-7\over 3-5} = {0}        &                                        &{{{-3\over 4} - {1\over12}}\over {7-9}} = {5\over 12}\\ 3 & 7 &                            & {{-3\over 2} - 0\over {7-5}} = {-3\over 4} \\ &  & {1-7\over 7-3} = {-3\over2} & \\ 7 & 1 &                            & \\ \end{matrix}$$ Thus, the four coefficients are $$x_{0}$$ and the upper diagonal of the calculated divided differences.

Using the Newton Forward Divided Difference formula, we find the polynomial is

 * $$p_3(x) = [y_0] + [y_0,y_1](x-x_0) + [y_0,y_1,y_2](x-x_0)(x-x_1) +  [y_0,y_1,y_2,y_3](x-x_0)(x-x_1)(x-x_2).$$
 * $$p_3(x) = 9 + \frac{1}{2}(x-9) + \frac{1}{12}(x-9)(x-5) +  \frac{5}{12}(x-9)(x-5)(x-3).$$
 * $$p_3(x) = \frac{5}{12}x^{3} - 7x^{2} + \frac{427}{12}x -48$$.

2. Find the interpolating polynomial passing through the four points given in the exercise above as well as through the point $$(x_{4},y_{4}) = (4,2)$$

Solution: To find the new coefficient, we can simply add the new point to the end and calculate $$[y_0,y_1,y_2,y_3,y_4]$$.

 * $$\begin{matrix}

x_i&y_i & [y_0,y_1]                  & [y_0,y_1,y_2]   & [y_0,y_1,y_2,y_3]    & [y_0,y_1,y_2,y_3,y_4] \\ 9 & 9 &                              & \\   &    & {1\over 2}                   & \\ 5 & 7 &                              & {1\over 12} \\ &   & {0}                          &                &{5\over 12}\\ 3 & 7 &                              & {-3\over 4}    &                       & {{{-23\over 12} - {5\over 12}}\over {4-9}} = {17\over 15} \\ &   & {-3\over2}                   &             & {{{7\over 6} - {-3\over 4}}\over {4-5}} = {-23\over 12} \\ 7 & 1 &                              & {{-1\over 3} - 0\over {-3\over 2}} = {7\over 6} \\ &   & {2-1\over 4-7} = {-1\over 3} & \\ 4 & 2 &                              & \\ \end{matrix}$$

Thus, we find the polynomial $$p_{4}$$ is

 * $$p_4(x) = p_3 + [y_0,y_1,y_2,y_3,y_4](x-x_0)(x-x_1)(x-x_2)(x-x_3).$$
 * $$p_3(x) = \frac{5}{12}x^{3} - 7x^{2} + \frac{427}{12}x -48 + \frac{17}{15}(x-9)(x-5)(x-3)(x-7).$$
 * $$p_3(x) = \frac{7}{15}x^{4} - \frac{647}{60}x^{3} + \frac{1337}{15}x^{2}-\frac{18697}{60}x + 393$$.

title: Numerical Analysis/Newton form exercise


text:
Numerical Analysis/Bisection Method Quiz

{2 and 4 such that f(2) = 4 and f(4) = 16 are appropriate initial points for the bisection method. - TRUE + FALSE
 * type=""}
 * The bisection method requires two initial points a and b such that f(a) and f(b) have opposite signs.

{We decide that 2 such that f(2) = 8 will be one of our initial points. Which of the following are appropriate choices for the second point? + -3 such that f(-3) = -27 - 1 such that f(1) = 1 + -1 such that f(-1) = -1 - 3 such that f(3) = 27
 * type="[]"}

{The function on which we want to use the bisection method must be continuous on the interval we choose. + TRUE - FALSE
 * type=""}

{After applying one iteration, by how much did our interval that might contain a zero of f decrease? - Almost half - More than half + 50%
 * type=""}

{The bisection method has... + linear convergence. - quadratic convergence. - cubic convergence.
 * type=""}
 * The absolute error halves at each step.


text:
User talk:EGM6321.F10.TEAM1.HW6

 Hello EGM6321.F10.TEAM1.HW6, and welcome to Wikiversity! If you need help, feel free to visit my talk page, or contact us and ask questions. After you leave a comment on a talk page, remember to sign and date; it helps everyone follow the threads of the discussion. The signature icon in the edit window makes it simple. To get started, you may

 * Take a guided tour and learn to edit.
 * Visit a (kind of) random project.
 * Browse Wikiversity, or visit a portal corresponding to your educational level: pre-school, primary, secondary, tertiary, non-formal education.
 * Find out about research activities on Wikiversity.

 * Read an introduction for teachers and find out how to write an educational resource for Wikiversity.
 * Give feedback about your initial observations
 * Discuss Wikiversity issues or ask questions at the colloquium.
 * Chat with other Wikiversitans on #wikiversity-en.

And don't forget to explore Wikiversity with the links to your left. Be bold to contribute and to experiment with the sandbox or your userpage, and see you around Wikiversity! If you're a twitter user, please follow http://twitter.com/Wikiversity. --Mu301Bot 08:17, 7 November 2010 (UTC)

Feel free to leave a message on my talk page if you have any questions or need help. --mikeu talk 08:17, 7 November 2010 (UTC)

title: User talk:EGM6321.F10.TEAM1.HW6


text:
User talk:Cj13

 Hello Cj13, and welcome to Wikiversity! If you need help, feel free to visit my talk page, or contact us and ask questions. After you leave a comment on a talk page, remember to sign and date; it helps everyone follow the threads of the discussion. The signature icon in the edit window makes it simple. To get started, you may

 * Take a guided tour and learn to edit.
 * Visit a (kind of) random project.
 * Browse Wikiversity, or visit a portal corresponding to your educational level: pre-school, primary, secondary, tertiary, non-formal education.
 * Find out about research activities on Wikiversity.

 * Read an introduction for teachers and find out how to write an educational resource for Wikiversity.
 * Give feedback about your initial observations
 * Discuss Wikiversity issues or ask questions at the colloquium.
 * Chat with other Wikiversitans on #wikiversity-en.

And don't forget to explore Wikiversity with the links to your left. Be bold to contribute and to experiment with the sandbox or your userpage, and see you around Wikiversity! If you're a twitter user, please follow http://twitter.com/Wikiversity. --Mu301Bot 08:17, 7 November 2010 (UTC)

Feel free to leave a message on my talk page if you have any questions or need help. --mikeu talk 08:17, 7 November 2010 (UTC)

title: User talk:Cj13


text:
User talk:Bunny204

 Hello Bunny204, and welcome to Wikiversity! If you need help, feel free to visit my talk page, or contact us and ask questions. After you leave a comment on a talk page, remember to sign and date; it helps everyone follow the threads of the discussion. The signature icon in the edit window makes it simple. To get started, you may

 * Take a guided tour and learn to edit.
 * Visit a (kind of) random project.
 * Browse Wikiversity, or visit a portal corresponding to your educational level: pre-school, primary, secondary, tertiary, non-formal education.
 * Find out about research activities on Wikiversity.

 * Read an introduction for teachers and find out how to write an educational resource for Wikiversity.
 * Give feedback about your initial observations
 * Discuss Wikiversity issues or ask questions at the colloquium.
 * Chat with other Wikiversitans on #wikiversity-en.

And don't forget to explore Wikiversity with the links to your left. Be bold to contribute and to experiment with the sandbox or your userpage, and see you around Wikiversity! If you're a twitter user, please follow http://twitter.com/Wikiversity. --Mu301Bot 08:17, 7 November 2010 (UTC)

Feel free to leave a message on my talk page if you have any questions or need help. --mikeu talk 08:17, 7 November 2010 (UTC)

title: User talk:Bunny204


text:
User talk:Sundeep.kumarreddy

 Hello Sundeep.kumarreddy, and welcome to Wikiversity! If you need help, feel free to visit my talk page, or contact us and ask questions. After you leave a comment on a talk page, remember to sign and date; it helps everyone follow the threads of the discussion. The signature icon in the edit window makes it simple. To get started, you may

 * Take a guided tour and learn to edit.
 * Visit a (kind of) random project.
 * Browse Wikiversity, or visit a portal corresponding to your educational level: pre-school, primary, secondary, tertiary, non-formal education.
 * Find out about research activities on Wikiversity.

 * Read an introduction for teachers and find out how to write an educational resource for Wikiversity.
 * Give feedback about your initial observations
 * Discuss Wikiversity issues or ask questions at the colloquium.
 * Chat with other Wikiversitans on #wikiversity-en.

And don't forget to explore Wikiversity with the links to your left. Be bold to contribute and to experiment with the sandbox or your userpage, and see you around Wikiversity! If you're a twitter user, please follow http://twitter.com/Wikiversity. --Mu301Bot 08:17, 7 November 2010 (UTC)

Feel free to leave a message on my talk page if you have any questions or need help. --mikeu talk 08:17, 7 November 2010 (UTC)

title: User talk:Sundeep.kumarreddy


text:
User:Jtneill/Sandbox/Quiz

{Question + The correct answer. - Distractor. - Distractor. - Distractor. }
 * type=""}

title: User:Jtneill/Sandbox/Quiz


text:
User:Jtneill/Books/Test

A free and open textbook

 * Front matter
 * Acknowledgements
 * Preface

 * Introduction
 * What is motivation?
 * What is emotion?
 * Relationship between motivation and emotion

 * Motivation - Biological motives
 * Hunger
 * Sexual motivation
 * Adultery
 * Alcohol consumption and sexual motivation
 * Gender differences in sexual motivation
 * Mate-seeking behaviour
 * Promiscuity
 * Sex offenders

 * Motivation - Mini-theories
 * Arousal
 * Flow theory
 * Intrinsic and extrinsic motivation
 * Learned helplessness

 * Motivation and behaviour
 * Dieting
 * Exercise motivation
 * Gambling
 * Overeating
 * Procrastination
 * Risk-taking
 * Sleep, energy and fatigue
 * Social inhibition
 * Violence

 * Motivation and cognition
 * Goal setting
 * Positive thinking

 * Motivation and culture
 * Motivation and indigenous Australians

 * Motivation and education
 * Student motivation theories
 * Goal-setting and educational achievement
 * Motivation and personality

 * Motivation and psychological disorders
 * Anxiety
 * Antisocial personality disorder
 * Dementia
 * Depression
 * Narcissism

 * Neurobiology of motivation
 * Motivational toxicity

 * Motivation and self
 * Self-concept
 * Self-sabotaging

 * Types of motivation
 * Achievement motivation
 * Self-actualisation
 * Spiritual
 * Unconscious motivation

title: User:Jtneill/Books/Test


text:
Template:Motivation and emotion/Textbook chapter/Title

==


text:
User:Ani~enwikiversity

hey people, i cannot help if i am irritating..but i don't know what to write here. let us start with my intro. i am ani, i am 13 years of age (hope its legal), i am from india. i joined wikiversity to learn new things and to help make it the best learner guide on the web with my knowledge.

title: User:Ani~enwikiversity


text:
User:Himacharya

A Practicing PEDIATRIC SURGEON from India

title: User:Himacharya


text:
History of the Crusades

Introduction
There were a series of military expeditions launched in the pursuit of religious goals. While the crusades were usually organized by the papacy, the military expeditions themselves were led by a mixture of nobles and monarchs, often lacking a formal leadership hierarchy. The crusades were started at the Council of Clermont where Pope Urban II called for the Christians in the west to defend their brethren in the east. This was framed as an armed pilgrimage and an act of penance through which one could earn salvation. Those who went on the crusade were said to have took the cross (sometimes literally with a cloth cross sown on their clothes) and the term crusade itself was derived from the word crucesignati which means "those signed by the cross".

Most of the crusading energy was directed toward the area known as the Holy Land, as Jerusalem was considered a sacred city. Many crusades were launched with the purpose of capturing Jerusalem from the Muslims. The first crusade succeeded in capturing Jerusalem in 1099 and establishing several independent political entities in the Levant called the Crusader States. These were the Kingdom of Jerusalem, of Antioch, County of Tripoli, and the County of Edessa. The crusaders' military presence in the region continued until the fall of Acre in 1291. There are generally accepted to be nine crusades to the Holy Land, but there were other crusades launched for religious reasons.

The Crusades

 * /The First Crusade/
 * /The Second Crusade/
 * /The Third Crusade/
 * /The Fourth Crusade/
 * /The Fifth Crusade/
 * /The Sixth Crusade/
 * /The Seventh Crusade/
 * /The Eighth Crusade/
 * /The Ninth Crusade/
 * /The Albigensian Crusade/
 * /The Northern Crusades/

quiz
{The Crusades were a series of military expeditions launched in the pursuit of ________ goals. - economic - territorial + religious - political
 * type="[]"}
 * Incorrect. The primary motivation was not economic.
 * Incorrect. While territories were involved, the primary motivation was different.
 * Correct! The Crusades were primarily motivated by religious goals.
 * Incorrect. Politics played a role, but the primary motivation was religious.

{The crusades were usually organized by the ________, but the military expeditions were led by nobles and monarchs. - monarchs + papacy - knights - commoners
 * type="[]"}
 * Incorrect. Monarchs were involved in leading, but not organizing.
 * Correct! The papacy usually organized the crusades.
 * Incorrect. Knights participated but did not usually organize the crusades.
 * Incorrect. Commoners did not organize the crusades.

{The crusades were started at the Council of ________ where Pope Urban II called for the Christians to defend their brethren in the east. - Rome + Clermont - Nicaea - Constantinople
 * type="[]"}
 * Incorrect. It was not the Council of Rome.
 * Correct! The crusades were started at the Council of Clermont.
 * Incorrect. The Council of Nicaea had a different purpose.
 * Incorrect. It was not the Council of Constantinople.

{Those who went on the crusade were said to have took the ________. - sword + cross - shield - pilgrimage
 * type="[]"}
 * Incorrect. While they did take weapons, this is not the symbolic term used.
 * Correct! They were said to have took the cross.
 * Incorrect. This is not the symbolic term used.
 * Incorrect. The crusade was framed as a pilgrimage, but they took the cross.

{The first crusade succeeded in capturing Jerusalem in ________. - 1055 + 1099 - 1123 - 1187
 * type="[]"}
 * Incorrect. That's not the correct year.
 * Correct! The first crusade captured Jerusalem in 1099.
 * Incorrect. That's not the correct year.
 * Incorrect. Jerusalem was recaptured by the Muslims in 1187, not by the crusaders.

{The crusaders' military presence in the region continued until the fall of ________ in 1291. - Jerusalem + Acre - Antioch - Edessa
 * type="[]"}
 * Incorrect. Jerusalem fell earlier.
 * Correct! The crusaders' presence continued until the fall of Acre in 1291.
 * Incorrect. Antioch was one of the Crusader States, but it's not the answer here.
 * Incorrect. Edessa was one of the Crusader States, but it's not the answer here.

title: History of the Crusades


title: File:Flowmodel1.png


text:
User:Slapshot136

Hi, this is my page, I create this account here to learn french, if anyone is interested in helping me, please e-mail me at slapshot136@msn.com

title: User:Slapshot136


title: File:Flowmodel2.png


text:
Numerical Analysis/Polynomial interpolation concept quiz

Choose the best answer for each question:

{Of the following polynomial interpolation methods, which is generally considered the method of choice due to its relative ease of use?} - Vandermonde matrix + Lagrange method - Newton form

{Which method is the best choice when the desired degree of the interpolating polynomial is known?} - Vandermonde matrix + Lagrange method - Newton form

{Which method is best suited when the desired degree of the interpolating polynomial is unknown?} - Vandermonde matrix - Lagrange method + Newton form

{Which method is best suited to the addition of points to the data set?} - Vandermonde matrix - Lagrange method + Newton form

{What is the computational cost of finding an interpolating polynomial through $$n$$ points using the Newton form?} - $$O(n)$$ + $$O(n^{2})$$ - $$O(n^{3})$$ - $$O(n^{4})$$

{What is the computational cost of the Vandermonde method, using Gaussian elimination?} - $$O(n)$$ - $$O(n^{2})$$ + $$O(n^{3})$$ - $$O(n^{4})$$

{Under what conditions can the Lagrange method of polynomial interpolation fail?} - When $$n > 10$$. - When $$n$$ is not a perfect square. - When two or more of your $$y$$-values are equal. + The Lagrange method cannot fail.

{Given a set of $$n$$ points, exactly how many interpolating polynomials can be found to pass through the points?} - $$0$$ + $$1$$ - $$n$$ - $$n-1$$

{ What is the error term of an interpolation polynomial? $$ f(x) - p_n(x) = $$ { $$\frac{f^{(n+1)}(\xi)}{(n+1)!} \prod_{i=0}^n (x-x_i) $$ }
 * type="{}"}


text:
User talk:Getgooglechrome

Your editing experiment
Thank you for experimenting with Wikiversity. Your test was recorded, and it has been reverted or removed. Please use the sandbox for further testing. You may wish to learn more about what Wikiversity is to help you contribute. If you haven't done it yet, consider registering if you are keen on our project. --Savh 18:55, 7 November 2010 (UTC)

title: User talk:Getgooglechrome


text:
User:Frozen Wind

simplewiki

title: User:Frozen Wind


text:
Motivation and emotion/Textbook/Emotion/Theories/Cognitive

Introduction
There are many theories which have been developed over time in order to explain the process of Emotion. Even scholars such as Aristotle and Plato were trying to figure out what made people feel the way that they do and experience specific emotions (Fortenbaugh, 1975). What makes some people happy and some people sad? Is it purely based on our facial expressions? Is it a chemical change in the body? Are emotions physically the same?

There are three key sets of theories revolving around the concept of emotion. These are biological theories of emotion, socio-cultural theories of emotion and cognitive theories of emotion (Johnson-Laird & Oatley, 1992). This chapter focuses specifically on cognitive theories of emotion.

Cognitive theories of emotion are theories which try to explain the concept of emotions by proving that they are based around a person’s perception, intuition and reasoning. Without an antecedent cognitive appraisal (an estimate of the personal significance) of a situation or event, emotions cannot occur (Hampson & Morris, 1996).

Psychologists interested in emotion have for some time been preoccupied with attempts to explain emotions in terms of cognitive processes. Relating emotions specifically to cognition seems to take something away from the concept of an emotion. This way of thinking means that emotions are just thoughts about situations we find ourselves in. However, cognitive theories appear to be the most recognised and accepted of the theories of emotion (LeDoux, 1996).

There are many key cognitive theories of emotion, but all theories centre around the same point, that it is the cognitive appraisal of a situation, not the event itself, that causes the emotion.

Schachter-Singer Cognitive Arousal theory
Schachter and Singer (1962) proposed that physiological arousal is necessary for an emotional response, but that similar patterns of arousal can occur for many different emotions (Rathus, 1989 and Woods, 1997). For example, our heart rate increases and we start to sweat when we are both afraid and angry. For this reason Schachter and Singer proposed that it is not the pattern of arousal that enables us to label an emotion, but our cognitive appraisal of the arousal and of the situation we are in (Schachter & Singer, 1962). Therefore, someone who experiences physiological arousal for no apparent reason will consider their surroundings and the situation they are in and then label the emotion using this cognitive appraisal (Rathus, 1989).

Schachter and Singer tested their theory in an experiment, the objective of which was to show that once the arousal is labelled as an emotion, the degree of a persons arousal determines the intensity of the emotion. The experiment consisted of four groups. Each participant was told that the purpose of the experiment was to assess the effects of vitamin injections on vision (Woods, 1997):
 * Group A was given an adrenaline injection and told of the side affects (such as increased heart rate and sweating)
 * Group B was given an adrenaline injection and told incorrect information about the side affects (such as itching)
 * Group C was given an adrenaline injection and told no information about the side effects
 * Group D was given a saline injection (acting as a placebo) and given no information about the side affects (Hassett, 1984 and Rathus, 1989).

Whilst waiting to go for their ‘vision test’ each participant was asked to wait in a room with a second person (a confederate). Half of the participants in each of the four conditions were placed with a happy person, who was laughing out loud and making paper aeroplanes, playing with a slingshot and building a fort out of manila folders. The other half were asked to fill out a questionnaire and placed in a room with an unhappy person, who was angry and irate and ended up ripping the questionnaire and storming out of the room (Hassett, 1984 and Rathus, 1989)

Researchers measured the participants' emotional responses according to the degree with which they participated in similar behaviour to the second person. The results showed that participants in groups B and C reflected the behaviour and mood of the confederate much more than the participants in groups A and D (Hassett, 1984).

Schachter and Singer’s explanation of the results is that group B and C participants could not explain the way that they were feeling physically, and so looked for alternative explanations. The cognitive appraisal of the participant’s environment meant that they associated their physical responses with the behaviour of the confederate and so felt either happy or angry, depending on which emotions the confederate was displaying (Schachter & Singer, 1962)

Later researchers have had trouble replicating the results of Schachter and Singer's experiment, and certain queries regarding the use of epinephrine injections and their elicited response have been raised. However, Schachter and Singers most important conclusion, that cognitive interpretations do affect emotional experience and behaviour, have been widely supported by further research (Woods, 1997).

For example, if Rachel was walking along and saw a snake, the snake (being the stimulus) would trigger a physical response such as increased heart rate and sweating. Rachel would subconsciously consider her surroundings and the situation she is in and then determine that she is afraid.

Appraisal theories of Emotion
Mary Arnold

Mary Arnold The appraisal concept was originally developed by Mary Arnold in the early 1960’s as a way to combat the elements she saw as missing from the Schachter-Singer cognitive arousal theory. Arnold believed that although Schachter and Singer had managed to explain how we deal with emotional responses once they occur, they had failed to address what generates the response in the first place. (Arnold, 1960)

Arnold proposed that appraisal is the mental assessment of the potential harm or benefit of a situation. She stated that emotion is the felt tendency towards anything that is appraised as good or away from anything that is appraised as bad (Arnold, 1960). The appraisal process happens unconsciously, however the resulting response is registered in consciousness as an emotional feeling (Ledoux, 1996)

For example, if Rachel was walking along and saw a snake, she would subconsciously appraise the situation and want to run away from the snake (her action tendency, either to stay if the stimulus is appraised as good or to flee if the stimulus is appraised as bad). The action tendency would be consciously registered as an emotional feeling (fear).

Richard Lazarus

Richard Lazarus was a clinical psychologist who used the appraisal concept to understand the way people react to and cope with stressful situations. In the 1960’s Lazarus conducted an experiment aimed at highlighting the fact that interpretations of situations strongly influence the emotion experienced (LeDoux, 1996). In his experiment, Lazarus showed a gruesome film clip of a circumcision ritual involving teenage members of an aboriginal Australian tribe to three groups of participants. The film clip was accompanied by one of three soundtracks. The first soundtrack emphasised the gruesome details of the film clip, saying things like “ Several men hold the boy so that he cannot escape” (Hassett, 1984 ).The second soundtrack denied the painfulness and suffering with phrases such as “The words of encouragement offered by the older men have their effect” (Hassett, 1984). The third soundtrack was more detached and focused on the intellectual perspective of the film clip, such as the surgical technique (Hassett, 1984 ).

Lazarus suggested that different soundtracks caused the participants to appraise the films in different ways and this led to different feelings about the situation (Lazarus, 1991). He argued that emotions can be started automatically unconsciously or consciously, but emphasised the role of higher thought processes and consciousness in coping with emotional reactions once they exist. Lazarus stated that “cognition is both a necessary and sufficient condition of emotion” (Lazarus, 1991).

For example, If Rachel was walking along and saw a snake, she would cognitively appraise the situation as bad, her heart beat would increase and she would start sweating and she would then feel the emotion and decide how to act.

Attribution Theory of Emotion
The first attribution theory was published by Fritz Heider in 1958. In his book, The Psychology of Interpersonal relations, Heider was concerned with they way that people use common sense to try and understand and explain events which occur, and the actions of other people and themselves (Hassett, 1984 ).

A Key element of Attribution theory is the concept of self perception, which is a term used to describe the ways in which people often draw conclusions from observations of their own behaviour. When people explain their own behaviour, they tend to overestimate the size of the effect of external factors (things which they have no control over) and underestimate the size of internal factors (personal responsibility) (Oatley & Johnson-Laird, 1987).

The concept of self perception has become one which is seemingly important in the study of emotions. It can be noted that in Schachter and Singer's cognitive theory of emotion, the concept of self- perception directly affects the emotions experienced by the participants in the study. Participants in Groups B and C who had received adrenaline injections, misattributed the cause of their physiological arousal to the happy or angry situation they were placed into. This has led to a series of studies on self-perception and the misattribution process, in which people are purposely given false information about internal states to see how they would interpret it and how it would affect their perceived emotions (Hassett, 1984 ).

One study conducted by Valins in 1966, male college students were asked to rate the attractiveness of women in sexually provocative pictures whilst they listened to what they were told was their heartbeat (Hammond, 2005). The sound they were listening to was a fake heart beat, which was controlled by the experimenter. Periodically the heartbeat rate was increased so that the participants believed that their own heart beats were increasing. Participants rated the women who they when they heard their heartbeat going faster as more attractive than those women whom which they did not hear their heart beat increasing for. The general assumption here is that on some unconscious level the men thought that since their heart rate was going faster, they must find the particular women more attractive (Oatley & Johnson-Laird, 1987).

The results of this experiment and the many others which have been conducted indicate that external information can lead people to draw conclusions about their own internal states. Peoples emotions can be affected by the factors which occur in their external environment (Hammond, 2005).

For example, Rachel is walking along with her friends when one of then hears a rustle in the bushes. Rachel’s heart beat increases and she starts to perspire. If her friends were to run away screaming, Rachel would interpret her physiological reaction to the environment as fear and her instinct would be to flee. However if her friends appeared excited and inquisitive about the rustle in the bushes, Rachel would interpret her physiological reaction as excitement.

Cannon-Bard Theory of Emotion
The cannon-Bard theory of emotion (also known as the Thalamic theory) was developed by physiologists Walter Cannon and Phillip Bard. It is a theory which combines both cognitive elements of emotion and biological elements of emotion(LeDoux, 1996).

The theory suggests that individuals experience emotions and have a physiological reaction to a stimulus at the same time. It was developed in order to contest the James- Lange Feedback theory of emotion, which stated that we physiologically respond to a stimulus first, and the feedback from this response then generates an emotion (Leahey, 1987). Cannon and Bard believed that this was incorrect, and that both a physiological response to a stimulus, and the experience of an emotion, occur at the same time. The concept behind the theory is that the thalamus is a central factor in generating emotional experience and behaviour(LeDoux, 1996).

The Cannon-Bard theory built up on the fact that the sensory systems that take in information from the outside world send the information to specialised regions of the cerebral cortex and onto specialised cortical areas. However, Cannon and Bard suggested that in their travels towards the specialised cortical areas the sensory messages make a stop in subcortical areas- in thalamic relay stations (Cannon, 1929). These thalamic relay stations are specialised for particular sensory processing (for example visual stimulus goes from the eyes to the visual thalamus and on to the visual cortex and auditory stimulus goes from the ears to the auditory thalamus and on to the auditory cortex). However some thalamic relay stations send the information to the hypothalamus, not to the cortical areas. This means that the hypothalamus receives sensory input at about the same time as the cortex, allowing the hypothalamus to activate the body to produce the automatic and behavioural responses characteristic of emotional reactions. Therefore, the emotional response is triggered by the hypothalamus at the same times as the physiological response to the stimulus is triggered by the cortex (Cannon, 1929).

For example, if Rachel was walking along and saw a snake she would immediately cognitively perceive the snake. The snake (being a visual stimulus) would trigger the thalamus to relay messages to both the visual cortex and to the hypothalamus. Rachel’s body would then start to feel the emotion fear at the same times as her body physically responds to the stimulus and starts to show the emotion fear (her heart beat accelerates and she starts to sweat).

Summary

Although all of the theories listed above seem to have at least some element of reality to them it is hard to determione which theory, if any, defines the process of emotion exactly. Scholars from many disciplines are constantly challenging the cognitive theories of emotion (and the biological and socio-cultural theories as well). However there seems to be a consensus that cognition is a necessary part of the emotion forming process. {{Hide in print|

Quiz
Answers }}
 * 1) Cognitive
 * 2) Cannon
 * 3) Thalamus
 * 4) Vision
 * 5) Appraisal
 * 6) Lazarus
 * 7) Self perception
 * 8) Heartbeat
 * 9) Stimulus
 * 10) Thalamic


text: File:A woman with a look of disgust crop.jpg


text: File:Little sadness crop.jpg

title: File:Little sadness crop.jpg


text: File:Sen. Barack Obama smiles crop.jpg

title: File:Sen. Barack Obama smiles crop.jpg


text: File:Surprised Kristen crop.jpg

title: File:Surprised Kristen crop.jpg


text:
Talk:Rising Tide

Critical path analysis notes list

 * Net present value accounting (long term economics)
 * as opposed to N-year (e.g. 5-year or 10-year) extrapolation
 * more accurate than N-year
 * insurance companies using N-year instead of NPV go out of business in 2N years under most conditions
 * government is the insurer of last resort

 * National security
 * mental health
 * obesity
 * in recruiting
 * government disinformation
 * epidemiological causes include psychological conditions of scarcity proportional to rising housing costs (people eat more when they don't know how to pay for housing)

 * Wind fuels http://dotyenergy.com
 * compete with OPEC
 * eliminate resource wars
 * fewer carcinogens from coal and petroleum
 * reduce extreme weather risks including flood damage

 * Amnesty for nonviolent offenders?
 * Parole reform?
 * Risk assessments
 * Accurate diagnoses

 * Single payer savings with 40% universal preventative care savings

 * Corporate top tax bracket rate should be larger than individuals'

 * Mortgage reform but preserving bankers' bonuses this time

 * Class size reduction
 * Increased teacher salaries
 * Higher education tuition reduction

 * Multilateral tax haven treaty -- already started at OECD Paris

 * Budget simulation (last but not least)
 * real property values
 * graduation rates
 * consumer spending

Work continues. 71.198.176.22 20:24, 9 November 2010 (UTC)

Health and inequality documentaries
On the documentary/educational video side, California Newsreel produces some exceptional work on the topic of health and inequality, among others. It's framed by scientific studies and includes some of the epidemiological findings discussed in the main article. Natural Causes is of particular note. Unfortunately, there are draconian licensing and pricing issues which limits the availability and reach of this work, but they really are a great resource in the video platform.
 * Thank you! 71.198.176.22 20:15, 9 November 2010 (UTC)

Misc to do items

 * talking about size of middle class or "level playing field" preferable to abstract stats on ultra rich
 * Humanities Reference Desk question (pre-archived) 71.198.176.22 20:15, 9 November 2010 (UTC)

Tax day link dump
Some subset of these articles also need to be incorporated:

http://yglesias.thinkprogress.org/2011/04/american-taxes-are-low/

http://www.huffingtonpost.com/2011/04/18/the-top-10-tax-breaks-_n_850534.html

http://wonkroom.thinkprogress.org/2011/04/17/allen-west-debt-ceiling-hostage/

http://www.altweeklies.com/aan/9-things-the-rich-dont-want-you-to-know-about-taxes/Story?oid=3971382

http://blogs.forbes.com/erikkain/2011/04/14/how-progressive-is-the-american-tax-code/

http://www.vanityfair.com/society/features/2011/05/top-one-percent-201105?currentPage=all

James Salsman 23:15, 18 April 2011 (UTC)

More sources
https://plus.google.com/u/0/113651533166317708938/posts/7BdFuEYXi8k

http://www.imf.org/external/pubs/ft/fandd/2011/09/Berg.htm

http://treasureislands.org/

http://en.wikipedia.org/w/index.php?title=Talk:99_Percent_Declaration&oldid=459644173#Proposed_inclusions

http://en.wikipedia.org/w/index.php?title=Wikipedia:Reference_desk/Humanities&oldid=459669535#Executive_order_for_single_payer_health_care

http://www.guardian.co.uk/commentisfree/2011/nov/07/one-per-cent-wealth-destroyers

http://blogs.wsj.com/marketbeat/2011/11/08/mutual-funds-top-hedge-funds-in-october

67.6.128.40 00:20, 9 November 2011 (UTC)

http://www.connectthedotsusa.com/

http://www.equalitytrust.org.uk/node/628

67.6.144.87 05:18, 25 November 2011 (UTC)

title: Talk:Rising Tide


text:
User:EdoDodo/vector.js

mw.loader.load('https://meta.wikimedia.org/w/index.php?title=User:EdoDodo/global.js&action=raw&ctype=text/javascript');

title: User:EdoDodo/vector.js


text:
Motivation and emotion/Textbook/Motivation/Social inhibition

Introduction
When you say someone is socially inhibited, this usually means that that individual has a shy nature, tends to be quite quiet in social environments, and will most likely not speak unless directly spoken to. But why are some people like this and not like others who find it easy to control a group conversation or who are comfortable in being quite loud in their opinion. There are many possible reasons for this, for example, they may be afraid of being embarrassed or afraid of being laughed at for what they have to say. Other explanations could be that they never developed the skills to work within a group situation and therefore have a limited idea of how to handle such a situation. There is an endless list of reasons why some people are socially inhibited because everyone has a different story that made them who they are. In this chapter we will talk about some of the main reasons and motivations behind social inhibition and try to discover how to give socially inhibited people a chance to change and become more confident.

History
It was Floyd Allport (1920) who first used the term social inhibition, while also studying social facilitation. Social facilitation is the opposite of social inhibition. It is the ability of people to perform better when they have an audience, or when they are in a group. In his paper he talks about how it is the presence of other people that is the cause for both social facilitation and inhibition. However, he found a vital difference between the two. Social facilitation could be caused in a relatively simple task like running or eating, however social inhibition takes control during relatively difficult tasks like maths or public speaking (Crisp & Turner, 2007). So from this we can say that either response could be invoked depending on the task complexity. After this initial theory came three main theories to describe social facilitation and inhibition and we will look at all three.

The Drive Theory
The first is the drive theory that was put forward by Zajonic (1965). He states that the two responses can be attributed to psychological arousal. There are two parts to this theory. The first part says that the mere presence of other people around you will cause psychological arousal. This stems from evolutionary factors which apply to all living beings, which mean that the presence of others either presents danger or a chance to reproduce. We cannot help but have this kind of response, however, perhaps these days there are different reasons behind these reactions, and this causes psychological arousal in us. The second part of this theory is the assumption that this psychological arousal will cause our well-learned response tendencies to become dominant (Hull, 1943), be they to become more confident in what we are doing or to regress and protect ourselves. This second part also makes sense when you consider that some types of arousal will cause us to narrow our field of focus and this in turn restricts our processing capacity (Kahneman, 1973). So if our well learned response becomes dominant and that is the only thing we can focus on then that is what shows to the outside world and becomes our normative response to that particular situation (Crisp & Turner, 2007). The important thing to note about drive theory is that it doesn’t say social facilitation or inhibition occurs whether the task at hand is simple or complex but rather that either one will occur depending on the match between our dominant response and the task requirements. For example, a correct match between these two would be social facilitation going with a task like running. Socially inhibited people often respond incorrectly in that their response to what should be a relatively relaxing time, in a group of friends, is to regress inwardly and remain quiet.

Evaluation Apprehension
The theory of evaluation apprehension holds with the idea that it is psychological arousal that makes us turn to our well-learned responses. However it does differ from the Drive Theory in that it states that it may not be the mere presence of others that is the trigger of the arousal. This theory instead claims that it is evaluation apprehension which causes the arousal. Evaluation apprehension is of course the concern that people have of being judged by the people around them and the fear that one may look a fool in front of an audience (Crisp & Turner, 2007). This idea of people clearly being worried about what others may think of them was clearly shown in the Asch Experiment: http://www.youtube.com/watch?v=iRh5qy09nNw.

As is stated in the video, humans by nature want to fit in with the group and are very aware of what the people around them are feeling. This experiment is a great example of how we will sometimes go completely against what we really think in order to avoid seeming ignorant even in front of people we may not even know.

Cottrell and his colleagues (1968) conducted an experiment in an attempt to show their theory of evaluation apprehension. In this experiment, participants were asked to pronounce out loud nonsense words that appeared on a screen every four seconds. The participants were split into two groups; one group had to read the words while in front of a blindfolded audience, while the other groups audience was not blindfolded. Now according to drive theory (Zajonic, 1965), the mere presence of an audience would be enough to evoke psychological arousal and cause the participants to revert to the dominant reaction. However what the evaluation apprehension theory says is that there should be a difference between the groups because one group knows that their audience can see them and judge them and the other group knows that their audience can’t. The results of this experiment supported the evaluation apprehension theory. There was found to be a large difference between the two groups and social facilitation was only found to occur within the group with the non-blindfolded audience. This shows that people need to actually know they are being watched and judged for the effects to be caused instead of just knowing that people are there.

Distraction Conflict
Evaluation apprehension theory (Cottrell et al, 1968) directly contradicts drive theory's statement of mere presence being the cause of the psychological arousal and in turn social facilitation and inhibition. However evaluation apprehension theory does not explain the social facilitation and inhibition effects seen in animals (Chen, 1937). So one theory accounts for the effects seen in humans and the other can explain the effects in animals but neither can explain both and that is what distraction conflict theory attempts to do. This theory puts forward the idea that neither mere presence nor evaluation apprehension is the cause of the psychological arousal. Instead it states that a conflict that is created between the task at hand and attending to others around you is the cause. The effect of becoming psychologically aroused when trying to attend to multiple things is an effect that is well established in psychology. So this conflict is created from people first worrying about what they are doing and then worrying about the people around them. This in turn causes psychological arousal which leads to social facilitation or inhibition. This theory does have great support because of the fact that it is the only theory so far that can explain the effects seen in both humans and animals (Crisp & Turner, 2007).

Therefore because distraction theory is the only theory that can explain both human and animal effects, should we take it as the one true theory or is there room for all the theories in explaining social facilitation and inhibition? The truth is that there is yet to be any major supporting evidence in favour of distraction conflict theory and so for now it is best to say that the best explanation we can give to the effects seen is that it is sometimes the combination of two of these theories or even all three working together. These theories are not mutually exclusive and it can be said, for now, that each main point of each theory all play a role in some way or another in making people either socially facilitated or inhibited (Crisp & Turner, 2007).

Artificial Means
An extreme lack of social inhibition may be seen as very antisocial, however on the other hand, a high level of inhibition may also create serious personal problems, including an inability to feel or express certain emotions. So everyone has some level of social inhibition, some more than others, but what if we were to take someone who normally has a healthy level of inhibition, and take it away from them? It may be quite difficult to predict their behavior but how might we do this in the first place? Alcohol is of course the most common method of reducing a person’s social inhibition. It has been well proven that alcohol will strip people of their inhibition to varying degrees and this does depend on the person. Other drugs do have a similar effect but none are more readily available then alcohol and none is more responsible for a large percentage of antisocial behavior in our society. One of the most interesting effects of alcohol influencing a person’s inhibition is that sometimes people don’t even need the alcohol in the first place in order to change. The classic "alcohol: without alcohol" experiment shows how some people just need to think they are drinking or maybe just look like they are drinking in order to begin to change. This type of person already wants a chance to change for whatever reason, and a chance to release the feelings inside them but because they are motivated by social norms to act in a certain way, they cannot do this during a normal day, in normal encounters with the people around them. So they think alcohol is the best way to gain that chance for release and sometimes all they need is the illusion of drinking alcohol, which automatically gives them an excuse to start to act differently. There are of course substances, such as stimulants, that can have the opposite effect to alcohol and actually strengthen people’s inhibition. An abuse of these kinds of drugs can lead to heightened anxiety and inhibition and as a result of these effects have the potential to cause great mental issues for the people taking them and can also add to antisocial behavior in society.

Natural Means
There are more natural and less invasive methods of changing ones inhibition. There are many different methods of doing something like this and it is almost impossible to name them all, so we will just focus on two. The first is to train a person to respond differently to difficult situations. This will not only show them that there is a different way but also start to train them to use the new response as their dominant one instead of the old response. So if both drive theory and evaluation apprehension theory hold true then when this person becomes psychologically aroused their new response will come into play instead of their old one and they will begin to feel more confident in their behaviour. The other method applies to people who are deeply afraid of being judged. It involves slowly beginning to show these people that those around them are not going to judge them in the way they think that they will be judged. If this were to happen they may begin to see that it is ok to speak up and be heard without being so afraid of the judgement of others. These methods are of course not as easy as they sound and it does make it easy to understand why people turn to alcohol instead. As has already been stated these two methods are just two examples of many natural ways of changing a person’s inhibition that can be used. They have been explored further as they are the most understandable and easy to explain.

Summary
The list of motivations and reasons people have today to be socially inhibited is still endless, however we may be able to say they all some how stem from at least one of the three theories discussed above. For example, most people might say they are shy because they are afraid of being embarrassed which is of course evaluation apprehension. Some people might say they don’t have the skills to work around other, this may be seen as drive theory because for these individuals their well learned response is to close up around others which is just what they will do when they become psychologically aroused. So although these theories may not be motivations to be socially inhibited it is save to say that the majority of people, who are socially inhibited, are so for some form of personal protection and that is a good motivation and there are of course others. The best way we have to describe these motivations is with these three theories. When it comes to changing a person, unfortunately it is increasingly hard to do so the older a person becomes due to them becoming stuck in their ways and all their responses becoming so deeply ingrained in their personality. However it is possible, it takes time and effort and above all else, the individual must be willing to change.


text:
Motivation and emotion/Textbook/Summary and conclusion

This hope for this book was to bring to life motivation and emotion topics that intrigued students - and to do so, for students. The chapters provide substantial overviews of many key theoretical and applied aspects of motivation and emotion, with links to a range of online resources.

This openly licensed book may be used, copied and remixed by anyone at no cost. In this spirit of open sharing, we hope this project will help inspire others to openly share and collaborate on our discoveries about the psychology of human beings.


text: File:Mindbrain.jpg

title: File:Mindbrain.jpg


text: File:Lovely.jpg

title: File:Lovely.jpg


title: File:Flood1.jpg


text: File:Brainunstable.jpg

title: File:Brainunstable.jpg