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Goal setting

Goal setting involves the development of an action plan designed in order to motivate and guide a person or group toward a goal. Goals are more deliberate than desires and momentary intentions. Therefore, setting goals means that a person has committed thought, emotion, and behavior towards attaining the goal. In doing so, the goal setter has established a desired future state which differs from their current state thus creating a mismatch which in turn spurs future actions. Goal setting can be guided by goal-setting criteria (or rules) such as SMART criteria. Goal setting is a major component of personal-development and management literature. Studies by Edwin A. Locke and his colleagues, most notably, Gary Latham have shown that more specific and ambitious goals lead to more performance improvement than easy or general goals. Difficult goals should be set ideally at the 90th percentile of performance, assuming that motivation and not ability is limiting attainment of that level of performance. As long as the person accepts the goal, has the ability to attain it, and does not have conflicting goals, there is a positive linear relationship between goal difficulty and task performance.

The theory of Locke and colleagues states that the simplest, most direct motivational explanation of why some people perform better than others is because they have different performance goals. The essence of the theory is:
 * 1) Difficult specific goals lead to significantly higher performance than easy goals, no goals, or even the setting of an abstract goal such as urging people to do their best.
 * 2) Holding ability constant, and given that there is goal commitment, the higher the goal the higher the performance.
 * 3) Variables such as praise, feedback, or the participation of people in decision-making about the goal only influence behavior to the extent that they lead to the setting of and subsequent commitment to a specific difficult goal.

History
Goal setting theory has been developed through both in the field and laboratory settings. Cecil Alec Mace carried out the first empirical studies in 1935.

Edwin A. Locke began to examine goal setting in the mid-1960s and continued researching goal setting for more than 30 years. He found that individuals who set specific, difficult goals performed better than those who set general, easy goals. Locke derived the idea for goal-setting from Aristotle's form of final causality. Aristotle speculated that purpose can cause action; thus, Locke began researching the impact goals have on human activity. Locke developed and refined his goal-setting theory in the 1960s, publishing his first article on the subject, "Toward a Theory of Task Motivation and Incentives", in 1968. This article established the positive relationship between clearly identified goals and performance.

Main idea
After controlling for ability, goals that are difficult to achieve and specific tend to increase performance far more than easy goals, no goals or telling people to do their best. It therefore follows that the simplest motivational explanation of why some individuals outperform others is that they have different goals. A goal can be made more specific by:

Setting goals can affect outcomes in four ways:

Choice: Goals may narrow someone's attention and direct their efforts toward goal-relevant activities and away from goal-irrelevant actions.

Effort: Goals may make someone more effortful. For example, if someone usually produces 4 widgets per hour but wants to produce 6 widgets per hour, then they may work harder to produce more widgets than without that goal.

Persistence: Goals may make someone more willing to work through setbacks.

Cognition: Goals may cause someone to develop and change their behavior.

Goal commitment
People perform better when they are committed to achieving certain goals. Through an understanding of the effect of goal setting on individual performance, organizations are able to use goal setting to benefit organizational performance. In addition, another aspect that goes with goal commitment is also goal acceptance. This is an individual's willingness to pursue their specific goal. Locke and Latham (2002) have indicated three moderators that indicate goal setting success:

Expanding the three from above, the level of commitment is influenced by external factors. Such as the person assigning the goal, setting the standard for the person to achieve/perform. This influences the level of commitment by how compliant the individual is with the one assigning the goal. An external factor can also be the role models of the individual. For example, say an individual looks up to their manager and cares about their opinion, the individual is more likely to listen to goal-setting strategies from that individual, and ultimately become more committed to their desired goal.

Internal factors can derive from their participation level in the work to achieve the goal. What they expect from themselves can either flourish their success, or destroy it. Also, the individual may want to appear superior to their peers or competitors. They want to achieve the goal the best and be known for it. The self-reward of accomplishing a goal is usually one of the main keys that keeps individuals committed. For example, if an individual was working toward becoming the president of their company, if they achieve their goal, they could reward themselves with something of importance to them.

Another route individuals can take to set their goals is to follow (STD) that is, setting their goals to be Specific, Time-bound, and Difficult. Specifically, an individual's goal should be set at the 90th percentile of difficulty.

 * Specific: goals should give a clear definition on what needs to be accomplished. Vague goals lack direction and can spread resources too thin. A goal should be specific enough that someone outside of or unfamiliar with the situation can determine if the goal was achieved or not.
 * Time-Bound: goals should have a specific end date. Goals should not have unrealistically short deadlines. This causes stress and difficulty beyond what is intended with setting a goal. Alternatively, if a deadline for a goal is set too far in the future, people will not be motivated to work on this goal, and instead will focus on short term deadlines. To combat this, short term sub-goals can be used to maintain motivation.
 * Difficult: goals should be difficult and reasonable. Easy goals do not motivate people to push their performance levels after the goal is quickly completed.

Goal–performance relationship
Locke and colleagues (1981) examined the behavioral effects of goal-setting, concluding that 90% of laboratory and field studies involving specific and challenging goals led to higher performance than did easy or no goals. This is because if an individual is intrinsically motivated by a goal, they will want to conquer the goal to receive internal rewards, and will be satisfied because of it.

Locke and Latham (2006) argue that it is not sufficient to urge employees to "do their best". "Doing one's best" has no external reference, which makes it useless in eliciting specific behavior. To elicit some specific form of behavior from another person, it is important that this person has a clear view of what is expected from them. A goal is thereby of vital importance because it helps an individual to focus their efforts in a specified direction. In other words, goals canalize behavior. However, when faced with complex tasks and directions that are difficult to specify, telling someone to "do their best", with a focus on learning, can sometimes lead to the discovery of better strategies whereby specific goals can then be set. A solution to this apparent contradiction where the "do your best" condition can lead to greater task performance than a high specific performance goal under certain conditions is resolved when task complexity is taken into account. Specifically, in a complex task where the prerequisite skills and knowledge to perform the task are not yet in place, the "do your best" condition can outperform the performance goal condition. If a high, specific learning goal is set instead then the goal-performance relationship is maintained and the (learning) goal setting condition outperforms the "do your best" condition.

Feedback
Feedback and goal setting are highly interrelated and more effective when used in conjunction with each other. Feedback cannot be given without goals in the same way that goals can not be established without providing feedback.

Goal setting can lead to the creation of feedback loops, either negative or positive comparison of the output to the goal. Negative feedback loops lead to increasing the input associated with goal attainment to improve output in the next loop cycle. Positive feedback loops, if not sufficiently reinforced, can lead to subsequent setting of goals at a less difficult level. Negative feedback can be reframed and errors seen as beneficial to the learning and goal achievement process and in turn increase participant resilience. This reframing process can be taught through error management training and with clear instructions about how to engage with errors. Error management training involves participants practicing metacognitive activities of planning, monitoring, and evaluation.

Negative feedback also interacts with goal type, perceived tension and conscientiousness. People with high conscientiousness and performance goals experience high tension following negative feedback which leads to lower performance. This is not the case with learning goals where the effect of negative feedback is less detrimental.

Without proper feedback channels it is impossible for employees to adapt or adjust to the required behavior. Managers should keep track of performance to allow employees to see how effective they have been in attaining their goals. Providing feedback on short-term objectives helps to sustain motivation and commitment to the goal. There are two forms of feedback in which the employee can receive (Outcome and Process feedback). Outcome feedback is after the goal or activity is finished, and process feedback is during the completion of a goal. Feedback should be provided on the strategies followed to achieve the goals and on the final outcomes achieved.

Honing goal setting using temporal motivation theory
Locke and Latham (2004) note that goal-setting theory lacks "the issue of time perspective". Taking this into consideration, Steel and Konig (2006) utilize their temporal motivation theory (TMT) to account for goal setting's effects, and suggest new hypotheses regarding a pair of its moderators: goal difficulty and proximity. The effectiveness of goal setting can be explained by two aspects of TMT: the principle of diminishing returns and temporal discounting. Similar to the expression "the sum of the parts can be greater than the whole", a division of a project into several, immediate, subgoals appears to take advantage of these two elements.

Superordinate goals
Goals can be modelled as forming a hierarchy, within such a hierarchy superordinate goals are the goals at a higher level of abstraction. Superordinate goals tend to less concrete and lack a specific endpoint but have some advantages of more concrete subordinate goal.

Self-regulated learning
While the literature on self-regulated learning covers a broad variety of theoretical perspectives and concepts such as control theory, self-efficacy, action regulation, and resource allocation, goal-setting is a crucial component of virtually all of these approaches as the initiator of self-regulation mechanisms such as planning, monitoring, metacognition, attention, learning strategies, persistence, time management, environmental structuring, help seeking, emotion control, motivation, effort, and self-efficacy.

Objectives and key results
Based initially on Drucker's management by objectives (MBO) model, a popular applied version of goal setting theory for business is the objectives and key results model (OKR). Originally developed at Intel by Andy Grove, the tool was designed to set individual and collaborative goal team goals that are specific, concrete, challenging, ambitious and have measurable and time bound key results. OKRs were later introduced to many other companies and foundations such as Google and the Gates Foundation by Grove disciple John Doerr whose book Measure what Matters outlines the use of OKRs across a wide range of organizational settings.

Goal setting in sports and fitness
The tenets of Goal setting theory generally hold true in physical domains. In a study of high school students using sit up tests all students set a specific and challenging goal out performed students with a non-specific goal supporting the principles of goal specificity and goal difficulty from general goal setting theory. Goal setting appears to be especially beneficial to athletes with self-inflated narcissism by protecting athletes with that personality trait from their tendency to underperform in tedious but necessary training with no audience present.

Goal setting in workplaces
In business, goal setting remains a popular evidence based approach to align efforts across organizations, communicate objectives, and improve motivation as well as task performance for individuals and groups. Goal setting encourages participants to put in substantial effort over and above a "do your best condition". Also, because every member has defined expectations for their role, little room is left for inadequate, marginal effort to go unnoticed

Employee motivation
The more employees are motivated, the more they are stimulated and interested in accepting goals. These success factors are interdependent. For example, the expected outcomes of goals are positively influenced when employees are involved in the goal setting process. Not only does participation increase commitment in attaining the goals that are set, participation influences self-efficacy as well. Additionally, feedback is necessary to monitor one's progress. When feedback is not presented, an employee might think they are not making enough progress. This can reduce self-efficacy and thereby harm the performance outcomes in the long run.

Workplace training
Goal setting is used to improve training outcomes. For example, Tomokazu Kishiki and colleagues performed a randomized controlled trial on surgical trainees to determine whether or not their participation in a goal-setting program would improve performance and testing scores; the addition of achievable goals appeared to be beneficial to the trainees. When goal setting is applied optimally during training processes, both employee motivation and organizational commitment can increase.

Furthermore, training in goal setting has been linked to higher levels of performance among adults and children with mild to severe intellectual disability.

Work-life balance
Goal setting is also used by some companies with a stated aim of ensuring that employee work life balance is maintained. The idea behind this is that employees set a non-work related goal to improve their well-being, and managers help team members stick to those goals. An example of this in practice is "One Simple Thing", a goal-setting well-being practice employed by Google.

Impact on individual performance
Managers cannot constantly drive motivation, or keep track of an employee's work on a continuous basis. Goals are therefore an important tool for managers, since goals have the ability to function as a self-regulatory mechanism that helps employees prioritize tasks.

Four mechanisms through which goal setting can affect individual performance are:

 * 1) Goals focus attention toward goal-relevant activities and away from goal-irrelevant activities.
 * 2) Goals serve as an energizer: Higher goals induce greater effort, while low goals induce lesser effort.
 * 3) Goals affect persistence; constraints with regard to resources affect work pace.
 * 4) Goals activate cognitive knowledge and strategies that help employees cope with the situation at hand.

Domain general benefits from goal setting
Goal setting research has shown positive results as an effective, and inexpensive to implement intervention for a broad range of academic purposes across a range of age groups. Beginning with struggling undergraduate students, there is some evidence that when compared to a control group, students who had set general rather than domain specific academic goals experienced the following benefits in the semester after the intervention namely, i) increased GPA, ii) a higher probability of maintaining a full course load, iii) a reduction in self reported negative affect. While goal setting research tends to be domain specific, these results among others, does suggest that benefits from goal setting may have broader benefits for goal setters even beyond the domain of the goal itself.

Further work with undergraduates has suggested that these broader benefits apply even if non-academic goals are set. This could imply that the original goal setting theory can be modified to include a more domain general "spread" effect from superordinate life goals and that writing about general life goals regardless of domain can improve academic performance. Developing these life goals can include linked procedures such as i) reflecting on/envisaging one's best possible life, ii) listing good quality goals relevant to achieving this best possible life, iii) strategizing on how to achieve their attainment, iv) reflect in writing about anticipated obstacles, v) developing specific plans for overcoming these anticipated obstacles.

Closing achievement gaps based on gender and ethnicity
Goal setting interventions have shown promising and scalable results in terms of closing persistent gender (ongoing male underperformance) and ethnicity achievement gaps in the areas of academic achievement and increased retention rates. In one study persistent male underperformance in tertiary education was almost entirely closed to achieve parity with females after one year of the intervention. Similar albeit slower positive impacts in closing the ethnicity achievement gap by the second year of the same goal setting intervention were reported. These findings suggest that a goal setting intervention early in students' academic careers can significantly and substantially reduce gender and ethnic minority inequalities in academic achievement at least at the tertiary level. More work remains to be done to see if similar conclusions can be drawn at the level of secondary education.

Use of reflection diaries for goal setting
Goal-setting activities with final-year university students focused around self-reflective and personal growth through setting three growth goals and recording progress in reflective diaries shows that goal setting and making progress towards the goals can have many positive impacts. These include increased self-esteem, time and improved stress management and self-monitoring skills as well as motivating, and energizing effects. These suggest that, at least with undergraduates, the setting of growth goals with a reflective diary element can be a useful addition to academic programs.

Online learning and massive open online courses (MOOCs)
Goal-setting activities including the setting of both performance and learning goals have been associated with both increased performance and completion rates for MOOC participants. Students who completed a goal setting writing activity at the start of a course achieved more over a longer period of time than those who did not set goals. For online learning more generally, students who have a better understanding of the tasks set better more detailed goals and in turn achieve higher performance suggesting that instructional time spent explaining learning tasks can be beneficial.

Behavior management
Properly implemented taught goal-setting programs are effective in K-12 schools for behavior and emotional management interventions. In particular, programs that included student input on the goal setting process as well as the collection of targeting data to monitor progress and ensure the delivery of high quality feedback to students on progress towards goals were more effective behavior management interventions. In order to ensure a properly designed goal setting intervention for behavior management some variation of a checklist can be an effective addition to behavioral management programs. A typical such checklist could include the following factors:

 * 1) Identify and define behavior
 * 2) Establish a behavior monitoring plan
 * 3) Collect baseline data
 * 4) Set goals
 * 5) Monitor progress towards goals
 * 6) Review data

Goal-setting also works effectively either or its own or as part of a package of other behavior management interventions.

Identifying sub-goals
Common personal goals include losing weight, achieving good grades, and saving money. The strategy for goal setting begins with the big picture; taking a look at the big picture before breaking it into smaller components allows one to focus on the primary goal. Once the main goal is set, breaking it up into smaller, more achievable components helps in the planning portion of setting the goal. These smaller, more obtainable objectives promote self-esteem and provide instant feedback to keep the individual on task.

Time and task management
Time management is the practice of systematically finishing tasks assigned by superiors or one's self in an efficient and timely manner. Time management steps require identifying the objective and laying out a plan that maximizes efficiency and execution of the objective. There are many useful mobile apps that help with personal goal setting; some of the categories include budgeting, wellness, calendar and productivity apps.

The book What They Don't Teach You in the Harvard Business School is known for citing a study which found that written goals have a significant effect on financial success, but in 1996 Fast Company determined that this study did not occur. In 2015, a research study on goals found those who wrote them down accomplished them at a significantly higher rate than those who did not.

Life goals
There is evidence that setting and reflecting on progress life goals are an effective intervention to provide both a sense of purpose and increase happiness. In particular, setting life goals based on others leads to more positive emotions and therefore has a more positive impact on happiness than goals focused on oneself. Further evidence for this effect is provided by the more broader personal benefits of prosocial behavior and acts of kindness towards others rather than self care/focusing on oneself.

Recovery from illness and injury
There is evidence from randomized control trials that goal setting treatments improved executive function, attention/working memory, and learning in stroke patients. As well as suggesting that there is a motivational element to vascular cognitive impairment caused by strokes, or at least in terms of recovering from them, goal setting does appear to be a useful, easy to implement and cost effective solution to improve cognitive outcomes in stroke patients.

Limitations and side effects
Goal-setting has limitations and there is some evidence of potentially harmful side effects to both individuals and organizations from misuse of goals and in particular the use of performance/outcome goals.

Leader and organization goal misalignment
In an organization, a goal of a manager may not align with the goals of the organization as a whole. In such cases, the goals of an individual may come into direct conflict with the employing organization. Without clearly aligning goals between the organization and the individual, overall performance may suffer.

Unethical behavior
Additionally, there is evidence that suggests that goal-setting can foster unethical behavior when people do not achieve their desired goals. Schweitzer et al. found empirical support for their hypotheses that specific goals, rather than "do your best" goals, would lead participants to overstate performance if their true performance fell short of the goal, with the overstatement frequency increasing as the performance-goal gap narrowed. Niven and Healy found that a subset of the population having a relatively high tendency to morally justify behavior was more likely to engage in the kind of cheating identified by Schweitzer et al. Particular side effects associated with goal setting include a narrow focus that neglects non-goal areas, more unethical behavior, distorted risk preferences, damage to organizational culture, and reduced intrinsic motivation. High performance goal seem particularly likely to induce unethical behaviour under certain circumstances by creating the desire to achieve the goal but also altering moral reasoning processes and in particular, moral disengagement and encourage moral motivated reasoning due to the focus on attaining the goal.

Goals may also result in overly singleminded competition within organizations if two or more people have goals that encourage competition rather than cooperation. This can include withholding information or ideas, obstructing the goal progress of others or becoming indifferent to their progress and so withdrawing completely from interaction with other organization members. A solution to some of these potential issues is to set a unifying organizational vision or superordinate goal.

Tunnel vision
Goal setting may have the drawback of inhibiting implicit learning if the required knowledge and strategic awareness are not in place: goal setting may encourage simple focus on an outcome without openness to exploration, understanding, or growth and result in lower performance than simply encouraging people to "do their best". A solution to this limitation is to set learning goals as well as performance goals, so that learning is expected as part of the process of reaching goals. The section on learning goals has more information on this effect and how to counter it. Goal setting also may impair performance in certain situations. Such situations include when an individual becomes overly focused on accomplishing a previously-set goal that they end up underperforming on current tasks.

Narrow focus
Goal setting theory has been criticized for being too narrow in focus to be a complete theory of work motivation as goals alone are not sufficient to address all aspects of workplace motivation. In particular, it does not address why some people choose goals they dislike or how to increase intrinsic rather than extrinsic motivation. This raises the possibility that goal setting may in turn be subsumed within a more comprehensive theory of work motivation in time.

Self efficacy
An important addition to goal setting theory was the incorporation of self-efficacy from Bandura's social cognitive theory. Broadly defined as task specific self-confidence, goal setting theory incorporates self-efficacy in the following ways:

Self-efficacy levels can also influence how people react to not meeting specific challenging goals. People with high self-efficacy redouble their efforts whereas people with low self-efficacy expend less effort and coast along. Goal achievement also interacts with self-efficacy and goal achievement does not necessarily lead to increased efforts as after meeting challenging goals some can be reluctant to expend a similar level of effort again and will settle for the goal they have.
 * 1) People with higher self-efficacy set harder goals which as per the terms of the theory lead to higher motivation and task performance
 * 2) People with higher self-efficacy are more committed to the goals they have set and in turn more likely to achieve them.
 * 3) People with higher self-efficacy are more likely to respond positively to negative feedback and use it productively rather than be discouraged.
 * 4) Appropriately challenging leader assigned goals and communicating these powerfully can increase follower self-efficacy as they directly imply that the leader has confidence in the employees' ability to achieve them.
 * 5) Leaders can also increase follower self-efficacy, and in turn goal commitment and task performance, by providing quality training and either modeling task performance themselves or providing appropriate performance models.

Goal choice
Self-efficacy, past experiences, and various other social factors influence goal setting. Failure to achieve previous goals often leads to setting more achievable goals.

Learning goals
There are times when having specific performance goals is not a best option; this is the case when the goal requires skills or knowledge that have not yet been acquired. Tunnel vision can be a consequence of specific performance goals; if a person is too focused on attaining a specific goal, they may ignore the need to learn new skills or acquire new information. This concept is illustrated well by the "basketball game task" study in which observers watched a video of a group of people wearing white shirts and black shirts who are passing a basketball back and forth, and the observers were instructed to count the number of times a basketball is passed between only the players wearing white shirts. During the video, a woman carrying an open umbrella walks across the screen. Of 28 observers who were focused on counting the number of passes between only the players wearing white shirts, only 6 reported noticing the woman carrying the umbrella. When observers watched the video without focusing on a specific task, all of the observers noticed the umbrella woman. In situations where the risk of tunnel vision is high, the best option is to set a learning goal. A learning goal is a generalized goal to achieve knowledge in a certain topic or field, but it can ultimately lead to better performance in more complex tasks related to the learning goals.

Further to the above, learning goals can be more specifically operationalized as "a desired number of strategies, processes, or procedures to be developed in order to master a task'" . Some specific examples of learning goals from the literature are below:

 * "Discover and implement four shortcuts to performing a scheduling task'
 * "Find ten ways of developing a relationship with end-users of our products."

Locke and Latham (2006) attribute this response to metacognition. They believe that "a learning goal facilitates or enhances metacognition—namely, planning, monitoring, and evaluating progress toward goal attainment". This is necessary in environments with little or no guidance and structure. Although jobs typically have set goals, individual goals and achievement can benefit from metacognition. Some possible uses of learning goals follow:

 * Learning goals are likely to help leaders of globally diverse organizations find ways to effectively manage social identity groups and minimize intolerance within a multicultural workforce.
 * Learning goals are likely to be effective when leaders confront a situation with a great deal of unknowns and need to make sense of problems, as the learning goals encourage employees to collaborate with others to bring multiple experiences to solve the problem.

Framing
Framing, or how goals are viewed, influences performance. When one feels threatened and or intimidated by a high goal they perform poorer than those who view the goal as a challenge. Individuals who identify situations as challenge perform better under difficult performance goal conditions. Individuals who view situations as threats get better results using learning goals focused on developing strategy to achieve the task. These results connect goal setting theory to Folkman and Lazurus' Transactional Model of Stress and Coping which focused on the subjective appraisal of stress as being crucial to performance under challenging conditions.

Habits
Habits, defined as "behavioral tendencies tied to specific contexts, such as time of day, location, the presence of particular people, preceding actions, or even one's mood", habits develop through context, repetition, and reward and interact closely with goals to impact (often negatively) goal attainment. While goal setting can initiate behaviour change, it does appear likely that interventions combining goal setting with habit management strategies through disrupting bad habits by making them harder to fall into are more likely to be successful. Habits also reduce cognitive load and therefore good habit formation may be of benefit in particular to learning goal achievement which is often associated with more complex tasks by freeing up the cognitive resources needed to work towards the learning goal.

Affect
Realization of goals has an effect on affect—that is, feelings of success and satisfaction. Achieving goals has a positive effect, and failing to meet goals has negative consequences. However, the effect of goals is not exclusive to one realm. Success in one's job can compensate for feelings of failure in one's personal life.

Group goals
The relationship between group goals and individual goals influences group performance; when goals are compatible there is a positive effect, but when goals are incompatible the effects can be detrimental to the group's performance. There is another factor at work in groups, and that is the sharing factor; a positive correlation exists between sharing information within the group and group performance. In the case of group goals, feedback needs to be related to the group, not individuals, in order for it to improve the group's performance.

Goal concordance (agreement) among members of groups as well as concordance across hierarchies in organizations has positive performance impacts. Research evaluating effects of goals on employee commitment found an indirect relationship mediated by employee perception of organizational support, suggesting that leaders directly support goal setting by individual employees.

Overall, the available evidence suggests that group goals can have a robust effect on group performance. Less clearly, individual goals may promote group performance if used cautiously as in interdependent groups there is a potential for goal conflict between individual and group goals which could hinder group performance. There does appear to be a need for more work in this area.

Values
In goal setting terms, values can be defined as trans-situational goals with goals being more specific than values which are higher order and more general. In this sense goals can be defined further as the mechanism by which values lead to action. Goals can provide a vehicle for closing the value-action gap.

Sub-goals or proximal goals
Goal setting theory generally, but not always, supports the use of sub-goals (also known as proximal goals) which are intermediate/stepping stone goals on the way to goals (also known as distal goals). Proximal goals work by providing immediate incentives to maintain current performance, whereas distal goals are too far removed to have the same effect. In complex problem-solving tasks, setting subgoals increased initial self-efficacy and attaining proximal goals increased self-efficacy, performance satisfaction, and task persistence. One of the ways to reduce self-defeating while accomplishing sub-goals is to make sure to have deadlines for each sub-goal. Setting these deadlines adds a factor of accountability and helps to check on ourselves. The main reason why we don't usually accomplish sub-goals is because we don't put a timeframe to them.

While generally positive, setting too many sub-goals can have negative impacts such as reduced satisfaction (it's not an achievement to complete a goal that is too easy) and send the signal that managers do not have faith in employee ability to achieve challenging goals.

Goals and traits
On a basic level, the two types of goals are learning goals and performance goals; each possesses different traits associated with the selected goal.

Learning goals involve tasks where skills and knowledge can be acquired, whereas performance goals involve easy-to-accomplish tasks that will make one appear successful (thus tasks where error and judgment may be possible are avoided).

A more complex trait-mediation study is the one conducted by Lee, Sheldon, and Turban (2003), which yielded the following results:

 * Amotivated orientation (low confidence in one's capabilities) is associated with goal-avoidance motivation, and more generally, associated with lower goals levels and lower performance.
 * Control orientation (extrinsic motivation) is associated with both avoidance and approach goals. Approach goals are associated with higher goal levels and higher performance.
 * Autonomy goals (intrinsic motivation) leads to mastery goals, enhanced focus, and therefore enhanced performance.

Goal orientation
Whereas goal setting theory was developed in the sub-domain organizational psychology and primarily focuses on motivation and measuring task performance, the related but distinct literature around goal orientation was developed in the sub-domain of educational psychology and tends to focus on ability and trait measurement, this division has led to attempts to integrate the two literatures which in turn has led to the following conclusions:

 * For complex tasks a specific, challenging learning goal has a significant positive impact on performance.
 * In contrast, goal orientation affects performance when goals are vague rather than specific and challenging.

These conclusions have led to the following inferences:

 * As goal setting skills, including how to set a hard, specific goal and when to set a performance rather than a learning goal, are trainable and have greater influence than goal orientation in terms of determining performance, then it follows that the usefulness of tests of goal orientation for recruitment are limited and perhaps most suitable for solitary jobs that offer little training.
 * As well crafted appropriate goals mask the effect of goal orientation it seems likely that new employees assigned specific, high learning goals rather than performance goals will have better job performance regardless of goal orientation.

Macro-level goals
Macro-level goals refer to goal setting that is applied to the company as a whole. Cooperative goals reduce the negative feelings that occur as a result of alliances and the formation of groups. The most common parties involved are the company and its suppliers. The three motivators for macro-level goals are: self-efficacy, growth goals, and organizational vision.

Sub-conscious goals
Recent reviews of the available evidence suggests that goal setting theory applies to subconscious goals as well as consciously set goals. Subconsciously priming achievement goals through achievement related words and/or suitably triumphal photographs can significantly improve task and therefore job performance. Further enhancing this effect, context specific primes appear to induce substantially stronger goal effects. Furthermore, primed goals and consciously set goals work better together in improving task/job performance. Inevitably the use of sub-conscious goal with employees to improve work performance carries with it many potential ethical issues and concerns.

General action and inaction goals
Action goals encourage people to engage in more active behaviors, whereas inactive goals tend to result as inactive behaviors. Common action goals can be to do something, perform a certain act, or to go someplace, whereas typical inaction goals can take the form of having a rest or to stop doing something.

Goal-regulated overall activity and inactivity tendency result from both biological conditions and social-cultural environment. Recent research revealed that most nations hold more favorable attitude towards action rather than inaction, even though some countries value action and inaction slightly differently than others.

Recent research suggested that people tend to choose inaction goals when they are making decisions among choices where uncertainty could result in negative outcomes, but they prefer action over inaction in their daily behaviors when no deliberation is needed. Timothy D. Wilson and colleagues found that many people "preferred to administer electric shocks to themselves instead of being left alone with their thoughts".

Combining learning and performance goals
In workplace settings employees are often expected to achieve performance outcomes at tasks that are moderately complex and require learning new things. As noted above, setting performance goals can cause difficulties and lower performance compared to a "do your best" condition when prerequisite skills, strategies and knowledge are not in place which may be due to a cognitive load effect arising from the demands of complex tasks for relative novices. For these sorts of complex task situations learning and performance goals can be used effectively in combination if logically connected. Furthermore, while learning goals do tend to be most effective for new and complex tasks requiring complex application of strategy to achieve the task this does not mean that learning goals will be motivational enough on their own to ensure that the new strategies are used and an additional performance goal could motivate employees to actually use the discovered or acquired strategies to attain the desired outcome.

Stretch goals
Stretch or extremely hard to reach goals remain a subject of considerable debate with arguments both for and against their use. Among the potential negative side effects of stretch goals include them being dismissed as absurd or ignored by employees. Even if taken seriously, stretch goals can lead to employee burn out attempting to achieve them. Caveats aside, there are some ways that stretch goals can be valuable in spurring creative solutions to problems and new directions especially if used alongside more normal goals and without the need to achieve them but instead measure how much progress was made towards them.

Controversies, responses and resolutions
As a theory developed through induction there have been, and continue to be, circumstances where goal setting theory has been challenged and/or conflicting results have been reported. Specific examples of these controversies and resolutions to them are discussed below.

Assigned goals versus participatively set goals
The question of whether or not participatively set goals are more motivating than supervisor set goals arose due to differences in findings between Erez and her colleagues and those of Latham and colleagues. Erez and her colleagues found evidence that under certain circumstances Latham's earlier conclusion that performance was the same regardless of whether or not goals were set by supervisors or participatively, was wrong. This disagreement was resolved through a process beginning with a conversation between Erez and Latham with Locke as the neutral interlocutor which in turn led to both Erez and Latham jointly designing an experiment to explore the reasons for their disagreement. This collaboration of two researchers with differing views is an early example of adversarial collaboration and led to the following published findings that resolved the disagreement:

 * 1) Supervisor set goals are equally as motivating if they are accompanied by a reason for the goal.
 * 2) Participation in goal setting and decision making improves employee performance through increasing self-efficacy and aiding the discovery of suitable task strategies to achieve goals.

The Dark Side of Goal Setting
A 2009 article, "Goals Gone Wild" by Ordonez et al., sparked controversy by suggesting goal setting might lead to unethical behavior. The authors argued that the benefits of goal setting are often overstated, while its downsides are underreported.

One concern is that specific, challenging goals can lead to narrow thinking. Employees become laser-focused on achieving the goal, potentially neglecting other important aspects of their job. For instance, the article cites the case of Ford Motor Company. Under pressure to build a lighter car, safety measures were overlooked. This exemplifies how a singular focus on achieving a goal can have negative consequences.

Ordonez et al. further argue that setting too many goals or offering excessive rewards for quick results can pressure employees to prioritize quantity over quality and even resort to unethical shortcuts. Additionally, the authors suggest that goal setting might decrease intrinsic motivation by emphasizing extrinsic rewards.

This perspective challenges the traditional view of goal setting as a universally positive tool. Locke and Latham countered these arguments, while leveling accusations of Ordonez et al. having violated principles of good scholarship. Locke and Latham emphasized the importance of goals in organizational behavior and for individual purpose setting. A further reply from Ordonez et al. disputed Locke and Latham's points. The debate continues, with some scholars proposing learning goals as a potential solution to ethical concerns arising from performance goals.

While goal setting is a powerful tool for motivation and performance, it requires careful handling. Overemphasizing specific, challenging goals without considering potential downsides can lead to ethical lapses and counterproductive behavior. A more balanced approach is necessary, involving thoughtful goal design, awareness of potential side effects, and ongoing monitoring. By taking a critical look at goal setting, organizations can ensure it fosters both ethical conduct and successful outcomes.

title: Goal setting


text:
Villers-le-Sec, Meuse

Villers-le-Sec is a commune in the Meuse department in Grand Est in north-eastern France. Its epithet "Sec" (dry) comes from the fact that no rivers pass through Villers-le-Sec, which is located on the plateau between the rivers Ornain and Saulx. However, the village is not completely deprived of water since it is located above a water table and there are many ponds.

History
Villers-le-Sec had more than 500 inhabitants in the middle of the 19th century, when an iron mine was operated near the village. Forestry remains a significant resource today. The only industry which remains today is a distillery, which produces mirabelle brandy during the low season.

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

title: Villers-le-Sec, Meuse


text: User talk:69.137.3.36

title: User talk:69.137.3.36


text: Category:Hansford County, Texas

title: Category:Hansford County, Texas


text:
Sheaf cohomology

In mathematics, sheaf cohomology is the application of homological algebra to analyze the global sections of a sheaf on a topological space. Broadly speaking, sheaf cohomology describes the obstructions to solving a geometric problem globally when it can be solved locally. The central work for the study of sheaf cohomology is Grothendieck's 1957 Tôhoku paper.

Sheaves, sheaf cohomology, and spectral sequences were introduced by Jean Leray at the prisoner-of-war camp Oflag XVII-A in Austria. From 1940 to 1945, Leray and other prisoners organized a "université en captivité" in the camp.

Leray's definitions were simplified and clarified in the 1950s. It became clear that sheaf cohomology was not only a new approach to cohomology in algebraic topology, but also a powerful method in complex analytic geometry and algebraic geometry. These subjects often involve constructing global functions with specified local properties, and sheaf cohomology is ideally suited to such problems. Many earlier results such as the Riemann–Roch theorem and the Hodge theorem have been generalized or understood better using sheaf cohomology.

Definition
The category of sheaves of abelian groups on a topological space X is an abelian category, and so it makes sense to ask when a morphism f: B → C of sheaves is injective (a monomorphism) or surjective (an epimorphism). One answer is that f is injective (respectively surjective) if and only if the associated homomorphism on stalks Bx → Cx is injective (respectively surjective) for every point x in X. It follows that f is injective if and only if the homomorphism B(U) → C(U) of sections over U is injective for every open set U in X. Surjectivity is more subtle, however: the morphism f is surjective if and only if for every open set U in X, every section s of C over U, and every point x in U, there is an open neighborhood V of x in U such that s restricted to V is the image of some section of B over V. (In words: every section of C lifts locally to sections of B.)

As a result, the question arises: given a surjection B → C of sheaves and a section s of C over X, when is s the image of a section of B over X? This is a model for all kinds of local-vs.-global questions in geometry. Sheaf cohomology gives a satisfactory general answer. Namely, let A be the kernel of the surjection B → C, giving a short exact sequence
 * $$ 0\to A\to B\to C\to 0$$

of sheaves on X. Then there is a long exact sequence of abelian groups, called sheaf cohomology groups:
 * $$ 0\to H^0(X,A) \to H^0(X,B) \to H^0(X,C) \to H^1(X,A) \to \cdots,$$

where H0(X,A) is the group A(X) of global sections of A on X. For example, if the group H1(X,A) is zero, then this exact sequence implies that every global section of C lifts to a global section of B. More broadly, the exact sequence makes knowledge of higher cohomology groups a fundamental tool in aiming to understand sections of sheaves.

Grothendieck's definition of sheaf cohomology, now standard, uses the language of homological algebra. The essential point is to fix a topological space X and think of cohomology as a functor from sheaves of abelian groups on X to abelian groups. In more detail, start with the functor E ↦ E(X) from sheaves of abelian groups on X to abelian groups. This is left exact, but in general not right exact. Then the groups Hi(X,E) for integers i are defined as the right derived functors of the functor E ↦ E(X). This makes it automatic that Hi(X,E) is zero for i < 0, and that H0(X,E) is the group E(X) of global sections. The long exact sequence above is also straightforward from this definition.

The definition of derived functors uses that the category of sheaves of abelian groups on any topological space X has enough injectives; that is, for every sheaf E there is an injective sheaf I with an injection E → I. It follows that every sheaf E has an injective resolution:
 * $$0\to E\to I_0\to I_1\to I_2\to \cdots.$$

Then the sheaf cohomology groups Hi(X,E) are the cohomology groups (the kernel of one homomorphism modulo the image of the previous one) of the chain complex of abelian groups:
 * $$ 0\to I_0(X) \to I_1(X) \to I_2(X)\to \cdots.$$

Standard arguments in homological algebra imply that these cohomology groups are independent of the choice of injective resolution of E.

The definition is rarely used directly to compute sheaf cohomology. It is nonetheless powerful, because it works in great generality (any sheaf of abelian groups on any topological space), and it easily implies the formal properties of sheaf cohomology, such as the long exact sequence above. For specific classes of spaces or sheaves, there are many tools for computing sheaf cohomology, some discussed below.

Functoriality
For any continuous map f: X → Y of topological spaces, and any sheaf E of abelian groups on Y, there is a pullback homomorphism
 * $$f^*\colon H^j(Y,E) \to H^j(X,f^*(E))$$

for every integer j, where f*(E) denotes the inverse image sheaf or pullback sheaf. If f is the inclusion of a subspace X of Y, f*(E) is the restriction of E to X, often just called E again, and the pullback of a section s from Y to X is called the restriction s|X.

Pullback homomorphisms are used in the Mayer–Vietoris sequence, an important computational result. Namely, let X be a topological space which is a union of two open subsets U and V, and let E be a sheaf on X. Then there is a long exact sequence of abelian groups:
 * $$ 0\to H^0(X,E) \to H^0(U,E)\oplus H^0(V,E) \to H^0(U\cap V, E) \to H^1(X,E) \to \cdots.$$

Sheaf cohomology with constant coefficients
For a topological space $$X$$ and an abelian group $$A$$, the constant sheaf $$A_X$$ means the sheaf of locally constant functions with values in $$A$$. The sheaf cohomology groups $$H^j(X,A_X)$$ with constant coefficients are often written simply as $$H^j(X,A)$$, unless this could cause confusion with another version of cohomology such as singular cohomology.

For a continuous map f: X → Y and an abelian group A, the pullback sheaf f*(AY) is isomorphic to AX. As a result, the pullback homomorphism makes sheaf cohomology with constant coefficients into a contravariant functor from topological spaces to abelian groups.

For any spaces X and Y and any abelian group A, two homotopic maps f and g from X to Y induce the same homomorphism on sheaf cohomology:
 * $$f^*=g^*: H^j(Y,A)\to H^j(X,A).$$

It follows that two homotopy equivalent spaces have isomorphic sheaf cohomology with constant coefficients.

Let X be a paracompact Hausdorff space which is locally contractible, even in the weak sense that every open neighborhood U of a point x contains an open neighborhood V of x such that the inclusion V → U is homotopic to a constant map. Then the singular cohomology groups of X with coefficients in an abelian group A are isomorphic to sheaf cohomology with constant coefficients, H*(X,AX). For example, this holds for X a topological manifold or a CW complex.

As a result, many of the basic calculations of sheaf cohomology with constant coefficients are the same as calculations of singular cohomology. See the article on cohomology for the cohomology of spheres, projective spaces, tori, and surfaces.

For arbitrary topological spaces, singular cohomology and sheaf cohomology (with constant coefficients) can be different. This happens even for H0. The singular cohomology H0(X,Z) is the group of all functions from the set of path components of X to the integers Z, whereas sheaf cohomology H0(X,ZX) is the group of locally constant functions from X to Z. These are different, for example, when X is the Cantor set. Indeed, the sheaf cohomology H0(X,ZX) is a countable abelian group in that case, whereas the singular cohomology H0(X,Z) is the group of all functions from X to Z, which has cardinality
 * $$2^{2^{\aleph_0}}.$$

For a paracompact Hausdorff space X and any sheaf E of abelian groups on X, the cohomology groups Hj(X,E) are zero for j greater than the covering dimension of X. (This does not hold in the same generality for singular cohomology: for example, there is a compact subset of Euclidean space R3 that has nonzero singular cohomology in infinitely many degrees. ) The covering dimension agrees with the usual notion of dimension for a topological manifold or a CW complex.

Flabby and soft sheaves
A sheaf E of abelian groups on a topological space X is called acyclic if Hj(X,E) = 0 for all j > 0. By the long exact sequence of sheaf cohomology, the cohomology of any sheaf can be computed from any acyclic resolution of E (rather than an injective resolution). Injective sheaves are acyclic, but for computations it is useful to have other examples of acyclic sheaves.

A sheaf E on X is called flabby (French: flasque) if every section of E on an open subset of X extends to a section of E on all of X. Flabby sheaves are acyclic. Godement defined sheaf cohomology via a canonical flabby resolution of any sheaf; since flabby sheaves are acyclic, Godement's definition agrees with the definition of sheaf cohomology above.

A sheaf E on a paracompact Hausdorff space X is called soft if every section of the restriction of E to a closed subset of X extends to a section of E on all of X. Every soft sheaf is acyclic.

Some examples of soft sheaves are the sheaf of real-valued continuous functions on any paracompact Hausdorff space, or the sheaf of smooth (C∞) functions on any smooth manifold. More generally, any sheaf of modules over a soft sheaf of commutative rings is soft; for example, the sheaf of smooth sections of a vector bundle over a smooth manifold is soft.

For example, these results form part of the proof of de Rham's theorem. For a smooth manifold X, the Poincaré lemma says that the de Rham complex is a resolution of the constant sheaf RX:
 * $$0\to\mathbf{R}_X\to\Omega^0_X\to\Omega^1_X\to\cdots,$$

where ΩXj is the sheaf of smooth j-forms and the map ΩXj → ΩXj+1 is the exterior derivative d. By the results above, the sheaves ΩXj are soft and therefore acyclic. It follows that the sheaf cohomology of X with real coefficients is isomorphic to the de Rham cohomology of X, defined as the cohomology of the complex of real vector spaces:
 * $$0\to \Omega^0_X(X)\to\Omega^1_X(X)\to\cdots.$$

The other part of de Rham's theorem is to identify sheaf cohomology and singular cohomology of X with real coefficients; that holds in greater generality, as discussed above.

Čech cohomology
Čech cohomology is an approximation to sheaf cohomology that is often useful for computations. Namely, let $$\mathcal{U}$$ be an open cover of a topological space X, and let E be a sheaf of abelian groups on X. Write the open sets in the cover as Ui for elements i of a set I, and fix an ordering of I. Then Čech cohomology $$H^j(\mathcal{U},E)$$ is defined as the cohomology of an explicit complex of abelian groups with jth group
 * $$C^j(\mathcal{U},E)=\prod_{i_0<\cdots 0, then the homomorphism from Čech cohomology $$H^j(\mathcal{U},E)$$ to sheaf cohomology is an isomorphism.

Another approach to relating Čech cohomology to sheaf cohomology is as follows. The Čech cohomology groups $$\check{H}^j(X,E)$$ are defined as the direct limit of $$H^j(\mathcal{U},E)$$ over all open covers $$\mathcal{U}$$ of X (where open covers are ordered by refinement). There is a homomorphism $$\check{H}^j(X,E)\to H^j(X,E)$$ from Čech cohomology to sheaf cohomology, which is an isomorphism for j ≤ 1. For arbitrary topological spaces, Čech cohomology can differ from sheaf cohomology in higher degrees. Conveniently, however, Čech cohomology is isomorphic to sheaf cohomology for any sheaf on a paracompact Hausdorff space.

The isomorphism $$\check{H}^1(X,E)\cong H^1(X,E)$$ implies a description of H1(X,E) for any sheaf E of abelian groups on a topological space X: this group classifies the E-torsors (also called principal E-bundles) over X, up to isomorphism. (This statement generalizes to any sheaf of groups G, not necessarily abelian, using the non-abelian cohomology set H1(X,G).) By definition, an E-torsor over X is a sheaf S of sets together with an action of E on X such that every point in X has an open neighborhood on which S is isomorphic to E, with E acting on itself by translation. For example, on a ringed space (X,OX), it follows that the Picard group of invertible sheaves on X is isomorphic to the sheaf cohomology group H1(X,OX*), where OX* is the sheaf of units in OX.

Relative cohomology
For a subset Y of a topological space X and a sheaf E of abelian groups on X, one can define relative cohomology groups:
 * $$H^j_Y(X,E)=H^j(X,X-Y;E)$$

for integers j. Other names are the cohomology of X with support in Y, or (when Y is closed in X) local cohomology. A long exact sequence relates relative cohomology to sheaf cohomology in the usual sense:
 * $$\cdots \to H^j_Y(X,E)\to H^j(X,E)\to H^j(X-Y,E)\to H^{j+1}_Y(X,E)\to\cdots.$$

When Y is closed in X, cohomology with support in Y can be defined as the derived functors of the functor
 * $$H^0_Y(X,E):=\{s\in E(X): s|_{X-Y}=0\},$$

the group of sections of E that are supported on Y.

There are several isomorphisms known as excision. For example, if X is a topological space with subspaces Y and U such that the closure of Y is contained in the interior of U, and E is a sheaf on X, then the restriction
 * $$H^j_Y(X,E)\to H^j_Y(U,E)$$

is an isomorphism. (So cohomology with support in a closed subset Y only depends on the behavior of the space X and the sheaf E near Y.) Also, if X is a paracompact Hausdorff space that is the union of closed subsets A and B, and E is a sheaf on X, then the restriction
 * $$H^j(X,B;E)\to H^j(A,A\cap B;E)$$

is an isomorphism.

Cohomology with compact support
Let X be a locally compact topological space. (In this article, a locally compact space is understood to be Hausdorff.) For a sheaf E of abelian groups on X, one can define cohomology with compact support Hcj(X,E). These groups are defined as the derived functors of the functor of compactly supported sections:
 * $$H^0_c(X,E)=\{s\in E(X): \text{there is a compact subset }K\text{ of }X\text{ with }s|_{X-K}=0\}.$$

There is a natural homomorphism Hcj(X,E) → Hj(X,E), which is an isomorphism for X compact.

For a sheaf E on a locally compact space X, the compactly supported cohomology of X × R with coefficients in the pullback of E is a shift of the compactly supported cohomology of X:
 * $$H^{j+1}_c(X\times\mathbf{R},E)\cong H^j_c(X,E).$$

It follows, for example, that Hcj(Rn,Z) is isomorphic to Z if j = n and is zero otherwise.

Compactly supported cohomology is not functorial with respect to arbitrary continuous maps. For a proper map f: Y → X of locally compact spaces and a sheaf E on X, however, there is a pullback homomorphism
 * $$f^*\colon H^j_c(X,E)\to H^j_c(Y,f^*(E))$$

on compactly supported cohomology. Also, for an open subset U of a locally compact space X and a sheaf E on X, there is a pushforward homomorphism known as extension by zero:
 * $$H^j_c(U,E)\to H^j_c(X,E).$$

Both homomorphisms occur in the long exact localization sequence for compactly supported cohomology, for a locally compact space X and a closed subset Y:
 * $$\cdots\to H^j_c(X-Y,E)\to H^j_c(X,E)\to H^j_c(Y,E)\to H^{j+1}_c(X-Y,E)\to\cdots.$$

Cup product
For any sheaves A and B of abelian groups on a topological space X, there is a bilinear map, the cup product
 * $$H^i(X,A)\times H^j(X,B)\to H^{i+j}(X,A\otimes B),$$

for all i and j. Here A⊗B denotes the tensor product over Z, but if A and B are sheaves of modules over some sheaf OX of commutative rings, then one can map further from Hi+j(X,A⊗ZB) to Hi+j(X,A⊗O X B). In particular, for a sheaf OX of commutative rings, the cup product makes the direct sum
 * $$H^*(X,O_X) = \bigoplus_j H^j(X,O_X)$$

into a graded-commutative ring, meaning that
 * $$vu=(-1)^{ij}uv$$

for all u in Hi and v in Hj.

Complexes of sheaves
The definition of sheaf cohomology as a derived functor extends to define cohomology of a topological space X with coefficients in any complex E of sheaves:
 * $$\cdots\to E_j\to E_{j+1}\to E_{j+2}\to \cdots$$

In particular, if the complex E is bounded below (the sheaf Ej is zero for j sufficiently negative), then E has an injective resolution I just as a single sheaf does. (By definition, I is a bounded below complex of injective sheaves with a chain map E → I that is a quasi-isomorphism.) Then the cohomology groups Hj(X,E) are defined as the cohomology of the complex of abelian groups
 * $$\cdots \to I_j(X)\to I_{j+1}(X)\to I_{j+2}(X)\to\cdots.$$

The cohomology of a space with coefficients in a complex of sheaves was earlier called hypercohomology, but usually now just "cohomology".

More generally, for any complex of sheaves E (not necessarily bounded below) on a space X, the cohomology group Hj(X,E) is defined as a group of morphisms in the derived category of sheaves on X:
 * $$H^j(X,E)=\operatorname{Hom}_{D(X)}(\mathbf{Z}_X,E[j]),$$

where ZX is the constant sheaf associated to the integers, and E[j] means the complex E shifted j steps to the left.

Poincaré duality and generalizations
A central result in topology is the Poincaré duality theorem: for a closed oriented connected topological manifold X of dimension n and a field k, the group Hn(X,k) is isomorphic to k, and the cup product
 * $$H^j(X,k)\times H^{n-j}(X,k)\to H^n(X,k)\cong k$$

is a perfect pairing for all integers j. That is, the resulting map from Hj(X,k) to the dual space Hn−j(X,k)* is an isomorphism. In particular, the vector spaces Hj(X,k) and Hn−j(X,k)* have the same (finite) dimension.

Many generalizations are possible using the language of sheaf cohomology. If X is an oriented n-manifold, not necessarily compact or connected, and k is a field, then cohomology is the dual of cohomology with compact support:
 * $$H^j(X,k)\cong H^{n-j}_c(X,k)^*.$$

For any manifold X and field k, there is a sheaf oX on X, the orientation sheaf, which is locally (but perhaps not globally) isomorphic to the constant sheaf k. One version of Poincaré duality for an arbitrary n-manifold X is the isomorphism:
 * $$H^j(X,o_X)\cong H^{n-j}_c(X,k)^*.$$

More generally, if E is a locally constant sheaf of k-vector spaces on an n-manifold X and the stalks of E have finite dimension, then there is an isomorphism
 * $$H^j(X,E^*\otimes o_X)\cong H^{n-j}_c(X,E)^*.$$

With coefficients in an arbitrary commutative ring rather than a field, Poincaré duality is naturally formulated as an isomorphism from cohomology to Borel–Moore homology.

Verdier duality is a vast generalization. For any locally compact space X of finite dimension and any field k, there is an object DX in the derived category D(X) of sheaves on X called the dualizing complex (with coefficients in k). One case of Verdier duality is the isomorphism:
 * $$H^j(X,D_X)\cong H^{-j}_c(X,k)^*.$$

For an n-manifold X, the dualizing complex DX is isomorphic to the shift oX[n] of the orientation sheaf. As a result, Verdier duality includes Poincaré duality as a special case.

Alexander duality is another useful generalization of Poincaré duality. For any closed subset X of an oriented n-manifold M and any field k, there is an isomorphism:
 * $$H^j_X(M,k)\cong H^{n-j}_c(X,k)^*.$$

This is interesting already for X a compact subset of M = Rn, where it says (roughly speaking) that the cohomology of Rn−X is the dual of the sheaf cohomology of X. In this statement, it is essential to consider sheaf cohomology rather than singular cohomology, unless one makes extra assumptions on X such as local contractibility.

Higher direct images and the Leray spectral sequence
Let f: X → Y be a continuous map of topological spaces, and let E be a sheaf of abelian groups on X. The direct image sheaf f*E is the sheaf on Y defined by
 * $$(f_*E)(U) = E(f^{-1}(U))$$

for any open subset U of Y. For example, if f is the map from X to a point, then f*E is the sheaf on a point corresponding to the group E(X) of global sections of E.

The functor f* from sheaves on X to sheaves on Y is left exact, but in general not right exact. The higher direct image sheaves Rif*E on Y are defined as the right derived functors of the functor f*. Another description is that Rif*E is the sheaf associated to the presheaf
 * $$U \mapsto H^i(f^{-1}(U),E)$$

on Y. Thus, the higher direct image sheaves describe the cohomology of inverse images of small open sets in Y, roughly speaking.

The Leray spectral sequence relates cohomology on X to cohomology on Y. Namely, for any continuous map f: X → Y and any sheaf E on X, there is a spectral sequence
 * $$ E_2^{ij} = H^i(Y,R^jf_*E) \Rightarrow H^{i+j}(X,E).$$

This is a very general result. The special case where f is a fibration and E is a constant sheaf plays an important role in homotopy theory under the name of the Serre spectral sequence. In that case, the higher direct image sheaves are locally constant, with stalks the cohomology groups of the fibers F of f, and so the Serre spectral sequence can be written as
 * $$ E_2^{ij} = H^i(Y,H^j(F,A)) \Rightarrow H^{i+j}(X,A)$$

for an abelian group A.

A simple but useful case of the Leray spectral sequence is that for any closed subset X of a topological space Y and any sheaf E on X, writing f: X → Y for the inclusion, there is an isomorphism
 * $$H^i(Y,f_*E)\cong H^i(X,E).$$

As a result, any question about sheaf cohomology on a closed subspace can be translated to a question about the direct image sheaf on the ambient space.

Finiteness of cohomology
There is a strong finiteness result on sheaf cohomology. Let X be a compact Hausdorff space, and let R be a principal ideal domain, for example a field or the ring Z of integers. Let E be a sheaf of R-modules on X, and assume that E has "locally finitely generated cohomology", meaning that for each point x in X, each integer j, and each open neighborhood U of x, there is an open neighborhood V ⊂ U of x such that the image of Hj(U,E) → Hj(V,E) is a finitely generated R-module. Then the cohomology groups Hj(X,E) are finitely generated R-modules.

For example, for a compact Hausdorff space X that is locally contractible (in the weak sense discussed above), the sheaf cohomology group Hj(X,Z) is finitely generated for every integer j.

One case where the finiteness result applies is that of a constructible sheaf. Let X be a topologically stratified space. In particular, X comes with a sequence of closed subsets
 * $$X=X_n\supset X_{n-1}\supset\cdots\supset X_{-1}=\emptyset$$

such that each difference Xi−Xi−1 is a topological manifold of dimension i. A sheaf E of R-modules on X is constructible with respect to the given stratification if the restriction of E to each stratum Xi−Xi−1 is locally constant, with stalk a finitely generated R-module. A sheaf E on X that is constructible with respect to the given stratification has locally finitely generated cohomology. If X is compact, it follows that the cohomology groups Hj(X,E) of X with coefficients in a constructible sheaf are finitely generated.

More generally, suppose that X is compactifiable, meaning that there is a compact stratified space W containing X as an open subset, with W–X a union of connected components of strata. Then, for any constructible sheaf E of R-modules on X, the R-modules Hj(X,E) and Hcj(X,E) are finitely generated. For example, any complex algebraic variety X, with its classical (Euclidean) topology, is compactifiable in this sense.

Cohomology of coherent sheaves
In algebraic geometry and complex analytic geometry, coherent sheaves are a class of sheaves of particular geometric importance. For example, an algebraic vector bundle (on a locally Noetherian scheme) or a holomorphic vector bundle (on a complex analytic space) can be viewed as a coherent sheaf, but coherent sheaves have the advantage over vector bundles that they form an abelian category. On a scheme, it is also useful to consider the quasi-coherent sheaves, which include the locally free sheaves of infinite rank.

A great deal is known about the cohomology groups of a scheme or complex analytic space with coefficients in a coherent sheaf. This theory is a key technical tool in algebraic geometry. Among the main theorems are results on the vanishing of cohomology in various situations, results on finite-dimensionality of cohomology, comparisons between coherent sheaf cohomology and singular cohomology such as Hodge theory, and formulas on Euler characteristics in coherent sheaf cohomology such as the Riemann–Roch theorem.

Sheaves on a site
In the 1960s, Grothendieck defined the notion of a site, meaning a category equipped with a Grothendieck topology. A site C axiomatizes the notion of a set of morphisms Vα → U in C being a covering of U. A topological space X determines a site in a natural way: the category C has objects the open subsets of X, with morphisms being inclusions, and with a set of morphisms Vα → U being called a covering of U if and only if U is the union of the open subsets Vα. The motivating example of a Grothendieck topology beyond that case was the étale topology on schemes. Since then, many other Grothendieck topologies have been used in algebraic geometry: the fpqc topology, the Nisnevich topology, and so on.

The definition of a sheaf works on any site. So one can talk about a sheaf of sets on a site, a sheaf of abelian groups on a site, and so on. The definition of sheaf cohomology as a derived functor also works on a site. So one has sheaf cohomology groups Hj(X, E) for any object X of a site and any sheaf E of abelian groups. For the étale topology, this gives the notion of étale cohomology, which led to the proof of the Weil conjectures. Crystalline cohomology and many other cohomology theories in algebraic geometry are also defined as sheaf cohomology on an appropriate site.

title: Sheaf cohomology


text:
Talk:Computer desk

The requirements of protection because of a conflict of physiology should be included in any discussion of a computer desk. HP and Compact have removed blinking lights from their Computer CPU's but there are still sources of stimulus your brain can mistake for threat movement. I will continue to investigate but the information is very closely held. Herman Miller Inc has not responded except to suggest that if I was worried about Cubicle Level Protection I should choose a Systems Furniture design that used solid panels rather that semi-opaque glass, plastic, or fabric panels. This type protection is not attached to the desk but is included in walls and moveable panels surrounding the desk, Cubicle.

If you have a source of information email me from an email link at VisionAndPsychosis.Net L K Tucker 10/10/04

As an anonymous user, L K Tucker (above) of http://visionandpsychosis.net contributed these paragraphs:

"The invention of the computer caused a change in the design of the modern office workstation. Computer desks are designed with different levels of Cubicle Level Protection depending on the perceived use. About fifty years ago engineers discovered a conflict of human physiology when workers in newly designed Compact-Close-Spaced office workstations began to have bizarre or psychotic episodes. The problem was peripheral vision reflexes. The solution was to block peripheral vision of workers that require deep mental investment to perform their jobs. Not all workstations or computer desks require 'Cubicle Level Protection. Accounts vary as to the history of this discovery. Recent articles give the date of Herman Miller's Action Office One as 1964. There were articles as early as 1961. Computer desks may or may not be designed to stand alone and supply protection for the user. Simple desks have only the bare necessities to support a computer and its peripherals. The theory seems to be that there is not enough human traffic in the home, dorm, student apartment, or small business office to require Cubicle Level Protection in those locations."

These ideas are based on original research - see this talk page for more discussion - and are not supported by mainstream psychology. I would go so far as to describe Mr Tuckers ideas as a crank theory. Even if it was of some value, it would be more appropriate on a separate article and isn't particularly relevant to the Computer Desk article. Hence, I have removed said material from this page. Ppe42 11:56, 5 October 2005 (UTC)

Proposal to merge.
I am suggesting that the computer table and ergonomic desk articles be merged into this one. There is considerable overlap. I suspect that any growth in the article will be common to all three. JonHarder 22:22, 26 January 2006 (UTC)

title: Talk:Computer desk


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Villers-Canivet

Villers-Canivet is a commune in the Calvados department in the Normandy region in northwestern France.

title: Villers-Canivet


text:
User talk:Writerite

ClockworkTroll 19:31, 10 Oct 2004 (UTC)

title: User talk:Writerite


text:
Sangaku

Sangaku or san gaku (算額) are Japanese geometrical problems or theorems on wooden tablets which were placed as offerings at Shinto shrines or Buddhist temples during the Edo period by members of all social classes.

History
The sangaku were painted in color on wooden tablets (ema) and hung in the precincts of Buddhist temples and Shinto shrines as offerings to the kami and buddhas, as challenges to the congregants, or as displays of the solutions to questions. Many of these tablets were lost during the period of modernization that followed the Edo period, but around nine hundred are known to remain.

Fujita Kagen (1765–1821), a Japanese mathematician of prominence, published the first collection of sangaku problems, his Shimpeki Sampo (Mathematical problems Suspended from the Temple) in 1790, and in 1806 a sequel, the Zoku Shimpeki Sampo.

During this period Japan applied strict regulations to commerce and foreign relations for western countries so the tablets were created using Japanese mathematics, developed in parallel to western mathematics. For example, the connection between an integral and its derivative (the fundamental theorem of calculus) was unknown, so sangaku problems on areas and volumes were solved by expansions in infinite series and term-by-term calculation.

Select examples

 * A typical problem, which is presented on an 1824 tablet in Gunma Prefecture, covers the relationship of three touching circles with a common tangent, a special case of Descartes' theorem. Given the size of the two outer large circles, what is the size of the small circle between them? The answer is:

 * $$\frac{1}{\sqrt{r_\text{middle}}} = \frac{1}{\sqrt{r_\text{left}}} + \frac{1}{\sqrt{r_\text{right}}}.$$

(See also Ford circle.)

 * Soddy's hexlet, thought previously to have been discovered in the west in 1937, had been discovered on a sangaku dating from 1822.
 * One sangaku problem from Sawa Masayoshi and other from Jihei Morikawa were solved only recently.

title: Sangaku


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Japanese mathematics

Japanese mathematics (和算) denotes a distinct kind of mathematics which was developed in Japan during the Edo period (1603–1867). The term wasan, from wa ("Japanese") and san ("calculation"), was coined in the 1870s and employed to distinguish native Japanese mathematical theory from Western mathematics (洋算 yōsan).

In the history of mathematics, the development of wasan falls outside the Western realm. At the beginning of the Meiji period (1868–1912), Japan and its people opened themselves to the West. Japanese scholars adopted Western mathematical technique, and this led to a decline of interest in the ideas used in wasan.

History
The Japanese mathematical schema evolved during a period when Japan's people were isolated from European influences, but instead borrowed from ancient mathematical texts written in China, including those from the Yuan dynasty and earlier. The Japanese mathematicians Yoshida Shichibei Kōyū, Imamura Chishō, and Takahara Kisshu are among the earliest known Japanese mathematicians. They came to be known to their contemporaries as "the Three Arithmeticians".

Yoshida was the author of the oldest extant Japanese mathematical text, the 1627 work called Jinkōki. The work dealt with the subject of soroban arithmetic, including square and cube root operations. Yoshida's book significantly inspired a new generation of mathematicians, and redefined the Japanese perception of educational enlightenment, which was defined in the Seventeen Article Constitution as "the product of earnest meditation".

Seki Takakazu founded enri (円理: circle principles), a mathematical system with the same purpose as calculus at a similar time to calculus's development in Europe. However Seki's investigations did not proceed from the same foundations as those used in Newton's studies in Europe.

Mathematicians like Takebe Katahiro played an important role in developing Enri (" circle principle"), a crude analog to the Western calculus. He obtained power series expansion of $$(\arcsin(x))^2$$ in 1722, 15 years earlier than Euler. He used Richardson extrapolation in 1695, about 200 years earlier than Richardson. He also computed 41 digits of π, based on polygon approximation and Richardson extrapolation.

Select mathematicians
The following list encompasses mathematicians whose work was derived from wasan.

 * Yoshida Mitsuyoshi (1598–1672)
 * Seki Takakazu (1642–1708)
 * Takebe Kenkō (1664–1739)
 * Matsunaga Ryohitsu (fl. 1718-1749)
 * Kurushima Kinai (d. 1757)
 * Arima Raido (1714–1783)
 * Fujita Sadasuke (1734-1807)
 * Ajima Naonobu (1739–1783)
 * Aida Yasuaki (1747–1817)
 * Sakabe Kōhan (1759–1824)
 * Fujita Kagen (1765–1821)
 * Hasegawa Ken (c. 1783-1838)
 * Wada Nei (1787–1840)
 * Shiraishi Chochu (1796–1862)
 * Koide Shuke (1797–1865)
 * Omura Isshu (1824–1871)

title: Japanese mathematics


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Lucas pseudoprime

Lucas pseudoprimes and Fibonacci pseudoprimes are composite integers that pass certain tests which all primes and very few composite numbers pass: in this case, criteria relative to some Lucas sequence.

Baillie-Wagstaff-Lucas pseudoprimes
Baillie and Wagstaff define Lucas pseudoprimes as follows: Given integers P and Q, where P > 0 and $$D=P^2-4Q$$, let Uk(P, Q) and Vk(P, Q) be the corresponding Lucas sequences.

Let n be a positive integer and let $$\left(\tfrac{D}{n}\right)$$ be the Jacobi symbol. We define

 * $$\delta(n)=n-\left(\tfrac{D}{n}\right).$$

If n is a prime that does not divide Q, then the following congruence condition holds:

If this congruence does not hold, then n is not prime. If n is composite, then this congruence usually does not hold. These are the key facts that make Lucas sequences useful in primality testing.

The congruence ($$) represents one of two congruences defining a Frobenius pseudoprime. Hence, every Frobenius pseudoprime is also a Baillie-Wagstaff-Lucas pseudoprime, but the converse does not always hold.

Some good references are chapter 8 of the book by Bressoud and Wagon (with Mathematica code), pages 142–152 of the book by Crandall and Pomerance, and pages 53–74 of the book by Ribenboim.

Lucas probable primes and pseudoprimes
A Lucas probable prime for a given (P, Q) pair is any positive integer n for which equation ($$) above is true (see, page 1398).

A Lucas pseudoprime for a given (P, Q) pair is a positive composite integer n for which equation ($$) is true (see, page 1391).

A Lucas probable prime test is most useful if D is chosen such that the Jacobi symbol $$\left(\tfrac{D}{n}\right)$$ is −1 (see pages 1401–1409 of, page 1024 of, or pages 266–269 of ). This is especially important when combining a Lucas test with a strong pseudoprime test, such as the Baillie–PSW primality test. Typically implementations will use a parameter selection method that ensures this condition (e.g. the Selfridge method recommended in and described below).

If $$\left(\tfrac{D}{n}\right)=-1,$$ then equation ($$) becomes

If congruence ($$) is false, this constitutes a proof that n is composite.

If congruence ($$) is true, then n is a Lucas probable prime. In this case, either n is prime or it is a Lucas pseudoprime. If congruence ($$) is true, then n is likely to be prime (this justifies the term probable prime), but this does not prove that n is prime. As is the case with any other probabilistic primality test, if we perform additional Lucas tests with different D, P and Q, then unless one of the tests proves that n is composite, we gain more confidence that n is prime.

Examples: If P = 3, Q = −1, and D = 13, the sequence of U's is : U0 = 0, U1 = 1, U2 = 3, U3 = 10, etc.

First, let n = 19. The Jacobi symbol $$\left(\tfrac{13}{19}\right)$$ is −1, so δ(n) = 20, U20 = 6616217487 = 19·348221973 and we have
 * $$ U_{20} = 6616217487 \equiv 0 \pmod {19} . $$

Therefore, 19 is a Lucas probable prime for this (P, Q) pair. In this case 19 is prime, so it is not a Lucas pseudoprime.

For the next example, let n = 119. We have $$\left(\tfrac{13}{119}\right)$$ = −1, and we can compute
 * $$ U_{120} \equiv 0 \pmod {119}. $$

However, 119 = 7·17 is not prime, so 119 is a Lucas pseudoprime for this (P, Q) pair. In fact, 119 is the smallest pseudoprime for P = 3, Q = −1.

We will see below that, in order to check equation ($$) for a given n, we do not need to compute all of the first n + 1 terms in the U sequence.

Let Q = −1, the smallest Lucas pseudoprime to P = 1, 2, 3, ... are
 * 323, 35, 119, 9, 9, 143, 25, 33, 9, 15, 123, 35, 9, 9, 15, 129, 51, 9, 33, 15, 21, 9, 9, 49, 15, 39, 9, 35, 49, 15, 9, 9, 33, 51, 15, 9, 35, 85, 39, 9, 9, 21, 25, 51, 9, 143, 33, 119, 9, 9, 51, 33, 95, 9, 15, 301, 25, 9, 9, 15, 49, 155, 9, 399, 15, 33, 9, 9, 49, 15, 119, 9, ...

Strong Lucas pseudoprimes
Now, factor $$\delta(n)=n-\left(\tfrac{D}{n}\right)$$ into the form $$d\cdot2^s$$ where $$d$$ is odd.

A strong Lucas pseudoprime for a given (P, Q) pair is an odd composite number n with GCD(n, D) = 1, satisfying one of the conditions

 * $$ U_d \equiv 0 \pmod {n}  $$

or
 * $$ V_{d \cdot 2^r} \equiv 0 \pmod {n}  $$

for some 0 ≤ r < s; see page 1396 of. A strong Lucas pseudoprime is also a Lucas pseudoprime (for the same (P, Q) pair), but the converse is not necessarily true. Therefore, the strong test is a more stringent primality test than equation ($$).

There are infinitely many strong Lucas pseudoprimes, and therefore, infinitely many Lucas pseudoprimes. Theorem 7 in states: Let $$P$$ and $$Q$$ be relatively prime positive integers for which $$P^2 - 4Q$$ is positive but not a square. Then there is a positive constant $$c$$ (depending on $$P$$ and $$Q$$) such that the number of strong Lucas pseudoprimes not exceeding $$x$$ is greater than $$c \cdot \log x$$, for $$x$$ sufficiently large.

We can set Q = −1, then $$U_n$$ and $$V_n$$ are P-Fibonacci sequence and P-Lucas sequence, the pseudoprimes can be called strong Lucas pseudoprime in base P, for example, the least strong Lucas pseudoprime with P = 1, 2, 3, ... are 4181, 169, 119, ...

An extra strong Lucas pseudoprime is a strong Lucas pseudoprime for a set of parameters (P, Q) where Q = 1, satisfying one of the conditions

 * $$ U_d \equiv 0 \pmod {n}  \text{  and  } V_d \equiv \pm 2 \pmod {n}  $$

or
 * $$ V_{d \cdot 2^r} \equiv 0 \pmod {n}  $$

for some $$ 0 \le r<s-1$$. An extra strong Lucas pseudoprime is also a strong Lucas pseudoprime for the same $$(P,Q)$$ pair.

Implementing a Lucas probable prime test
Before embarking on a probable prime test, one usually verifies that n, the number to be tested for primality, is odd, is not a perfect square, and is not divisible by any small prime less than some convenient limit. Perfect squares are easy to detect using Newton's method for square roots.

We choose a Lucas sequence where the Jacobi symbol $$\left(\tfrac{D}{n}\right)=-1$$, so that δ(n) = n + 1.

Given n, one technique for choosing D is to use trial and error to find the first D in the sequence 5, −7, 9, −11, ... such that $$\left(\tfrac{D}{n}\right)=-1$$. Note that $$\left(\tfrac{k}{n}\right)\left(\tfrac{-k}{n}\right)=-1$$. (If D and n have a prime factor in common, then $$\left(\tfrac{D}{n}\right)=0$$). With this sequence of D values, the average number of D values that must be tried before we encounter one whose Jacobi symbol is −1 is about 1.79; see, p. 1416. Once we have D, we set $$P=1$$ and $$Q=(1-D)/4$$. It is a good idea to check that n has no prime factors in common with P or Q. This method of choosing D, P, and Q was suggested by John Selfridge.

(This search will never succeed if n is square, and conversely if it does succeed, that is proof that n is not square. Thus, some time can be saved by delaying testing n for squareness until after the first few search steps have all failed.)

Given D, P, and Q, there are recurrence relations that enable us to quickly compute $$U_{n+1}$$ and $$V_{n+1}$$ in $$O(\log_2 n)$$ steps; see. To start off,
 * $$U_{1}=1$$
 * $$V_{1}=P=1$$

First, we can double the subscript from $$k$$ to $$2k$$ in one step using the recurrence relations
 * $$U_{2k}=U_k\cdot V_k$$
 * $$V_{2k}=V_k^2-2Q^k=\frac{V_k^2+DU_k^2}{2}$$.

Next, we can increase the subscript by 1 using the recurrences
 * $$U_{2k+1}=(P\cdot U_{2k}+V_{2k})/2$$
 * $$V_{2k+1}=(D\cdot U_{2k}+P\cdot V_{2k})/2$$.

If $$P\cdot U_{2k}+V_{2k}$$ is odd, replace it with $$P\cdot U_{2k}+V_{2k}+n$$; this is even so it can now be divided by 2. The numerator of $$V_{2k+1}$$ is handled in the same way. (Adding n does not change the result modulo n.) Observe that, for each term that we compute in the U sequence, we compute the corresponding term in the V sequence. As we proceed, we also compute the same, corresponding powers of Q.

At each stage, we reduce $$U$$, $$V$$, and the power of $$Q$$, mod n.

We use the bits of the binary expansion of n to determine which terms in the U sequence to compute. For example, if n+1 = 44 (= 101100 in binary), then, taking the bits one at a time from left to right, we obtain the sequence of indices to compute: 12 = 1, 102 = 2, 1002 = 4, 1012 = 5, 10102 = 10, 10112 = 11, 101102 = 22, 1011002 = 44. Therefore, we compute U1, U2, U4, U5, U10, U11, U22, and U44. We also compute the same-numbered terms in the V sequence, along with Q1, Q2, Q4, Q5, Q10, Q11, Q22, and Q44.

By the end of the calculation, we will have computed Un+1, Vn+1, and Qn+1, (mod n). We then check congruence ($$) using our known value of Un+1.

When D, P, and Q are chosen as described above, the first 10 Lucas pseudoprimes are (see page 1401 of ): 323, 377, 1159, 1829, 3827, 5459, 5777, 9071, 9179, and 10877

The strong versions of the Lucas test can be implemented in a similar way.

When D, P, and Q are chosen as described above, the first 10 strong Lucas pseudoprimes are: 5459, 5777, 10877, 16109, 18971, 22499, 24569, 25199, 40309, and 58519

To calculate a list of extra strong Lucas pseudoprimes, set $$Q = 1$$. Then try P = 3, 4, 5, 6, ..., until a value of $$D=P^2-4Q$$ is found so that the Jacobi symbol $$\left(\tfrac{D}{n}\right)=-1$$. With this method for selecting D, P, and Q, the first 10 extra strong Lucas pseudoprimes are 989, 3239, 5777, 10877, 27971, 29681, 30739, 31631, 39059, and 72389

Checking additional congruence conditions
If we have checked that congruence ($$) is true, there are additional congruence conditions we can check that have almost no additional computational cost. If n happens to be composite, these additional conditions may help discover that fact.

If n is an odd prime and $$\left(\tfrac{D}{n}\right)=-1$$, then we have the following (see equation 2 on page 1392 of ):

Although this congruence condition is not, by definition, part of the Lucas probable prime test, it is almost free to check this condition because, as noted above, the value of Vn+1 was computed in the process of computing Un+1.

If either congruence ($$) or ($$) is false, this constitutes a proof that n is not prime. If both of these congruences are true, then it is even more likely that n is prime than if we had checked only congruence ($$).

If Selfridge's method (above) for choosing D, P, and Q happened to set Q = −1, then we can adjust P and Q so that D and $$\left(\tfrac{D}{n}\right)$$ remain unchanged and P = Q = 5 (see Lucas sequence-Algebraic relations). If we use this enhanced method for choosing P and Q, then 913 = 11·83 is the only composite less than 108 for which congruence ($$) is true (see page 1409 and Table 6 of; ). More extensive calculations show that, with this method of choosing D, P, and Q, there are only five odd, composite numbers less than 1015 for which congruence ($$) is true.

If $$Q \neq \pm 1 $$, then a further congruence condition that involves very little additional computation can be implemented.

Recall that $$Q^{n+1}$$ is computed during the calculation of $$U_{n+1}$$, and we can easily save the previously computed power of $$Q$$, namely, $$Q^{(n+1)/2}$$.

If n is prime, then, by Euler's criterion,
 * $$ Q^{(n-1)/2} \equiv \left(\tfrac{Q}{n}\right) \pmod {n}  $$.

(Here, $$\left(\tfrac{Q}{n}\right)$$ is the Legendre symbol; if n is prime, this is the same as the Jacobi symbol).

Therefore, if n is prime, we must have,

The Jacobi symbol on the right side is easy to compute, so this congruence is easy to check. If this congruence does not hold, then n cannot be prime. Provided GCD(n, Q) = 1 then testing for congruence ($$) is equivalent to augmenting our Lucas test with a "base Q" Solovay–Strassen primality test.

Additional congruence conditions that must be satisfied if n is prime are described in Section 6 of. If any of these conditions fails to hold, then we have proved that n is not prime.

Comparison with the Miller–Rabin primality test
k applications of the Miller–Rabin primality test declare a composite n to be probably prime with a probability at most (1/4)k.

There is a similar probability estimate for the strong Lucas probable prime test.

Aside from two trivial exceptions (see below), the fraction of (P,Q) pairs (modulo n) that declare a composite n to be probably prime is at most (4/15).

Therefore, k applications of the strong Lucas test would declare a composite n to be probably prime with a probability at most (4/15)k.

There are two trivial exceptions. One is n = 9. The other is when n = p(p+2) is the product of two twin primes. Such an n is easy to factor, because in this case, n+1 = (p+1)2 is a perfect square. One can quickly detect perfect squares using Newton's method for square roots.

By combining a Lucas pseudoprime test with a Fermat primality test, say, to base 2, one can obtain very powerful probabilistic tests for primality, such as the Baillie–PSW primality test.

Fibonacci pseudoprimes
When P = 1 and Q = −1, the Un(P,Q) sequence represents the Fibonacci numbers.

A Fibonacci pseudoprime is often defined as a composite number n not divisible by 5 for which congruence ($$) holds with P = 1 and Q = −1. By this definition, the Fibonacci pseudoprimes form a sequence:
 * 323, 377, 1891, 3827, 4181, 5777, 6601, 6721, 8149, 10877, ....

The references of Anderson and Jacobsen below use this definition.

If n is congruent to 2 or 3 modulo 5, then Bressoud, and Crandall and Pomerance  point out that it is rare for a Fibonacci pseudoprime to also be a Fermat pseudoprime base 2. However, when n is congruent to 1 or 4 modulo 5, the opposite is true, with over 12% of Fibonacci pseudoprimes under 1011 also being base-2 Fermat pseudoprimes.

If n is prime and GCD(n, Q) = 1, then we also have

This leads to an alternative definition of Fibonacci pseudoprime:
 * a Fibonacci pseudoprime is a composite number n for which congruence ($$) holds with P = 1 and Q = −1.

This definition leads the Fibonacci pseudoprimes form a sequence:
 * 705, 2465, 2737, 3745, 4181, 5777, 6721, 10877, 13201, 15251, ... ,

which are also referred to as Bruckman-Lucas pseudoprimes. Hoggatt and Bicknell studied properties of these pseudoprimes in 1974. Singmaster computed these pseudoprimes up to 100000. Jacobsen lists all 111443 of these pseudoprimes less than 1013.

It has been shown that there are no even Fibonacci pseudoprimes as defined by equation (5). However, even Fibonacci pseudoprimes do exist under the first definition given by ($$).

A strong Fibonacci pseudoprime is a composite number n for which congruence ($$) holds for Q = −1 and all P. It follows that an odd composite integer n is a strong Fibonacci pseudoprime if and only if:
 * 1) n is a Carmichael number
 * 2) 2(p + 1) | (n − 1) or 2(p + 1) | (n − p) for every prime p dividing n.

The smallest example of a strong Fibonacci pseudoprime is 443372888629441 = 17·31·41·43·89·97·167·331.

Pell pseudoprimes
A Pell pseudoprime may be defined as a composite number n for which equation ($$) above is true with P = 2 and Q = −1; the sequence Un then being the Pell sequence. The first pseudoprimes are then 35, 169, 385, 779, 899, 961, 1121, 1189, 2419, ...

This differs from the definition in which may be written as:

 * $$ \text{ } U_n \equiv \left(\tfrac{2}{n}\right) \pmod {n}$$

with (P, Q) = (2, −1) again defining Un as the Pell sequence. The first pseudoprimes are then 169, 385, 741, 961, 1121, 2001, 3827, 4879, 5719, 6215 ...

A third definition uses equation (5) with (P, Q) = (2, −1), leading to the pseudoprimes 169, 385, 961, 1105, 1121, 3827, 4901, 6265, 6441, 6601, 7107, 7801, 8119, ...

title: Lucas pseudoprime


text:
User talk:Vogon/Paranoia rewrite

By the way, anyone who happens by this page should feel free to edit it if they've got something about it to improve. Once it reaches a complete enough state, I hope to move it to Paranoia (role-playing game). Vogon 19:35, Oct 10, 2004 (UTC)

I think that less space needs to be given to the history of the setting (Rise of Alpha Complex etc) and more to the things that make Paranoia so unique - the humour, the perversity, the sadism. That said, the history as it stands is very well written, but needs to at least be preceded by some text which explains the tone and feel of the game, which will give the reader a more accurate reading of the history. IMHO, of course.

Straight vs Classic
Isn't the version that is supposed to be darker/more serious "Straight", as opposed to "Classic"?

Also, instead of wiping it out by replacing it with this one, why not make gradual changes to the already existing article? Brutulf 21:05, Jan 22, 2005 (UTC)

The Wobblies
The already existing article has an interesting paragraph on the wobblies secret society. That is quite in the spirit of the game, and thus I think it should be added. Also, from which edition is this society? I can't find it in my book.

Location
I don't think there is any offica/canonical place the Alpha Complex is located. It's all up to the GM.

title: User talk:Vogon/Paranoia rewrite