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Psychology of religion

Psychology of religion consists of the application of psychological methods and interpretive frameworks to the diverse contents of religious traditions as well as to both religious and irreligious individuals. The various methods and frameworks can be summarized according to the classic distinction between the natural-scientific and human-scientific approaches. The first cluster amounts to objective, quantitative, and preferably experimental procedures for testing hypotheses about causal connections among the objects of one's study. In contrast, the human-scientific approach accesses the human world of experience using qualitative, phenomenological, and interpretive methods. This approach aims to discern meaningful, rather than causal, connections among the phenomena one seeks to understand.

Psychologists of religion pursue three major projects: The psychology of religion first arose as a self-conscious discipline in the late 19th century, but all three of these tasks have a history going back many centuries before that.
 * 1) systematic description, especially of religious contents, attitudes, experiences, and expressions
 * 2) explanation of the origins of religion, both in the history of the human race and in individual lives, taking into account a diversity of influences
 * 3) mapping out the consequences of religious attitudes and conduct, both for the individual and for society at large.

Overview
The challenge for the psychology of religion is essentially threefold:
 * 1) to provide a thoroughgoing description of the objects of investigation, whether they be shared religious content (e.g., a tradition's ritual observances) or individual experiences, attitudes, or conduct
 * 2) to account in psychological terms for the rise of such phenomena, whether they be in individual lives
 * 3) to clarify the outcomes – the fruits, as William James put it – of these phenomena, for individuals, and the larger society. These fruits may be both positive and negative.

The first, descriptive task naturally requires a clarification of one's terms – above all, the word religion. Historians of religion have long underscored the problematic character of this term. They note that its usage over the centuries has changed in significant ways, generally in the direction of reification. The early psychologists of religion were fully aware of these difficulties, typically acknowledging that the definitions they chose were to some degree arbitrary. With the rise of positivistic trends in psychology over the 20th century, especially the demand that all phenomena be operationalized by quantitative procedures, psychologists of religion developed a multitude of scales, most of them developed for use by Protestant Christians. Factor analysis was also brought into play by both psychologists and sociologists of religion, to establish a fixed core of dimensions and a corresponding set of scales. The justification and adequacy of these efforts, especially in the light of constructivist and other postmodern viewpoints, remains a matter of debate.

In the last several decades, especially among clinical psychologists, a preference for the terms "spirituality" and "spiritual" has emerged, along with efforts to distinguish them from "religion" and "religious." Especially in the United States, "religion" has for many become associated with sectarian institutions and their obligatory creeds and rituals, thus giving the word a negative cast; "spirituality," in contrast, is positively constructed as deeply individual and subjective, as a universal capacity to apprehend and accord one's life with higher realities. In fact, "spirituality" has likewise undergone an evolution in the West, from a time when it was essentially a synonym for religion in its original, subjective meaning. Today, efforts are ongoing to "operationalize" these terms, with little regard for their history in their Western context, and with the apparent realist assumption that underlying them are fixed qualities identifiable using empirical procedures.

Schnitker and Emmons theorized that the understanding of religion as a search for meaning makes implications in the three psychological areas of motivation, cognition and social relationships. The cognitive aspects relate to God and a sense of purpose, the motivational ones to the need to control, and the religious search for meaning is also woven into social communities.

Edwin Diller Starbuck
Edwin Diller Starbuck is considered a pioneer of the psychology of religion and his book Psychology of Religion (1899) has been described as the first book in the genre. This book had the endorsement of William James who wrote a preface to it. Starbuck's work would influence James' own book The Varieties of Religious Experience, with James thanking him in the preface for having "made over to me his large collection of manuscript material". In the book itself James mentions Starbuck's name 46 times and cites him on dozens of occasions.

William James
American psychologist and philosopher William James (1842–1910) is regarded by most psychologists of religion as the founder of the field. He served as president of the American Psychological Association, and wrote one of the first psychology textbooks. In the psychology of religion, James' influence endures. His Varieties of Religious Experience is considered to be the classic work in the field, and references to James' ideas are common at professional conferences.

James distinguished between institutional religion and personal religion. Institutional religion refers to the religious group or organization and plays an important part in a society's culture. Personal religion, in which the individual has mystical experience, can be experienced regardless of the culture. James was most interested in understanding personal religious experience.

In studying personal religious experiences, James made a distinction between healthy-minded and sick-souled religiousness. Individuals predisposed to healthy-mindedness tend to ignore the evil in the world and focus on the positive and the good. James used examples of Walt Whitman and the "mind-cure" religious movement to illustrate healthy-mindedness in The Varieties of Religious Experience. In contrast, individuals predisposed to having a sick-souled religion are unable to ignore evil and suffering and need a unifying experience, religious or otherwise, to reconcile good and evil. James included quotations from Leo Tolstoy and John Bunyan to illustrate the sick soul.

William James' hypothesis of pragmatism stems from the efficacy of religion. If an individual believes in and performs religious activities, and those actions happen to work, then that practice appears the proper choice for the individual. However, if the processes of religion have little efficacy, then there is no rationality for continuing the practice.

G.W.F. Hegel
Georg Wilhelm Friedrich Hegel (1770–1831) described all systems of religion, philosophy, and social science as expressions of the basic urge of consciousness to learn about itself and its surroundings, and record its findings and hypotheses. Thus, religion is only a form of that search for knowledge, within which humans record various experiences and reflections. Others, compiling and categorizing these writings in various ways, form the consolidated worldview as articulated by that religion, philosophy, social science, etc. His work The Phenomenology of Spirit was a study of how various types of writing and thinking draw from and re-combine with the individual and group experiences of various places and times, influencing the current forms of knowledge and worldviews that are operative in a population. This activity is the functioning of an incomplete group mind, where each is accessing the recorded wisdom of others. His works often include detailed descriptions of the psychological motivations involved in thought and behavior, e.g., the struggle of a community or nation to know itself and thus correctly govern itself. In Hegel's system, Religion is one of the major repositories of wisdom to be used in these struggles, representing a huge body of recollections from humanity's past in various stages of its development.

Sigmund Freud
Sigmund Freud (1856–1939) gave explanations of the genesis of religion in his various writings. In Totem and Taboo, he applied the idea of the Oedipus complex (involving unresolved sexual feelings of, for example, a son toward his mother and hostility toward his father) and postulated its emergence in the primordial stage of human development.

In Moses and Monotheism, Freud reconstructed biblical history by his general theory. His ideas were also developed in The Future of an Illusion. When Freud spoke of religion as an illusion, he maintained that it "is a fantasy structure from which a man must be set free if he is to grow to maturity."

Freud views the idea of God as being a version of the father image, and religious belief as at bottom infantile and neurotic. Authoritarian religion, Freud believed, is dysfunctional and alienates man from himself.

Carl Jung
The Swiss psychoanalyst Carl Jung (1875–1961) adopted a very different posture, one that was more sympathetic to religion and more concerned with a positive appreciation of religious symbolism. Jung considered the question of the metaphysical existence of God to be unanswerable by the psychologist and adopted a kind of agnosticism.

Jung postulated, in addition to the personal unconscious (roughly adopting Freud's concept), the collective unconscious, which is the repository of human experience and which contains "archetypes" (i.e. basic images that are universal in that they recur regardless of culture). The irruption of these images from the unconscious into the realm of consciousness he viewed as the basis of religious experience and often of artistic creativity. Some of Jung's writings have been devoted to elucidating some of the archetypal symbols, and include his work in comparative mythology.

Alfred Adler
Austrian psychiatrist Alfred Adler (1870–1937), who parted ways with Freud, emphasized the role of goals and motivation in his Individual Psychology. One of Adler's most famous ideas is that we try to compensate for inferiorities that we perceive in ourselves. A lack of power often lies at the root of feelings of inferiority. One way that religion enters into this picture is through our beliefs in God, which are characteristic of our tendency to strive for perfection and superiority. For example, in many religions, God is considered to be perfect and omnipotent, and commands people likewise to be perfect. If we, too, achieve perfection, we become one with God. By identifying with God in this way, we compensate for our imperfections and feelings of inferiority.

Our ideas about God are important indicators of how we view the world. According to Adler, these ideas have changed over time, as our vision of the world – and our place in it – has changed. Consider this example that Adler offers: the traditional belief that people were placed deliberately on earth as God's ultimate creation is being replaced with the idea that people have evolved by natural selection. This coincides with a view of God not as a real being, but as an abstract representation of nature's forces. In this way, our view of God has changed from one that was concrete and specific to one that is more general. From Adler's vantage point, this is a relatively ineffective perception of God because it is so general that it fails to convey a strong sense of direction and purpose.

An important thing for Adler is that God (or the idea of God) motivates people to act and that those actions do have real consequences for ourselves and others. Our view of God is important because it embodies our goals and directs our social interactions.

Compared to science, another social movement, religion is more efficient because it motivates people more effectively. According to Adler, only when science begins to capture the same religious fervor, and promotes the welfare of all segments of society, will the two be more equal in peoples' eyes.

Gordon Allport
In his 1950 book The Individual and His Religion, Gordon Allport (1897–1967) illustrates how people may use religion in different ways. He makes a distinction between Mature religion and Immature religion. Mature religious sentiment is how Allport characterized the person whose approach to religion is dynamic, open-minded, and able to maintain links between inconsistencies. In contrast, immature religion is self-serving and generally represents the negative stereotypes that people have about religion.

More recently, this distinction has been encapsulated in the terms "intrinsic religion", referring to a genuine, heartfelt devout faith, and "extrinsic religion", referring to a more utilitarian use of religion as a means to an end, such as church attendance to gain social status. These dimensions of religion were measured on the Religious Orientation Scale of Allport and Ross. The third form of religious orientation has been described by Daniel Batson. This refers to treatment of religion as an open-ended search.

More specifically, it has been seen by Batson as comprising a willingness to view religious doubts positively, acceptance that religious orientation can change and existential complexity, the belief that one's religious beliefs should be shaped from personal crises that one has experienced in one's life. Batson refers to extrinsic, intrinsic and quests respectively as religion-as-means, religion-as-end, and religion-as-quest, and measures these constructs on the Religious Life Inventory.

Erik H. Erikson
Erik Erikson (1902–1994) is best known for his theory of psychological development, which has its roots in the psychoanalytic importance of identity in personality. His biographies of Gandhi and Martin Luther reveal Erikson's positive view of religion. He considered religions to be important influences in successful personality development because they are the primary way that cultures promote the virtues associated with each stage of life. Religious rituals facilitate this development. Erikson's theory has not benefited from systematic empirical study, but it remains an influential and well-regarded theory in the psychological study of religion.

Erich Fromm
The American scholar Erich Fromm (1900–1980) modified the Freudian theory and produced a more complex account of the functions of religion. In his book Psychoanalysis and Religion he responded to Freud's theories by explaining that part of the modification is viewing the Oedipus complex as based not so much on sexuality as on a "much more profound desire", namely, the childish desire to remain attached to protecting figures. The right religion, in Fromm's estimation, can, in principle, foster an individual's highest potentialities, but religion in practice tends to relapse into being neurotic.

According to Fromm, humans need a stable frame of reference. Religion fills this need. In effect, humans crave answers to questions that no other source of knowledge has an answer to, which only religion may seem to answer. However, a sense of free will must be given for religion to appear healthy. An authoritarian notion of religion appears detrimental.

Rudolf Otto
Rudolf Otto (1869–1937) was a German Protestant theologian and scholar of comparative religion. Otto's most famous work, The Idea of the Holy (published first in 1917 as Das Heilige), defines the concept of the holy as that which is numinous. Otto explained the numinous as a "non-rational, non-sensory experience or feeling whose primary and immediate object is outside the self." It is a mystery (Latin: mysterium tremendum) that is both fascinating (fascinans) and terrifying at the same time; A mystery that causes trembling and fascination, attempting to explain that inexpressible and perhaps supernatural emotional reaction of wonder drawing us to seemingly ordinary and/or religious experiences of grace. This sense of emotional wonder appears evident at the root of all religious experiences. Through this emotional wonder, we suspend our rational mind for non-rational possibilities.

The Idea of the Holy also set out a paradigm for the study of religion that focuses on the need to realize the religious as a non-reducible, original category in its own right. This paradigm was under much attack between approximately 1950 and 1990 but has made a strong comeback since then.

Modern thinkers
Autobiographal accounts of 20th-century psychology of religion as a field have been supplied by numerous modern psychologists of religion, primarily based in Europe, but also by several US-based psychologists such as Ralph W. Hood and Donald Capps.

Allen Bergin
Allen Bergin is noted for his 1980 paper "Psychotherapy and Religious Values," which is known as a landmark in scholarly acceptance that religious values do, in practice, influence psychotherapy. He received the Distinguished Professional Contributions to Knowledge award from the American Psychological Association in 1989 and was cited as challenging "psychological orthodoxy to emphasize the importance of values and religion in therapy."

Robert A. Emmons
Robert A. Emmons offered a theory of "spiritual strivings" in his 1999 book, The Psychology of Ultimate Concerns. With support from empirical studies, Emmons argued that spiritual strivings foster personality integration because they exist at a higher level of the personality.

Ralph W. Hood Jr.
Ralph W. Hood Jr. is a professor of psychology at the University of Tennessee at Chattanooga. He is a former editor of the Journal for the Scientific Study of Religion and a former co-editor of the Archive for the Psychology of Religion and The International Journal for the Psychology of Religion. He is Past President of Division 36 of the American Psychological Association and a recipient of its William James Award. He has published several hundred articles and book chapters on the psychology of religion and has authored, co-authored, or edited thirteen volumes, all dealing with the psychology of religion.

Kenneth Pargament
Kenneth Pargament is noted for his book Psychology of Religion and Coping (1997), as well as for a 2007 book on religion and psychotherapy, and a sustained research program on religious coping. He is professor of psychology at Bowling Green State University (Ohio, US), and has published more than 100 papers on the subject of religion and spirituality in psychology. Pargament led the design of a questionnaire called the "RCOPE" to measure Religious Coping strategies. Pargament has distinguished between three types of styles for coping with stress:
 * 1) Collaborative, in which people co-operate with God to deal with stressful events;
 * 2) Deferring, in which people leave everything to God; and
 * 3) Self-directed, in which people do not rely on God and try exclusively to solve problems by their own efforts. He also describes four major stances toward religion that have been adopted by psychotherapists in their work with clients, which he calls the religiously rejectionist, exclusivist, constructivist, and pluralist stances.

James Hillman
James Hillman, at the end of his book Re-Visioning Psychology, reverses James' position of viewing religion through psychology, urging instead that we view psychology as a variety of religious experience. He concludes: "Psychology as religion implies imagining all psychological events as effects of Gods in the soul. "

Julian Jaynes
Julian Jaynes, primarily in his book The Origin of Consciousness in the Breakdown of the Bicameral Mind, proposed that religion (and some other psychological phenomena such as hypnosis and schizophrenia) is a remnant of a relatively recent time in human development, prior to the advent of consciousness. Jaynes hypothesized that hallucinated verbal commands helped non-conscious early man to perform tasks promoting human survival. Starting about 10,000 BCE, selective pressures favored the hallucinated verbal commands for social control, and they came to be perceived as an external, rather than internal, voice commanding the person to take some action. These were hence often explained as originating from invisible gods, spirits, and ancestors.

Hypotheses on the role of religion
There are three primary hypotheses on the role of religion in the modern world.

Secularization
The first hypothesis, secularization, holds that science and technology will take the place of religion. Secularization supports the separation of religion from politics, ethics, and psychology. Taking this position even further, Taylor explains that secularization denies transcendence, divinity, and rationality in religious beliefs.

Religious transformation
Challenges to the secularization hypothesis led to significant revisions, resulting in the religious transformation hypothesis. This perspective holds that general trends towards individualism and social disintegration will produce changes in religion, making religious practice more individualized and spiritually focused. This in turn is expected to produce more spiritual seeking, although not exclusive to religious institutions. Eclecticism, which draws from multiple religious/spiritual systems and New Age movements are also predicted to result.

Cultural divide
In response to the religious transformation hypothesis, Ronald Inglehart piloted the renewal of the secularization hypothesis. His argument hinges on the premise that religion develops to fill the human need for security. Therefore, the development of social and economic security in Europe explains its corresponding secularization due to a lack of need for religion. However, religion continues in the third world where social and economic insecurity is rampant. The overall effect is expected to be a growing cultural disparity.

The idea that religiosity arises from the human need for security has also been furthered by studies examining religious beliefs as a compensatory mechanism of control. These studies are motivated by the idea that people are invested in maintaining beliefs in order and structure to prevent beliefs in chaos and randomness.

In the experimental setting, researchers have also tested compensatory control in regard to individuals' perceptions of external systems, such as religion or government. For example, Kay and colleagues found that in a laboratory setting, individuals are more likely to endorse broad external systems (e.g., religion or sociopolitical systems) that impose order and control on their lives when they are induced with lowered levels of personal control. In this study, researchers suggest that when a person's personal control is lessened, their motivation to believe in order is threatened, resulting in compensation of this threat through adherence to other external sources of control.

Psychometric approaches to religion
Since the 1960s psychologists of religion have used the methodology of psychometrics to assess ways in which a person may be religious. An example is the Religious Orientation Scale of Allport and Ross, which measures how respondents stand on intrinsic and extrinsic religion as described by Allport. More recent questionnaires include the Age-Universal I-E Scale of Gorsuch and Venable, the Religious Life Inventory of Batson, Schoenrade and Ventis, and the Spiritual Experiences Index-Revised of Genia. The first provides an age-independent measure of Allport and Ross's two religious orientations. The second measures three forms of religious orientation: religion as means (intrinsic), religion as end (extrinsic), and religion as quest. The third assesses spiritual maturity using two factors: Spiritual Support and Spiritual Openness.

Religious orientations and religious dimensions
Some questionnaires, such as the Religious Orientation Scale, relate to different religious orientations, such as intrinsic and extrinsic religiousness, referring to different motivations for religious allegiance. A rather different approach, taken, for example, by Glock and Stark (1965), has been to list different dimensions of religion rather than different religious orientations, which relates to how an individual may manifest different forms of being religious. Glock and Stark's typology described five dimensions of religion – the doctrinal, the intellectual, the ethical-consequential, the ritual, and the experiential. In later works, these authors subdivided the ritual dimension into devotional and public ritual, and also clarified that their distinction of religion along multiple dimensions was not identical to distinguishing religious orientations. Although some psychologists of religion have found it helpful to take a multidimensional approach to religion for the purpose of psychometric scale design, there has been, as Wulff explains, considerable controversy about whether religion should really be seen as multidimensional.

Questionnaires to assess religious experience
What we call religious experiences can differ greatly. Some reports exist of supernatural happenings that it would be difficult to explain from a rational, scientific point of view. On the other hand, there also exist the sort of testimonies that simply seem to convey a feeling of peace or oneness – something which most of us, religious or not, may possibly relate to. In categorizing religious experiences it is perhaps helpful to look at them as explicable through one of two theories: the objectivist thesis or the subjectivist thesis.

An objectivist would argue that the religious experience is a proof of God's existence. However, others have criticised the reliability of religious experiences. The English philosopher Thomas Hobbes asked how it was possible to tell the difference between talking to God in a dream, and dreaming about talking to God.

The Subjectivist view argues that it is not necessary to think of religious experiences as evidence for the existence of an actual being whom we call God. From this point of view, the important thing is the experience itself and the effect that it has on the individual.

Developmental approaches to religion
Many have looked at stage models, like those of Jean Piaget and Lawrence Kohlberg, to explain how children develop ideas about God and religion in general.

James Fowler's model
The best-known stage model of spiritual or religious development is that of James W. Fowler, a developmental psychologist at the Candler School of Theology, in his Stages of Faith. He follows Piaget and Kohlberg and has proposed a holistic staged development of faith (or spiritual development) across the lifespan. These stages of faith development were along the lines of Piaget's theory of cognitive development and Kohlberg's stages of moral development.

The book-length study contains six stages of faith development proposed by James Fowler:
 * Stage 0 – "Primal or Undifferentiated" faith (birth to 2 years), is characterized by an early learning of the safety of their environment (i.e. warm, safe and secure vs. hurt, neglect and abuse). If consistent nurture is experienced, one will develop a sense of trust and safety about the universe and the divine. Conversely, negative experiences will cause one to develop distrust about the universe and the divine. Transition to the next stage begins with integration of thought and language which facilitates the use of symbols in speech and play.
 * Stage 1 – "Intuitive-Projective" faith (ages of three to seven), is characterized by the psyche's unprotected exposure to the Unconscious, and marked by a relative fluidity of thought patterns. Religion is learned mainly through experiences, stories, images, and the people that one comes in contact with.
 * Stage 2 – "Mythic-Literal" faith (mostly in school children), is characterized by persons have a strong belief in the justice and reciprocity of the universe, and their deities are almost always anthropomorphic. During this time metaphors and symbolic language are often misunderstood and are taken literally.
 * Stage 3 – "Synthetic-Conventional" faith (arising in adolescence; aged 12 to adulthood), is characterized by conformity to authority and the religious development of a personal identity. Any conflicts with one's beliefs are ignored at this stage due to the fear of threat from inconsistencies.
 * Stage 4 – "Individuative-Reflective" faith (usually mid-twenties to late thirties), is a stage of angst and struggle. The individual takes personal responsibility for his or her beliefs and feelings. As one is able to reflect on one's own beliefs, there is an openness to a new complexity of faith, but this also increases the awareness of conflicts in one's belief.
 * Stage 5 – "Conjunctive" faith (mid-life crisis), acknowledges paradox and transcendence relating reality behind the symbols of inherited systems. The individual resolves conflicts from previous stages by a complex understanding of a multidimensional, interdependent "truth" that cannot be explained by any particular statement.
 * Stage 6 – "Universalizing" faith: The individual would treat any person with compassion as he or she views people as from a universal community, and should be treated with universal principles of love and justice.

Fowler's model has inspired a considerable body of empirical research into faith development, although little of such research was ever conducted by Fowler himself. Gary Leak's Faith Development Scale, or FDS, has been subject to factor analysis by Leak.

Other hypotheses
Other theorists in developmental psychology have suggested that religiosity comes naturally to young children. Specifically, children may have a natural-born conception of mind-body dualism, which lends itself to beliefs that the mind may live on after the body dies. In addition, children have a tendency to see agency and human design where there is not, and prefer a creationist explanation of the world even when raised by parents who do not.

Researchers have also investigated attachment system dynamics as a predictor of the religious conversion experience throughout childhood and adolescence. One hypothesis is the correspondence hypothesis, which posits that individuals with secure parental attachment are more likely to experience a gradual conversion experience. Under the correspondence hypothesis, internal working models of a person's attachment figure is thought to perpetuate his or her perception of God as a secure base. Another hypothesis relating attachment style to the conversion experience is the compensation hypothesis, which states that individuals with insecure attachments are more likely to have a sudden conversion experience as they compensate for their insecure attachment relationship by seeking a relationship with God. Researchers have tested these hypotheses using longitudinal studies and individuals' self narratives of their conversion experience. For example, one study investigating attachment styles and adolescent conversions at Young Life religious summer camps resulted in evidence supporting the correspondence hypothesis through analysis of personal narratives and a prospective longitudinal follow-up of Young Life campers, with mixed results for the compensation hypothesis.

James Alcock summarizes a number of components of what he calls the "God engine," a "number of automatic processes and cognitive biases [that] combine to make supernatural belief the automatic default." These include magical thinking, agency detection, theory of mind that leads to dualism, the notion that "objects and events [serve] an intentional purpose," etc.

Evolutionary and cognitive psychology of religion
Evolutionary psychology is based on the hypothesis that, just like the cardiac, pulmonary, urinary, and immune systems, cognition has a functional structure with a genetic basis, and therefore appeared through natural selection. Like other organs and tissues, this functional structure should be universally shared among humans and should solve important problems of survival and reproduction. Evolutionary psychologists seek to understand cognitive processes by understanding the survival and reproductive functions they might serve.

Pascal Boyer is one of the leading figures in the cognitive psychology of religion, a new field of inquiry that is less than fifteen years old, which accounts for the psychological processes that underlie religious thought and practice. In his book Religion Explained, Boyer shows that there is no simple explanation for religious consciousness. Boyer is mainly concerned with explaining the various psychological processes involved in the acquisition and transmission of ideas concerning the gods. Boyer builds on the ideas of cognitive anthropologists Dan Sperber and Scott Atran, who first argued that religious cognition represents a by-product of various evolutionary adaptations, including folk psychology, and purposeful violations of innate expectations about how the world is constructed (for example, bodiless beings with thoughts and emotions) that make religious cognitions striking and memorable.

Religious persons acquire religious ideas and practices through social exposure. The child of a Zen Buddhist will not become an evangelical Christian or a Zulu warrior without the relevant cultural experience. While mere exposure does not cause a particular religious outlook (a person may have been raised a Roman Catholic but leave the church), nevertheless some exposure seems required – this person will never invent Roman Catholicism out of thin air. Boyer says cognitive science can help us to understand the psychological mechanisms that account for these manifest correlations and in so doing enable us to better understand the nature of religious belief and practice.

Boyer moves outside the leading currents in mainstream cognitive psychology and suggests that we can use evolutionary biology to unravel the relevant mental architecture. Our brains are, after all, biological objects, and the best naturalistic account of their development in nature is Darwin's theory of evolution. To the extent that mental architecture exhibits intricate processes and structures, it is plausible to think that this is the result of evolutionary processes working over vast periods of time. Like all biological systems, the mind is optimised to promote survival and reproduction in the evolutionary environment. On this view all specialised cognitive functions broadly serve those reproductive ends.

For Steven Pinker the universal propensity toward religious belief is a genuine scientific puzzle. He thinks that adaptationist explanations for religion do not meet the criteria for adaptations. An alternative explanation is that religious psychology is a by-product of many parts of the mind that originally evolved for other purposes.

Religion and prayer
Religious practice often manifests itself in some form of prayer. Recent studies have focused specifically on the effects of prayer on health. Measures of prayer and the above measures of spirituality evaluate different characteristics and should not be considered synonymous.

Prayer is fairly prevalent in the United States. About 55% of Americans report praying daily. However, the practice of prayer is more prevalent and practiced more consistently among Americans who perform other religious practices. There are four primary types of prayer in the West. Poloma and Pendleton, utilized factor analysis to delineate these four types of prayer: meditative (more spiritual, silent thinking), ritualistic (reciting), petitionary (making requests to God), and colloquial (general conversing with God). Further scientific study of prayer using factor analysis has revealed three dimensions of prayer. Ladd and Spilka's first factor was awareness of self, inward reaching. Their second and third factors were upward reaching (toward God) and outward reaching (toward others). This study appears to support the contemporary model of prayer as connection (whether to the self, higher being, or others).

Dein and Littlewood (2008) suggest that an individual's prayer life can be viewed on a spectrum ranging from immature to mature. A progression on the scale is characterized by a change in the perspective of the purpose of prayer. Rather than using prayer as a means of changing the reality of a situation, a more mature individual will use prayer to request assistance in coping with immutable problems and draw closer to God or others. This change in perspective has been shown to be associated with an individual's passage through adolescence.

Prayer appears to have health implications. Empirical studies suggest that mindfully reading and reciting the Psalms (from scripture) can help a person calm down and focus. Prayer is also positively correlated with happiness and religious satisfaction. A study conducted by Francis, Robbins, Lewis, and Barnes investigated the relationship between self-reported prayer frequency and measures of psychoticism and neuroticism according to the abbreviated form of the Revised Eysenck Personality Questionnaire (EPQR-A). The study included a sample size of 2306 students attending Protestant and Catholic schools in the highly religious culture of Northern Ireland. The data shows a negative correlation between prayer frequency and psychoticism. The data also shows that, in Catholic students, frequent prayer has a positive correlation to neuroticism scores. Ladd and McIntosh suggest that prayer-related behaviors, such as bowing the head and clasping the hands together in an almost fetal position, are suggestive of "social touch" actions. Prayer in this manner may prepare an individual to carry out positive pro-social behavior after praying, due to factors such as increased blood flow to the head and nasal breathing. Overall, slight health benefits have been found fairly consistently across studies.

Three main pathways to explain this trend have been offered: placebo effect, focus and attitude adjustment, and activation of healing processes. These offerings have been expanded by Breslin and Lewis (2008) who have constructed a five pathway model between prayer and health with the following mediators: physiological, psychological, placebo, social support, and spiritual. The spiritual mediator is a departure from the rest in that its potential for empirical investigation is not currently feasible. Although the conceptualizations of chi, the universal mind, divine intervention, and the like breach the boundaries of scientific observation, they are included in this model as possible links between prayer and health so as to not unnecessarily exclude the supernatural from the broader conversation of psychology and religion.

Religion and ritual
Another significant form of religious practice is ritual. Religious rituals encompass a wide array of practices, but can be defined as the performance of similar actions and vocal expressions based on prescribed tradition and cultural norms.

Scheff suggests that ritual provides catharsis, emotional purging, through distancing. This emotional distancing enables an individual to experience feelings with an amount of separation, and thus with less intensity. However, the conception of religious ritual as an interactive process has since matured and become more scientifically established. From this view, ritual offers a means to catharsis through behaviors that foster connection with others, allowing for emotional expression. This focus on connection contrasts to the separation that seems to underlie Scheff's view.

Additional research suggests a social component of ritual. For instance, findings suggest that ritual performance indicates group commitment and prevents the uncommitted from gaining membership benefits. Ritual may aid in emphasizing moral values that serve as group norms and regulate societies. It may also strengthen commitment to moral convictions and the likelihood of upholding these social expectations. Thus, performance of rituals may foster social-group stability.

Robert Sapolsky sees a similarity between the rituals accompanying obsessive–compulsive disorder and religious rituals. According to him, religious ritual reduces the tension and anxiety associated with the disorder and provides relief resulting from practicing in a social community.

Religion and health
There is considerable literature on the relationship between religion and health. More than 3000 empirical studies have examined relationships between religion and health, including more than 1200 in the 20th century, and more than 2000 additional studies between 2000 and 2009.

Psychologists consider that religion may benefit both physical and mental health in various ways, including encouraging healthy lifestyles, providing social support networks and encouraging an optimistic outlook on life; prayer and meditation may also benefit physiological functioning. Nevertheless, religion is not a unique source of health and well-being, and there are benefits to nonreligiosity as well. Haber, Jacob and Spangler have considered how different dimensions of religiosity may relate to health benefits in different ways.

Religion and personality
Some studies have examined whether there is a "religious personality." Research on the five factor model of personality suggests that people who identify as religious are more likely to be agreeable and conscientious. Similarly, people who identify as spiritual are more likely to be extrovert and open, although this varies based on the type of spirituality endorsed. For example, people endorsing fundamentalist religious beliefs are more likely to measure low on the Openness factor.

Religion and prejudice
To investigate the salience of religious beliefs in establishing group identity, researchers have also conducted studies looking at religion and prejudice. Some studies have shown that greater religious attitudes may be significant predictors of negative attitudes towards racial or social outgroups. These effects are often conceptualized under the framework of intergroup bias, where religious individuals favor members of their ingroup (ingroup favoritism) and exhibit disfavor towards members of their outgroup (outgroup derogation). Evidence supporting religious intergroup bias has been supported in multiple religious groups, including non-Christian groups, and is thought to reflect the role of group dynamics in religious identification. Many studies regarding religion and prejudice implement religious priming both in the laboratory and in naturalistic settings with evidence supporting the perpetuation of ingroup favoritism and outgroup derogation in individuals who are high in religiosity.

Recently, reparative or conversion therapy, a religiously motivated process intended to change an individual's sexuality, has been the subject of scrutiny and has been condemned by some governments, LGBT charities, and therapy/counselling professional bodies.

Religion and drugs
The American psychologist James H. Leuba (1868–1946), in A Psychological Study of Religion, accounts for mystical experience psychologically and physiologically, pointing to analogies with certain drug-induced experiences. Leuba argued forcibly for a naturalistic treatment of religion, which he considered to be necessary if religious psychology were to be looked at scientifically. Shamans all over the world and in different cultures have traditionally used drugs, especially psychedelics, for their religious experiences. In these communities the absorption of drugs leads to dreams (visions) through sensory distortion. The psychedelic experience is often compared to non-ordinary forms of consciousness such as those experienced in meditation, and mystical experiences. Ego dissolution is often described as a key feature of the psychedelic experience.

William James was also interested in mystical experiences from a drug-induced perspective, leading him to make some experiments with nitrous oxide and even peyote. He concludes that while the revelations of the mystic hold true, they hold true only for the mystic; for others they are certainly ideas to be considered, but hold no claim to truth without personal experience of such.

Religion and mental illness
Although many researchers have brought evidence for a positive role that religion plays in health, others have shown that religious beliefs, practices, and experiences may be linked to mental illnesses of various kinds (mood disorders, personality disorders, and psychiatric disorders). In 2012 a team of psychiatrists, behavioral psychologists, neurologists, and neuropsychiatrists from the Harvard Medical School published research which suggested the development of a new diagnostic category of psychiatric disorders related to religious delusion and hyperreligiosity.

They compared the thoughts and behaviors of the most important figures in the Bible (Abraham, Moses, Jesus Christ, and Paul) with patients affected by mental disorders related to the psychotic spectrum using different clusters of disorders and diagnostic criteria (DSM-IV-TR), and concluded that these Biblical figures "may have had psychotic symptoms that contributed inspiration for their revelations", such as schizophrenia, schizoaffective disorder, manic depression, delusional disorder, delusions of grandeur, auditory-visual hallucinations, paranoia, Geschwind syndrome (Paul especially), and abnormal experiences associated with temporal lobe epilepsy (TLE). The authors suggest that Jesus sought to condemn himself to death ("suicide by proxy").

The research went further and also focused on social models of psychopathology, analyzing new religious movements and charismatic cult leaders such as David Koresh, leader of the Branch Davidians, and Marshall Applewhite, founder of the Heaven's Gate cult. The researchers concluded that "If David Koresh and Marshall Applewhite are appreciated as having psychotic-spectrum beliefs, then the premise becomes untenable that the diagnosis of psychosis must rigidly rely upon an inability to maintain a social group. A subset of individuals with psychotic symptoms appears able to form intense social bonds and communities despite having an extremely distorted view of reality. The existence of a better socially functioning subset of individuals with psychotic-type symptoms is corroborated by research indicating that psychotic-like experiences, including both bizarre and non-bizarre delusion-like beliefs, are frequently found in the general population. This supports the idea that psychotic symptoms likely lie on a continuum."

Religion and psychotherapy
Clients' religious beliefs are increasingly being considered in psychotherapy with the goal of improving service and effectiveness of treatment. A resulting development was theistic psychotherapy. Conceptually, it consists of theological principles, a theistic view of personality, and a theistic view of psychotherapy. Following an explicit minimizing strategy, therapists attempt to minimize conflict by acknowledging their religious views while being respectful of client's religious views. This is argued to up the potential for therapists to directly utilize religious practices and principles in therapy, such as prayer, forgiveness, and grace. In contrast to such an approach, psychoanalyst Robin S. Brown argues for the extent to which our spiritual commitments remain unconscious. Drawing from the work of Jung, Brown suggests that "our biases can only be suspended in the extent to which they are no longer our biases".

Pastoral psychology
One application of the psychology of religion is in pastoral psychology, the use of psychological findings to improve the pastoral care provided by pastors and other clergy, especially in how they support ordinary members of their congregations. Pastoral psychology is also concerned with improving the practice of chaplains in healthcare and in the military. One major concern of pastoral psychology is to improve the practice of pastoral counseling. Pastoral psychology is a topic of interest for professional journals such as the Journal of Psychology and Christianity and the Journal of Psychology and Theology. In 1984, Thomas Oden severely criticized mid-20th-century pastoral care and the pastoral psychology that guided it as having entirely abandoned its classical/traditional sources, and having become overwhelmingly dominated by modern psychological influences from Freud, Rogers, and others. More recently, others have described pastoral psychology as a field that experiences a tension between psychology and theology.

title: Psychology of religion


text:
Baron Sanderson of Ayot

Baron Sanderson of Ayot, of Welwyn in the County of Hertford, is a title in the Peerage of the United Kingdom. It was created in 1960 for the businessman and public servant, Basil Sanderson (1894–1971). He was the son of the shipping magnate Harold Arthur Sanderson, general manager of the White Star Line, and himself was chairman and president of the Shipping Federation. On Lord Sanderson of Ayot's death in 1971, he was succeeded by his elder twin son, Alan Lindsay Sanderson (1931–2022), a psychiatrist (M.R.C.Psych.), who disclaimed the peerage for life the same year. , the title is held by the 2nd Baron's eldest son, Michael Sanderson, 3rd Baron Sanderson of Ayot.

Barons Sanderson of Ayot (1960)

 * Basil Sanderson, 1st Baron Sanderson of Ayot (1894–1971)
 * Alan Lindsay Sanderson, 2nd Baron Sanderson of Ayot (1931–2022) (disclaimed 1971)
 * Michael Sanderson, 3rd Baron Sanderson of Ayot (b. 1959)

The heir presumptive is the present holder's first cousin, Basil Sanderson (b. 1974), who is the son of the Hon. Murray Lee Sanderson (1931–2017), second son of the 1st Baron.

Line of succession

 * Coronet of a British Baron.svg Basil Sanderson, 1st Baron Sanderson of Ayot (1894–1971)
 * Alan Lindsay Sanderson, 2nd Baron Sanderson of Ayot (1931–2022) (title disclaimed 1971)
 * Coronet of a British Baron.svg Michael Sanderson, 3rd Baron Sanderson of Ayot (b. 1959)
 * Hon. Murray Lee Sanderson (1931–2017)
 * (1) Basil Sanderson (b. 1974)

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

title: Baron Sanderson of Ayot


text:
Talk:Pancreas

More common names for diabetes
Why ever does this article refer to "Type One diabetes" (a widely used term), but then go on to refer to "secondary diabetes" (which is not a widely used term at all)? Surely the term which should have been used here was the more commonly used term of "Type Two diabetes"? ACEOREVIVED (talk) 22:14, 12 November 2012 (UTC)

Hear, hear! John Mark Wagnon (talk) 01:26, 27 June 2021 (UTC)

Proposed merge with Exocrine component of pancreas
There is no reason for this article to be separate; the short size and nature of the Pancreas article bears witness to this; this article would be of better quality and more utility to readers if it was provided with the context that the pancreas article bestows LT910001 (talk) 02:02, 4 January 2014 (UTC)
 * I have completed this merge. --LT910001 (talk) 14:58, 28 February 2014 (UTC)

Anatomy: size error!
There is no way the pancreas is "5.75–9.5 cm long"! The German Wikipedia says 16-20 cm and I think this is more realistic. — Preceding unsigned comment added by 62.202.184.53 (talk) 12:47, 5 November 2014 (UTC)
 * Thanks for pointing this out, IP editor. I'd encourage you to create an account and contribute further to the encyclopedia! --Tom (LT) (talk) 21:25, 5 November 2014 (UTC)

Proposed merge of Uncinate process of pancreas into Pancreas
Most of this article relates simply to the pancreas. Better to have it located in the parent article than separately - making life easier for readers and more likely that the article will be maintained. Tom (LT) (talk) 23:50, 28 December 2019 (UTC)

 * Agreed, that looks to be a stub and would be better in this article. |→ Spaully ~talk~
 * ❌ I'm withdrawing this merge proposal. I'm just not certain that Uncinate process can be merged into pancreas whilst easily retaining all the information in uncinate process. Open to hearing the opinion of others but will withdraw this for the time being. --Tom (LT) (talk) 21:10, 13 March 2020 (UTC)

Harrison's book, wrong page reference
I downloaded the ebook, same edition (19th), the reference points to pages  pp. 2437–8, but those pages are about lipoproteins and the liver.

So if I try to fix this reference, the pancreas is mentioned in "Part 14: Disorders of the gastrointestinal System", "Section 3: Disorders of the Pancreas", those are pages 2086−2102.

I don't think I'm wrong in this, so I will fix the reference, but if I am, feel free to let me know and explain. Thanks :) —Arthurfragoso (talk) 01:15, 9 June 2022 (UTC)

title: Talk:Pancreas


text:
Paul Jacob Alexander

Paul Jacob Alexander (March 11, 1904 – May 6, 1969) was a newspaper publisher and Seattle City Councilman.

Paul Jacob Alexander was born in Seattle, Washington on March 11, 1904, to Alfred and Lillian (Wooding) Alexander. He graduated from Ballard High School and spent a semester at the University of Washington. He worked for The Seattle Times in the 1920s, and purchased the Rainier District Times, a community newspaper in the Rainier Valley, in 1929. He sold the paper in 1965.

He ran unsuccessfully for the Seattle City Council in 1952 and 1954. He was elected in 1956 and re-elected in 1960 and 1964. He was a Republican, and although he was a strong supporter of freedom of the press, he considered himself a conservative. In 1963, he succeeded in removing an emergency clause from Seattle's proposed open housing ordinance that would have allowed it to take effect without a public vote, and in 1964 he ran for re-election as an opponent of open housing.

As the chairman of the Council's Utilities Committee, he was attending a reception at the American Public Power Association in Washington, D.C. when he died of a heart attack. Liem Tuai was appointed to fill his seat.

He lived in Rainier Valley, a block from Lake Washington.

title: Paul Jacob Alexander


text:
User talk:Tallkmac

Kmac, Welcome to Wikipedia! Definitely add those PG links -- sounds like a great idea! Please, when you have time, post something at New user log to let us know who you are. If you want more info about formatting, see that welcome page and How to edit a page. As you've found, the Village pump is a good place for questions. You can also ask me things on my talk page, if you like. Have fun here, Jwrosenzweig 22:02, 5 Mar 2004 (UTC)

Hi there Kyle, and welcome from me as well. I noticed your message on the new user log, and I figured I'd suggest a few things you could do.

 * 1) Even if you don't know anything about anything (which is obviously not the case for anyone), you could just keep hitting the Random page link, and format the articles that are not formatted according to
 * 2) *How to edit a page
 * 3) *How to write a great article
 * 4) *Naming conventions
 * 5) *Manual of Style
 * as well hit google and find a fact or a home page for the articles on cities. This way you also learn the formatting and perhaps a fact or two :) You could also have a look at Pages needing attention and Cleanup and see if you could do something. But since this can get pretty boring, pretty fast, here are some more things...
 * 1) If you have a digital camera, you could take some pictures of your home town, or of random objects that are needed (you could pretty much think of picture that could go with an article that doesn't have one). To give you some ideas, have a look at Featured pictures and then at Requested pictures (most of these have been fulfilled though :) Lacking a camera, you could scour the web for some nice pictures that could be used in Wikipedia (have to be in public domain or under the GFDL, so take a look at Public domain image resources).
 * 2) Another more laid back way, is to "sit" on Recent changes, have a look at the edits, let the good editing seep in, and notify of bad editing (i.e. vandalisms) at Cleanup.

These are just some suggestions, you're free to do as you wish here, there are no assignments :)

I forgot to mention, you can sign messages using either (just name) or ~ (adds timestamp)

Happy editing, and don't hesitate to ask me if you have any questions

Dori | Talk 21:44, Mar 16, 2004 (UTC)

Just to echo Dori - a lot of us sit around and reload Special:Recent Changes until we see something interesting going on, or to catch vandalism. There's also Special:Newpages which displays some recently created pages (some of which always need work.) You could also take a look at some ongoing projects at WikiProject. If you want to write a brand new article, you can find suggestions at Requested articles or Most wanted articles.

Basically, do whatever you enjoy. People around here get satisfaction from very different things, so play around until you find the niche that suits you. Isomorphic 03:31, 17 Mar 2004 (UTC)

title: User talk:Tallkmac


text:
Multifactorial

Multifactorial (having many factors) can refer to:
 * The multifactorial in mathematics.
 * Multifactorial inheritance, a pattern of predisposition for a disease process.

title: Multifactorial


text:
Charles-Jean-François Hénault

Charles-Jean-François Hénault (8 February 1685 – 24 November 1770) was a French writer and historian.

Early years
Hénault was born in Paris. His father, René Jean Rémy Hénault de Cantobre (1648–1737) a farmer-general of taxes, Ferme générale, was a man of literary tastes, and young Hénault obtained a good education at the Jesuit college, Lycée Louis-le-Grand. His mother was Françoise de Ponthon (died 1738).

Captivated by the eloquence of Massillon, in his fifteenth year he entered the Oratory with the view of becoming a preacher but after two-year's residence he changed his intention, and, inheriting a position which secured him access to the most select society of Paris, he achieved distinction at an early period by his witty and graceful manners.

Career
Hénault's literary talent, manifested in the composition of various light poetical pieces, an opera, a tragedy (Cornélie, vestale, 1713), etc., obtained his entrance to the Academy (1723). Petit-maître as he was, he had also serious capacity, for he became councillor of the parlement of Paris (1705), and in 1710 he was chosen president of the chamber of inquiries (French: Chambre des Enquêtes). He also hosted the Club de l'Entresol (1724–1731) in his house on the Place Vendôme. After the death of the count de Rieux (son of the famous financier, Samuel Bernard) he became (1753) superintendent of the household of Queen Marie Leszczynska, whose intimate friendship he had previously enjoyed. On Hénault's recovery in his eightieth year from a dangerous malady (1765) he professed to have undergone religious conversion and retired into private life, devoting the remainder of his days to study and devotion. His religion was, however, according to the marquis d'Argenson, exempt from fanaticism, persecution, bitterness and intrigue; and it did not prevent him from continuing his friendship with Voltaire, to whom it is said he had formerly rendered the service of saving the manuscript of La Henriade, when its author was about to commit it to the flames.

The literary work on which Hénault bestowed his chief attention was the Abrégé chronologique de l'histoire de France, first published in 1744 without the author's name. In the compass of two volumes he comprised the whole history of France from the earliest times to the death of Louis XIV. The work has no originality. Hénault had kept his note-books of the history lectures at the Jesuit college, of which the substance was taken from Mézeray and P Daniel. He revised them first in 1723, and later put them in the form of question and answer on the model of P le Ragois, and by following Dubos and Boulainvilliers and with the aid of the abbé Boudot he compiled his Abrégé.

The research is all on the surface and is only borrowed. But the work had a prodigious success, and was translated into several languages, even into Chinese. This was due partly to Hénault's popularity and position, partly to the agreeable style which made the history readable. He inserted, according to the fashion of the period, moral and political reflections, which are always brief and generally as fresh and pleasing as they are just. A few masterly strokes reproduced the leading features of each age and the characters of its illustrious men; accurate chronological tables set forth the most interesting events in the history of each sovereign and the names of the great men who flourished during his reign; and interspersed throughout the work are occasional chapters on the social and civil state of the country at the close of each era in its history. Continuations of the work have been made at separate periods by Fantin des Odoards, by Anguis with notes by Walckenaer, and by Michaud. He died at Paris on 24 November 1770.

Hénault's Mémoires have come down to us in two different versions, both are said to be authentic. One was published in 1855 by M. du Vigan; the other was owned by the Comte de Coutades, who permitted Lucien Perey to give long extracts in his work on Président Hénault (Paris, 1893). The memoirs are fragmentary and disconnected, but contain interesting anecdotes and details concerning persons of note. See the Correspondance of Grimm, of Madame de Deffand and of Voltaire; the notice by Walckenaer in the edition of the Abrégé; Sainte-Beuve, Causeries du lundi, vol. xi.; and the Origines de l'abrégé (Ann. Bulletin de la Société de l'histoire de France, 1901). Also H Lion, Le Président Hénault (Paris, 1903).

title: Charles-Jean-François Hénault


text:
Green Lake (Seattle)

Green Lake is a freshwater lake in north central Seattle, Washington, within Green Lake Park. The park is surrounded by the Green Lake neighborhood to the north and east, the Wallingford neighborhood to the south, the Phinney Ridge neighborhood to the west, and Woodland Park to the southwest. It is a glacial lake, its basin having been dug 50,000 years ago by the Vashon glacier, which also created Lake Washington, Union, Bitter and Haller Lakes.

History
Green Lake is called  in Lushootseed, the native language of Seattle. The etymology of the name is unknown. It was given its English name by David Phillips, who surveyed the area in September 1855 for the United States Surveyor General. His first notes referred to it as "Lake Green" because even in its natural state the lake is prone to algae blooms.

The lake has a surface area of 259 acre, a mean depth of 13 ft, and a maximum depth of 30 ft. The lake has been dredged in order to maintain its depth. Green Lake lacks both surface water inflows and outflows. It once drained into Lake Washington via Ravenna Creek, but in 1911 the water level was lowered by 7 ft to create parkland, causing the creek to dry up between Green Lake and Cowen Park. Ravenna Boulevard and its wide, grassy median were constructed over the creek bed. The lake is now fed by rainfall, storm runoff, and Seattle's municipal water supply.

The area was originally homesteaded by various pioneers, the first being Erhart Sarfried, "Green Lake John." Sarfried subdivided his homestead in 1888 and sold the land to various entrepreneurs. W.D. Wood built an "amusement park" on the west side of the lake (which never amounted to more than a glorified lawn for picnics). On the east side of the lake, A.L. Parker logged the woods and built a sawmill. Edward C. Kilbourne built the first trolley line connecting the area to the city, the route of which is now Green Lake Way North. The trolley lines kept growing, until by 1910 they extended completely around the lake and a round trip could be made on a separate line going back to the city.

In July 2008, several metal spikes up to 3 ft in length were found along the bottom of the lake, injuring one person who accidentally stepped on them. The Seattle Parks Department later confirmed that the spikes were from a forgotten pilot program to control the growth of Eurasian water milfoil in Green Lake. The invasive aquatic plant had first appeared in the lake in the early 1980s. In 1984, the Parks Department installed 120000 sqft of black plastic sheeting along the bottom of the lake, using the spikes to hold the sheeting in place. The spikes originally had curved tips to prevent injury, but the tips had corroded and broken off. The city paid a team of divers to remove the remaining spikes in late July 2008. Throughout the course of the investigation, it was revealed that the city had also tried combating the milfoil with weed-eating fish, but hadn't made any serious efforts in the past 15 years aside from occasionally paying divers to pull the weeds by hand.

Green Lake Park
After 1903 the area became part of Seattle's grand Olmsted Plan to create a series of interconnected greenspaces around the entire city. The park design still reflects the Olmsted vision.

Green Lake is surrounded by a 2.8 mi paved pedestrian pathway. Bicycles, scooters, roller skates, and other wheeled vehicles were barred from using the paved pathway during the COVID-19 pandemic due to high pedestrian use and the construction of new protected bicycle lanes on Green Lake Way. This restriction, made by the Seattle Parks and Recreation department, is currently classified as a "long-term temporary use restriction".

The path attracts people seeking exercise and relaxation, and can be crowded. There is also an outer non-paved 3.2 mi path along the edge of the park. The park is a popular spot for qigong classes, roller hockey, soccer, baseball, golf, the Derek Baker Memorial Boccie Ball Club, and lawn bowls, part of the Woodland Park Lawn Bowling Club, and a monthly midnight bicycle race.

The bathhouse was built in 1927 next to an outdoor swimming area with concrete steps leading into the water. A lifeguard station with a boat was built next to this area in 1930 after several drownings in 1929. The bathhouse is now home to the Seattle Public Theater at the Bathhouse, a small venue for plays.

Across the lake from the bathhouse, in the northeast part of the park, Green Lake's first community center was built in 1929 at a cost of $95,598. As it was built on the fill land from the 1911 draining, the community center was built on pilings. It contains two conference rooms, a gym with showers and bathrooms, and a stage. Toward the lake, another stepped swimming area was built. The tennis courts were added in 1945. In 1955 a 150,000 gallon (568 cubic meter) swimming pool was added. It was named the Evans Pool in honor of two brothers, Ben and Lou Evans, for their long service to athletics at Seattle parks.

The children's wading pool was a Works Progress Administration project, as was the drainage ditch and the arched stone bridge providing a path over the ditch. The wading pool is staffed in the summer by the Seattle Parks department, and operates daily from June 23–September 3, from 11 a.m.–8 p.m.

South of the bathhouse is a lawn and fishing pier. Since 1984 this part of the lake has hosted an annual floating lantern memorial dedicated to the victims of the atomic bombings of Hiroshima and Nagasaki and all other victims of war.

Prospect Point, a spit of land that points toward Duck Island, protects a small area of water from high winds. This once was a popular spot for model boats, though model boating is no longer allowed on the lake.

The Green Lake Aqua Theater was built in 1950 for the first Seafair in order to house an attraction called the Aqua Follies and their "swimusicals," a combination of aqua ballet, stage dancing, and comedy. The theater included a round stage and floating (though still recessed below the stage) orchestra pit, encircling a section of the lake with high diving platforms on each side. The grandstand capacity was more than 5,000 seats. The Aqua Follies continued to run during Seafair until 1965. Outside of the Seafair schedule the theater was the stage for plays and musicals whose directors always took advantage of the unique setting. In the summer of 1962, coinciding with the Century 21 Exposition, the Aqua Theater stage was host to a jazz festival, popular performers such as Bob Hope, two plays, and a special presentation of the Aqua Follies with 100 performers. After the World's Fair, summer productions languished, which is usually blamed on Seattle's unpredictable weather, until the Aqua Theater was mostly abandoned. A 1969 concert by the Grateful Dead revealed that the grandstand was crumbling and dangerous. Beginning in 1970 the theater was dismantled, stage right now serving as a pedestrian pier and stage left providing dock and storage for crew shells. Some sections of the grandstand were left in place.

The southwest portion of the park connects with adjoining Woodland Park on land that is also mostly fill, much of which came from the excavation of a route for Aurora Avenue. The southwest portion of the lake once extended to what is now N. 54th Street.

In the summer, Green Lake is also popular for swimming and boating. Although public use of motorized boats has been banned since at least 1968, the lake was the site of hydroplane races from 1929 to 1984. Today many forms of motor-less boats, including sailboarding, pedal boats, rowboats, skiffs, and canoes, are commonly seen on the lake. The Milk Carton Derby is held annually on the lake as one of the opening events of Seafair. While remnants of boat launches still exist, all launches have been removed from the lake; all boats must be hand carried to the water.

The Green Lake Small Craft Center, a Seattle Parks facility, is located on the south end of the lake. It houses both Green Lake Crew and the Seattle Canoe and Kayak Club. From August 10–13, 2006, Green Lake hosted the USRowing Masters National Championship Regatta, which included an estimated 2,000 competitors ranging in age from 23 to 86 years of age.

To the east of the lake across from the park, sits the Green Lake Library, built in 1910 with funds from Andrew Carnegie.

Within the lake is an artificial island that is commonly called Duck Island but was originally named Swan Island. It was built by the Works Progress Administration in 1936. The island was built (with dumped gravel) as a wildlife sanctuary and later housed some swan gifted to the city by Vancouver, British Columbia. The park board named the island Waldo's Wildlife Sanctuary, after Waldo J Dahl, who took care of the swan. The swans, which were intended to start a small flock, did not have proper breeding habitat around the island. Future attempts to breed swans on the island (at least until the 1970s) were also failures. The state game commission officially made the island a reserve, off limits to people, in 1956. That designation ended in 1983. Though not noted as supporting any unique wildlife, the Washington Department of Fish and Wildlife notes the island is part of a bald eagle management buffer encompassing the south end of the lake. The parks department still considers the island off-limits to people for safety reasons. But there is an occasional presence of rope swings and empty alcohol containers.

Green Lake Crew
The Green Lake Small Craft Center, which is a Seattle Parks facility, is the training ground for Green Lake Crew, which has a history of successful rowing at the national level.

The Green Lake Small Craft Center also is home to the Seattle Canoe and Kayak Club. This Olympic style sprint paddling also competes at a national level.

Animal and plant life
Green Lake is a beacon for wildlife. Many types of wildlife, toads, ducks, cormorants, loons, herons, geese, turtles, raccoons, rats, squirrels, bats, hawks, eagles, and osprey are among the wild creatures commonly viewed there.

Green Lake is a popular dumping ground for unwanted house pets and developed a large population of feral rabbits. The rabbits created problems by burrowing under streets and retaining walls and into the Woodland Park Zoo. Over the years volunteers adopted or removed the rabbits but not fast enough to keep the population under control. In 2005 the city and zoo combined with the Rabbit Sanctuary to remove all of the rabbits and present an educational outreach campaign to teach people not to abandon rabbits. The practice of abandoning pets into or around Green Lake has resulted in several other non-native species needing removal, mostly ordinary goldfish but including exotic species such as sturgeon and caiman at various times.

Recently, a program has been implemented to substantially reduce the number of ducks and geese. Their droppings raise the level of phosphorus in the lake, leading to excessive growths of algae and milfoil. In 2003 the lake was treated with aluminum sulfate to encapsulate the phosphorus. A paddle boat, moored in the lake, is used to cut the milfoil. The fowl also leave pathogens in the lake which can cause Swimmer's itch. This shows the difficulty of maintaining water quality in a stagnant lake.

Many species of fish live in Green Lake. Trout, mainly rainbow trout, are stocked for recreational fishing. A large population of common carp are present, and can be seen spawning in the shallows in spring. Tiger muskellunge were stocked to control the carp population, as carp are perceived as an invasive species and can contribute to eutrophication. The lake also contains largemouth bass, yellow perch, bluegill, yellow bullhead, channel catfish, and small populations of many different unexpected species.

During the spring, Green Lake Park is in bloom with pink and white cherry trees. Planted along the west side of the lake in 1931 and 1932, they were a gift from the Japanese Association of North America.

Species List of Common Names
The park is home to many species of birds and mammals as well as a few reptile and amphibian species. The most notable species are feral rabbits, Barred Owls, and Red Eared Sliders (released turtles). The most common bird species include American Crow, American Goldfinch, American Robin, Anna's Hummingbird, Bald Eagle, Barn Owl, Barred Owl, Bewick's Wren, Black-Capped Chickadee, Bushtit, European Starling, House Finch, Hairy/Downy Woodpeckers, Northern Flicker, Oregon Junco, Red-winged Blackbird, Eurasian Rock Dove, Spotted Towhee, Steller's Jay, various Gull species, various Sparrow species including Song Sparrow, and various Swallow species. Common waterfowl species consists of Buffleheads, Common Merganser, Gadwalls, Mallards, North American Coots, North American Widgeons, and Pied Billed grebes. Mammal species include Nutria, feral rabbits, Norway Rats, Gray Squirrel, Mountain Beaver, and American Beaver. Reptile and Amphibian species include Red Eared Sliders, Spring Peepers and Garter Snakes.

title: Green Lake (Seattle)


text:
Double factorial

In mathematics, the double factorial of a number $n$, denoted by $15 = (6 − 1)‼$, is the product of all the positive integers up to $n$ that have the same parity (odd or even) as $n$. That is,

$$n!! = \prod_{k=0}^{\left\lceil\frac{n}{2}\right\rceil - 1} (n-2k) = n (n-2) (n-4) \cdots.$$

Restated, this says that for even $n$, the double factorial is $$n!! = \prod_{k=1}^\frac{n}{2} (2k) = n(n-2)(n-4)\cdots 4\cdot 2 \,,$$ while for odd $n$ it is $$n!! = \prod_{k=1}^\frac{n+1}{2} (2k-1) = n(n-2)(n-4)\cdots 3\cdot 1 \,.$$ For example, $n‼$. The zero double factorial $9‼ = 9 × 7 × 5 × 3 × 1 = 945$ as an empty product.

The sequence of double factorials for even $n$ = $0‼ = 1$ starts as

The sequence of double factorials for odd $n$ = $0, 2, 4, 6, 8,...$ starts as

The term odd factorial is sometimes used for the double factorial of an odd number.

The term semifactorial is also used by Knuth as a synonym of double factorial.

History and usage
In a 1902 paper, the physicist Arthur Schuster wrote: "The symbolical representation of the results of this paper is much facilitated by the introduction of a separate symbol for the product of alternate factors, $n \cdot n-2 \cdot n-4 \cdots 1$, if $n$ be odd, or $n \cdot n-2 \cdots 2$ if $n$ be odd [sic]. I propose to write $n!!$ for such products, and if a name be required for the product to call it the "alternate factorial" or the "double factorial"."

states that the double factorial was originally introduced in order to simplify the expression of certain trigonometric integrals that arise in the derivation of the Wallis product. Double factorials also arise in expressing the volume of a hypersphere, and they have many applications in enumerative combinatorics. They occur in Student's $t$-distribution (1908), though Gosset did not use the double exclamation point notation.

Relation to the factorial
Because the double factorial only involves about half the factors of the ordinary factorial, its value is not substantially larger than the square root of the factorial $1, 2, 8, 48, 384, 3840, 46080, 645120, ...$, and it is much smaller than the iterated factorial $1, 3, 5, 7, 9,...$.

The factorial of a positive $n$ may be written as the product of two double factorials: $$n! = n!! \cdot (n-1)!!\,,$$ and therefore $$n!! = \frac{n!}{(n-1)!!} = \frac{(n+1)!}{(n+1)!!}\,,$$ where the denominator cancels the unwanted factors in the numerator. (The last form also applies when $1, 3, 15, 105, 945, 10395, 135135, ...$.)

For an even non-negative integer $n!$ with $(n!)!$, the double factorial may be expressed as $$(2k)!! = 2^k k!\,.$$

For odd $n = 0$ with $n = 2k$, combining the two previous formulas yields $$(2k-1)!! = \frac{(2k)!}{2^k k!} = \frac{(2k-1)!}{2^{k-1} (k-1)!}\,.$$

For an odd positive integer $k ≥ 0$ with $n = 2k − 1$, the double factorial may be expressed in terms of $k ≥ 1$-permutations of $n = 2k − 1$ or a falling factorial as $$(2k-1)!! = \frac {_{2k}P_k} {2^k} = \frac {(2k)^{\underline k}} {2^k}\,.$$

Applications in enumerative combinatorics
Double factorials are motivated by the fact that they occur frequently in enumerative combinatorics and other settings. For instance, $k ≥ 1$ for odd values of $n$ counts
 * Perfect matchings of the complete graph $k$ for odd $n$. In such a graph, any single vertex v has $n$ possible choices of vertex that it can be matched to, and once this choice is made the remaining problem is one of selecting a perfect matching in a complete graph with two fewer vertices. For instance, a complete graph with four vertices a, b, c, and d has three perfect matchings: ab and cd, ac and bd, and ad and bc. Perfect matchings may be described in several other equivalent ways, including involutions without fixed points on a set of $2k$ items (permutations in which each cycle is a pair) or chord diagrams (sets of chords of a set of $15 = (2 × 4 − 3)‼$ points evenly spaced on a circle such that each point is the endpoint of exactly one chord, also called Brauer diagrams). The numbers of matchings in complete graphs, without constraining the matchings to be perfect, are instead given by the telephone numbers, which may be expressed as a summation involving double factorials.
 * Stirling permutations, permutations of the multiset of numbers $n‼$ in which each pair of equal numbers is separated only by larger numbers, where $K_{n + 1}$. The two copies of $k$ must be adjacent; removing them from the permutation leaves a permutation in which the maximum element is $n + 1$, with $n$ positions into which the adjacent pair of $k$ values may be placed. From this recursive construction, a proof that the Stirling permutations are counted by the double permutations follows by induction. Alternatively, instead of the restriction that values between a pair may be larger than it, one may also consider the permutations of this multiset in which the first copies of each pair appear in sorted order; such a permutation defines a matching on the $n + 1$ positions of the permutation, so again the number of permutations may be counted by the double permutations.
 * Heap-ordered trees, trees with $1, 1, 2, 2, ..., k, k$ nodes labeled $k = n + 1⁄2$, such that the root of the tree has label 0, each other node has a larger label than its parent, and such that the children of each node have a fixed ordering. An Euler tour of the tree (with doubled edges) gives a Stirling permutation, and every Stirling permutation represents a tree in this way.
 * Unrooted binary trees with $k − 1$ labeled leaves. Each such tree may be formed from a tree with one fewer leaf, by subdividing one of the $n$ tree edges and making the new vertex be the parent of a new leaf.
 * Rooted binary trees with $2k$ labeled leaves. This case is similar to the unrooted case, but the number of edges that can be subdivided is even, and in addition to subdividing an edge it is possible to add a node to a tree with one fewer leaf by adding a new root whose two children are the smaller tree and the new leaf.

and list several additional objects with the same counting sequence, including "trapezoidal words" (numerals in a mixed radix system with increasing odd radixes), height-labeled Dyck paths, height-labeled ordered trees, "overhang paths", and certain vectors describing the lowest-numbered leaf descendant of each node in a rooted binary tree. For bijective proofs that some of these objects are equinumerous, see and.

The even double factorials give the numbers of elements of the hyperoctahedral groups (signed permutations or symmetries of a hypercube)

Asymptotics
Stirling's approximation for the factorial can be used to derive an asymptotic equivalent for the double factorial. In particular, since $$n! \sim \sqrt{2 \pi n}\left(\frac{n}{e}\right)^n,$$ one has as $$n$$ tends to infinity that

$$n!! \sim \begin{cases} \displaystyle \sqrt{\pi n}\left(\frac{n}{e}\right)^{n/2} & \text{if } n \text{ is even}, \\[5pt] \displaystyle \sqrt{2 n}\left(\frac{n}{e}\right)^{n/2} & \text{if } n \text{ is odd}. \end{cases}$$

Negative arguments
The ordinary factorial, when extended to the gamma function, has a pole at each negative integer, preventing the factorial from being defined at these numbers. However, the double factorial of odd numbers may be extended to any negative odd integer argument by inverting its recurrence relation $$n!! = n \times (n-2)!!$$ to give $$n!! = \frac{(n+2)!!}{n+2}\,.$$ Using this inverted recurrence, (−1)!! = 1, (−3)!! = −1, and (−5)!! = $1⁄3$; negative odd numbers with greater magnitude have fractional double factorials. In particular, when $n$ is an odd number, this gives $$(-n)!! \times n!! = (-1)^\frac{n-1}{2} \times n\,.$$

Complex arguments
Disregarding the above definition of $k + 1$ for even values of $n$, the double factorial for odd integers can be extended to most real and complex numbers $z$ by noting that when $z$ is a positive odd integer then

$$\begin{align} z!! &= z(z-2)\cdots 5 \cdot 3 \\[3mu] &= 2^\frac{z-1}{2}\left(\frac z2\right)\left(\frac{z-2}2\right)\cdots \left(\frac{5}{2}\right) \left(\frac{3}{2}\right) \\[5mu] &= 2^\frac{z-1}{2} \frac{\Gamma\left(\tfrac z2+1\right)}{\Gamma\left(\tfrac12+1\right)} \\[5mu] &= \sqrt{\frac{2}{\pi}} 2^\frac{z}{2} \Gamma\left(\tfrac z2+1\right) \,,\end{align}$$ where $$\Gamma(z)$$ is the gamma function.

The final expression is defined for all complex numbers except the negative even integers and satisfies $0, 1, 2, ... k$ everywhere it is defined. As with the gamma function that extends the ordinary factorial function, this double factorial function is logarithmically convex in the sense of the Bohr–Mollerup theorem. Asymptotically, $n!! \sim \sqrt{2 n^{n+1} e^{-n}}\,.$

The generalized formula $$\sqrt{\frac{2}{\pi}} 2^\frac{z}{2} \Gamma\left(\tfrac z2+1\right)$$ does not agree with the previous product formula for $n + 5⁄2$ for non-negative even integer values of $z$. Instead, this generalized formula implies the following alternative: $$(2k)!! = \sqrt{\frac{2}{\pi}} 2^k \Gamma\left(k+1\right) = \sqrt{ \frac{2}{\pi} } \prod_{i=1}^k (2i) \,,$$ with the value for 0!! in this case being

 * $$0!! = \sqrt{ \frac{2}{\pi} } \approx 0.797\,884\,5608\dots$$.

Using this generalized formula as the definition, the volume of an $n$-dimensional hypersphere of radius $R$ can be expressed as

$$V_n=\frac{2 \left(2\pi\right)^\frac{n-1}{2}}{n!!} R^n\,.$$

Additional identities
For integer values of $n$,

$$\int_{0}^\frac{\pi}{2}\sin^n x\,dx=\int_{0}^\frac{\pi}{2}\cos^n x\,dx=\frac{(n-1)!!}{n!!}\times \begin{cases}1 & \text{if } n \text{ is odd} \\ \frac{\pi}{2} & \text{if } n \text{ is even.}\end{cases}$$ Using instead the extension of the double factorial of odd numbers to complex numbers, the formula is $$\int_{0}^\frac{\pi}{2}\sin^n x\,dx=\int_{0}^\frac{\pi}{2}\cos^n x\,dx=\frac{(n-1)!!}{n!!} \sqrt{\frac{\pi}{2}}\,.$$

Double factorials can also be used to evaluate integrals of more complicated trigonometric polynomials.

Double factorials of odd numbers are related to the gamma function by the identity:

$$(2n-1)!! = 2^n \cdot \frac{\Gamma\left(\frac{1}{2} + n\right)} {\sqrt{\pi}} = (-2)^n \cdot \frac{\sqrt{\pi}} { \Gamma\left(\frac{1}{2} - n\right)}\,.$$

Some additional identities involving double factorials of odd numbers are:

$$\begin{align} (2n-1)!! &= \sum_{k=0}^{n-1} \binom{n}{k+1} (2k-1)!! (2n-2k-3)!! \\ &= \sum_{k=1}^{n} \binom{n}{k} (2k-3)!! (2(n-k)-1)!! \\ &= \sum_{k=0}^{n} \binom{2n-k-1}{k-1} \frac{(2k-1)(2n-k+1)}{k+1}(2n-2k-3)!! \\ &= \sum_{k=1}^{n} \frac{(n-1)!}{(k-1)!} k(2k-3)!!\,. \end{align}$$

An approximation for the ratio of the double factorial of two consecutive integers is $$ \frac{(2n)!!}{(2n-1)!!} \approx \sqrt{\pi n}. $$ This approximation gets more accurate as $n$ increases, which can be seen as a result of the Wallis Integral.

Definitions
In the same way that the double factorial generalizes the notion of the single factorial, the following definition of the integer-valued multiple factorial functions (multifactorials), or $α$-factorial functions, extends the notion of the double factorial function for positive integers $$\alpha$$:

$$ n!_{(\alpha)} = \begin{cases} n \cdot (n-\alpha)!_{(\alpha)} & \text{ if } n > \alpha \,; \\ n & \text{ if } 1 \leq n \leq \alpha \,; \text{and} \\ (n+\alpha)!_{(\alpha)} / (n+\alpha) & \text{ if } n \leq 0 \text{ and is not a negative multiple of } \alpha \,; \end{cases} $$

Alternative extension of the multifactorial
Alternatively, the multifactorial $n + 3⁄2$ can be extended to most real and complex numbers $n!!$ by noting that when $(z + 2)!! = (z + 2) · z!!$ is one more than a positive multiple of the positive integer $z!!$ then

$$\begin{align} z!_{(\alpha)} &= z(z-\alpha)\cdots (\alpha+1) \\ &= \alpha^\frac{z-1}{\alpha}\left(\frac{z}{\alpha}\right)\left(\frac{z-\alpha}{\alpha}\right)\cdots \left(\frac{\alpha+1}{\alpha}\right) \\ &= \alpha^\frac{z-1}{\alpha} \frac{\Gamma\left(\frac{z}{\alpha}+1\right)}{\Gamma\left(\frac{1}{\alpha}+1\right)}\,. \end{align}$$

This last expression is defined much more broadly than the original. In the same way that $z!_{(α)}$ is not defined for negative integers, and $z$ is not defined for negative even integers, $z$ is not defined for negative multiples of $α$. However, it is defined for and satisfies $z!$ for all other complex numbers $z‼$. This definition is consistent with the earlier definition only for those integers $z!_{(α)}$ satisfying $α$.

In addition to extending $(z+α)!_{(α)} = (z+α)·z!_{(α)}$ to most complex numbers $z$, this definition has the feature of working for all positive real values of $z$. Furthermore, when $z ≡ 1 mod α$, this definition is mathematically equivalent to the $z!_{(α)}$ function, described above. Also, when $z$, this definition is mathematically equivalent to the alternative extension of the double factorial.

Generalized Stirling numbers expanding the multifactorial functions
A class of generalized Stirling numbers of the first kind is defined for $α$ by the following triangular recurrence relation:

$$\left[\begin{matrix} n \\ k \end{matrix} \right]_{\alpha} = (\alpha n+1-2\alpha) \left[\begin{matrix} n-1 \\ k \end{matrix} \right]_{\alpha} + \left[\begin{matrix} n-1 \\ k-1 \end{matrix} \right]_{\alpha} + \delta_{n,0} \delta_{k,0}\,. $$

These generalized $α$-factorial coefficients then generate the distinct symbolic polynomial products defining the multiple factorial, or $α$-factorial functions, $α = 1$, as

$$ \begin{align} (x-1|\alpha)^{\underline{n}} & := \prod_{i=0}^{n-1} \left(x-1-i\alpha\right) \\ & = (x-1)(x-1-\alpha)\cdots\bigl(x-1-(n-1)\alpha\bigr) \\ & = \sum_{k=0}^n \left[\begin{matrix} n \\ k \end{matrix} \right] (-\alpha)^{n-k} (x-1)^k \\ & = \sum_{k=1}^n \left[\begin{matrix} n \\ k \end{matrix} \right]_{\alpha} (-1)^{n-k} x^{k-1}\,. \end{align} $$

The distinct polynomial expansions in the previous equations actually define the $α$-factorial products for multiple distinct cases of the least residues $Π(z)$ for $α = 2$.

The generalized $α$-factorial polynomials, $α > 0$ where $(x − 1)!_{(α)}$, which generalize the Stirling convolution polynomials from the single factorial case to the multifactorial cases, are defined by

$$\sigma_n^{(\alpha)}(x) := \left[\begin{matrix} x \\ x-n \end{matrix} \right]_{(\alpha)} \frac{(x-n-1)!}{x!}$$

for $x ≡ n_{0} mod α$. These polynomials have a particularly nice closed-form ordinary generating function given by

$$\sum_{n \geq 0} x \cdot \sigma_n^{(\alpha)}(x) z^n = e^{(1-\alpha)z} \left(\frac{\alpha z e^{\alpha z}}{e^{\alpha z}-1}\right)^x\,. $$

Other combinatorial properties and expansions of these generalized $α$-factorial triangles and polynomial sequences are considered in.

Exact finite sums involving multiple factorial functions
Suppose that $n_{0} ∈ {0, 1, 2, ..., α − 1 }$ and $σ(α) n(x)$ are integer-valued. Then we can expand the next single finite sums involving the multifactorial, or $α$-factorial functions, $σ(1) n(x) ≡ σ_{n}(x)$, in terms of the Pochhammer symbol and the generalized, rational-valued binomial coefficients as

$$ \begin{align} (\alpha n-1)!_{(\alpha)} & = \sum_{k=0}^{n-1} \binom{n-1}{k+1} (-1)^k \times \left(\frac{1}{\alpha}\right)_{-(k+1)} \left(\frac{1}{\alpha}-n\right)_{k+1} \times \bigl(\alpha(k+1)-1\bigr)!_{(\alpha)} \bigl(\alpha(n-k-1)-1\bigr)!_{(\alpha)} \\ & = \sum_{k=0}^{n-1} \binom{n-1}{k+1} (-1)^k \times \binom{\frac{1}{\alpha}+k-n}{k+1} \binom{\frac{1}{\alpha}-1}{k+1} \times \bigl(\alpha(k+1)-1\bigr)!_{(\alpha)} \bigl(\alpha(n-k-1)-1\bigr)!_{(\alpha)}\,, \end{align} $$

and moreover, we similarly have double sum expansions of these functions given by

$$ \begin{align} (\alpha n-1)!_{(\alpha)} & = \sum_{k=0}^{n-1} \sum_{i=0}^{k+1} \binom{n-1}{k+1} \binom{k+1}{i} (-1)^k \alpha^{k+1-i} (\alpha i-1)!_{(\alpha)} \bigl(\alpha(n-1-k)-1\bigr)!_{(\alpha)} \times (n-1-k)_{k+1-i} \\ & = \sum_{k=0}^{n-1} \sum_{i=0}^{k+1} \binom{n-1}{k+1} \binom{k+1}{i} \binom{n-1-i}{k+1-i} (-1)^k \alpha^{k+1-i} (\alpha i-1)!_{(\alpha)} \bigl(\alpha(n-1-k)-1\bigr)!_{(\alpha)} \times (k+1-i)!. \end{align} $$

The first two sums above are similar in form to a known non-round combinatorial identity for the double factorial function when $0 ≤ n ≤ x$ given by.

$$(2n-1)!! = \sum_{k=0}^{n-1} \binom{n}{k+1} (2k-1)!! (2n-2k-3)!!.$$

Similar identities can be obtained via context-free grammars. Additional finite sum expansions of congruences for the $α$-factorial functions, $n ≥ 1$, modulo any prescribed integer $α ≥ 2$ for any $(αn − 1)!_{(α)}$ are given by.

title: Double factorial


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Hyperfactorial

In mathematics, and more specifically number theory, the hyperfactorial of a positive integer $$n$$ is the product of the numbers of the form $$x^x$$ from $$1^1$$ to $n^n$.

Definition
The hyperfactorial of a positive integer $$n$$ is the product of the numbers $$1^1, 2^2, \dots, n^n$$. That is, $$ H(n) = 1^1\cdot 2^2\cdot \cdots n^n = \prod_{i=1}^{n} i^i = n^n H(n-1).$$ Following the usual convention for the empty product, the hyperfactorial of 0 is 1. The sequence of hyperfactorials, beginning with $$H(0)=1$$, is:

Interpolation and approximation
The hyperfactorials were studied beginning in the 19th century by Hermann Kinkelin and James Whitbread Lee Glaisher. As Kinkelin showed, just as the factorials can be continuously interpolated by the gamma function, the hyperfactorials can be continuously interpolated by the K-function.

Glaisher provided an asymptotic formula for the hyperfactorials, analogous to Stirling's formula for the factorials: $$H(n) = An^{(6n^2+6n+1)/12}e^{-n^2/4}\left(1+\frac{1}{720n^2}-\frac{1433}{7257600n^4}+\cdots\right)\!,$$ where $$A\approx 1.28243$$ is the Glaisher–Kinkelin constant.

Other properties
According to an analogue of Wilson's theorem on the behavior of factorials modulo prime numbers, when $$p$$ is an odd prime number $$H(p-1)\equiv(-1)^{(p-1)/2}(p-1)!!\pmod{p},$$ where $$!!$$ is the notation for the double factorial.

The hyperfactorials give the sequence of discriminants of Hermite polynomials in their probabilistic formulation.

title: Hyperfactorial


text:
Superfactorial

In mathematics, and more specifically number theory, the superfactorial of a positive integer $$n$$ is the product of the first $$n$$ factorials. They are a special case of the Jordan–Pólya numbers, which are products of arbitrary collections of factorials.

Definition
The $$n$$th superfactorial $$\mathit{sf}(n)$$ may be defined as: $$\begin{align} \mathit{sf}(n) &= 1!\cdot 2!\cdot \cdots n! = \prod_{i=1}^{n} i! = n!\cdot\mathit{sf}(n-1)\\ &= 1^n \cdot 2^{n-1} \cdot \cdots n = \prod_{i=1}^{n} i^{n+1-i}.\\ \end{align}$$ Following the usual convention for the empty product, the superfactorial of 0 is 1. The sequence of superfactorials, beginning with $$\mathit{sf}(0)=1$$, is:

Properties
Just as the factorials can be continuously interpolated by the gamma function, the superfactorials can be continuously interpolated by the Barnes G-function.

According to an analogue of Wilson's theorem on the behavior of factorials modulo prime numbers, when $$p$$ is an odd prime number $$\mathit{sf}(p-1)\equiv(p-1)!!\pmod{p},$$ where $$!!$$ is the notation for the double factorial.

For every integer $$k$$, the number $$\mathit{sf}(4k)/(2k)!$$ is a square number. This may be expressed as stating that, in the formula for $$\mathit{sf}(4k)$$ as a product of factorials, omitting one of the factorials (the middle one, $$(2k)!$$) results in a square product. Additionally, if any $$n+1$$ integers are given, the product of their pairwise differences is always a multiple of $$\mathit{sf}(n)$$, and equals the superfactorial when the given numbers are consecutive.

title: Superfactorial


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Yezhov

Yezhov or Ezhov (Ежов) is a Russian masculine surname derived from the word yozh (ёж), meaning hedgehog; its feminine counterpart is Yezhova or Ezhova. It may refer to:

 * Denis Ezhov (born 1985), Russian ice hockey player
 * Elena Ezhova (born 1977), Ukraine-born Russian volleyball player
 * Elena Yezhova (1793–1853), Russian stage actress and opera singer
 * Ilya Ezhov (born 1987), Russian ice hockey goaltender
 * Ludmila Ezhova (born 1982), Russian artistic gymnast
 * Nikolay Yezhov (1895–1940), Soviet secret police official
 * Pyotr Yezhov (1900–1975), Russian football player
 * Roman Yezhov (born 1997), Russian football player
 * Valentin Yezhov (1921–2004), Russian screenwriter and playwright
 * Yekaterina Yezhova (1787–1837), Russian stage actress
 * Yevgeni Yezhov (born 1995), Russian football defender

title: Yezhov


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Moritzburg, Saxony

Moritzburg is a municipality in the district of Meissen in Saxony, Germany, between Meissen itself, an early centre of Saxony, and today's capital Dresden. It is most famous for its Baroque castle, Schloss Moritzburg.

The village, which was originally known as Eisenburg, was first mentioned in 1294. It became a market in 1675. The Saxon state stud has been located here since 1828. In 1884, a narrow gauge railway, the Radebeul–Radeburg line, was built connecting the town to the district capital Radebeul and Radeburg. It was renamed Moritzburg, after the nearby castle, in 1934. German artist Käthe Kollwitz lived at the invitation of Prince Ernst Heinrich von Wettin in the Rüdenhof in Moritzburg from 1944 until her death on 22 April 1945. In 1995, a small museum was opened in the Rüdenhof.

Twin towns

 * 🇩🇪 Cochem, Germany.

Sons of the place

 * Albert Casimir, Duke of Teschen (1738–1822), artistic patron
 * Martin Dulig, Minister of State in Saxony

Other personalities associated with the place

 * Maurice, Elector of Saxony (1521–1553), Elector
 * Augustus II the Strong (1670–1733), Elector of Saxony, King of Poland
 * Frederick Augustus I of Saxony (1750–1827), first king of Saxony
 * Käthe Kollwitz (1867–1945), artist
 * Prince Ernst Heinrich of Saxony (1896–1971), Wettiner
 * Jan Vogler (born 1964), cellist and artistic director of the Moritzburg Festival

title: Moritzburg, Saxony


text:
Talk:John Sweeney (labor leader)

Untitled
Is it just a coincidence that this guy and a New York Representative share the same name? This needs to be disambiguated... ugen64 22:10, Mar 5, 2004 (UTC)

LabourProject Importance
I'd argue for a High-Level importance rating. Sweeney is the current leader of the largest labor movement in the United States. If he dies or retires, I'd argue it would be Mid-Level. But he's one of two primary voices for organized labor in the U.S. I think that ranks as High-Level. Tim1965 03:00, 25 November 2006 (UTC)

title: Talk:John Sweeney (labor leader)


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François Eudes de Mézeray

François Eudes de Mézeray (1610 – 10 July 1683) was a French historian.

Mézeray was born at Ri near Argentan, where his father was a surgeon. He had two brothers, one of whom, Jean-Eudes, was the founder of the order of the Eudists. François studied at the University of Caen, and completed his education at the College of Ste Barbe at Paris. His Histoire de France depuis Faramond jusqu'au règne de Louis le juste (3 vols., 1643–1651), is a fairly accurate summary of French and Latin chronicles.

Mézeray was appointed to a committee that supervised La Gazette.

Mézeray won the favor of Pierre Séguier and was given the title "Historiographer to the King of France". In 1649, on the death of Vincent Voiture, he was admitted to the Académie française. His Abrégé chronologique (3 vols., 1667–1668) went through fifteen editions between 1668 and 1717; and he used it to attack the financiers, with the result that his salary as historiographer was diminished by Colbert.

Mézeray succeeded Valentin Conrart as permanent secretary to the Académie française (1675), and died at Paris. He translated Grotius's Traité de la religion chretienne (1640), and a Histoire des Turcs depuis 1612 jusqu'en 1649 (1650), which is an addition to a continuation of Laonikos Chalkokondyles.

title: François Eudes de Mézeray