ideals

Fields are important objects of study in algebra since they provide a useful generalization of many number systems, such as the rational numbers, real numbers, and complex numbers. In particular, the usual rules of associativity, commutativity and distributivity hold. Fields also appear in many other areas of mathematics; see the examples below. When abstract algebra was first being developed, the definition of a field usually did not include commutativity of multiplication, and what we today call a field would have been called either ...

See also:

Field mathematics, Field mathematics - Introduction, Field mathematics - Definition, Field mathematics - Examples of fields, Field mathematics - Some first theorems

Read more here: » Field mathematics: Encyclopedia II - Field mathematics - Introduction

Formally, given two categories C and D, an equivalence of categories consists of a functor F : C -> D, a functor G : D -> C, and two natural isomorphisms η: FG->ID and ε: IC->GF. Here FG: D->D and GF: C->C, denote the respective compositions of F and G, and IC< ...

See also:

Equivalence of categories, Equivalence of categories - Definition, Equivalence of categories - Equivalent characterizations, Equivalence of categories - Examples, Equivalence of categories - Properties

Read more here: » Equivalence of categories: Encyclopedia II - Equivalence of categories - Definition

The traditional morphisms between complete lattices are the complete homomorphisms (or complete lattice homomorphisms). These are characterized as functions that preserve all joins and all meets. Explicitly, this means that a function f: L→M between two complete lattices L and M is a complete homomorphism if and , for all subsets A of L. Such functions are automatically monotonic, but the condition of being a complete homomorphism is in fact much more spe ...

See also:

Complete lattice, Complete lattice - Formal definition, Complete lattice - Complete semilattices, Complete lattice - Examples, Complete lattice - Morphisms of complete lattices, Complete lattice - Free construction and completion, Complete lattice - Free complete semilattices, Complete lattice - Free complete lattices, Complete lattice - Completion, Complete lattice - Representation, Complete lattice - Further results, Complete lattice - Literature

Read more here: » Complete lattice: Encyclopedia II - Complete lattice - Morphisms of complete lattices

Christian left - Australia. Frank Brennan Jesuit and advocate for Australia's Indigenous peoples Tim Costello Baptist minister Peter Garrett, rock singer and politician Christian left - Canada. Richard Allen, politician and historian of Christian socialism Bill Blaikie, minister and politician Andrew Brewin, politician and author Lorne Calvert, minister and politician and Premier of Saskatchewan Tommy Doug ...

See also:

Christian left, Christian left - History, Christian left - Early Christianity as anti-establishment movement, Christian left - Movements, Christian left - Early antagonism between the Left and Christianity, Christian left - Alliance of the Left and Christianity, Christian left - Christian Left and campaigns for peace and human rights, Christian left - Christian Left in the United States, Christian left - Liberation Theology, Christian left - Christian Left and Homosexuality, Christian left - The Consistent Life Ethic, Christian left - Notable Christian leftists, Christian left - Australia, Christian left - Canada, Christian left - Colombia, Christian left - El Salvador, Christian left - Germany, Christian left - Italy, Christian left - Netherlands, Christian left - Nicaragua, Christian left - New Zealand, Christian left - Russia, Christian left - Slovenia, Christian left - Switzerland, Christian left - United States, Christian left - United Kingdom, Christian left - Venezuela, Christian left - Parties of the Christian left

Read more here: » Christian left: Encyclopedia II - Christian left - Notable Christian leftists

Mitch Albom wrote about his dying professor Morrie and their last lessons together in the bestseller Tuesdays with Morrie in which some interesting questions were raised. Albom's life as a writer was until then in vain because he chased the wrong things in life: bigger houses, bigger SUVs, and bigger paychecks. No matter how big they were, they still could not fill his emptiness. The reality that we all have to confront eventually is the same thing Morrie realized when he learned he had Lou Gehrig's disease: that the world was as gree ...

See also:

Meaning of life, Meaning of life - Popular beliefs, Meaning of life - Scientific approaches and theories, Meaning of life - Philosophical views, Meaning of life - Value as meaning, Meaning of life - Atheist views, Meaning of life - Existentialist views, Meaning of life - Humanist views, Meaning of life - Nihilist views, Meaning of life - Positivist views, Meaning of life - Pragmatist views, Meaning of life - Transhumanist views, Meaning of life - Religious beliefs, Meaning of life - Spiritual views, Meaning of life - Humorous treatments, Meaning of life - General philosophy topics, Meaning of life - General philosophy lists

Read more here: » Meaning of life: Encyclopedia II - Meaning of life - Spiritual views

It is often suggested that religion is a response to humanity's need to simply stop being confused, or to the fear of death (and the concomitant desire not to die). By defining a realm outside of life (the spiritual realm), these needs are fulfilled by providing a meaning, purpose, and hope for our otherwise pointless, directionless, and finite lives. Most people who believe in a personal God would agree that it is God "in whom we live and move and have our being". The notion here is that we ought to seek a higher authority who will give our ...

See also:

Meaning of life, Meaning of life - Popular beliefs, Meaning of life - Scientific approaches and theories, Meaning of life - Philosophical views, Meaning of life - Value as meaning, Meaning of life - Atheist views, Meaning of life - Existentialist views, Meaning of life - Humanist views, Meaning of life - Nihilist views, Meaning of life - Positivist views, Meaning of life - Pragmatist views, Meaning of life - Transhumanist views, Meaning of life - Religious beliefs, Meaning of life - Spiritual views, Meaning of life - Humorous treatments, Meaning of life - General philosophy topics, Meaning of life - General philosophy lists

Read more here: » Meaning of life: Encyclopedia II - Meaning of life - Religious beliefs

Where scientists and philosophers converge on the quest for the meaning of life is an assumption that the mechanics of life (i.e., the universe) are determinable, thus the meaning of life may eventually be derived through our understanding of the mechanics of the universe in which we live, including the mechanics of the human body. There are, however, strictly speaking, no scientific views on the meaning of biological life other than its observable biological function: to continue and to reproduce itself. In this regard, science simply addresses quantitative questions such as: "What does it do?", "By wha ...

See also:

Meaning of life, Meaning of life - Popular beliefs, Meaning of life - Scientific approaches and theories, Meaning of life - Philosophical views, Meaning of life - Value as meaning, Meaning of life - Atheist views, Meaning of life - Existentialist views, Meaning of life - Humanist views, Meaning of life - Nihilist views, Meaning of life - Positivist views, Meaning of life - Pragmatist views, Meaning of life - Transhumanist views, Meaning of life - Religious beliefs, Meaning of life - Spiritual views, Meaning of life - Humorous treatments, Meaning of life - General philosophy topics, Meaning of life - General philosophy lists

Read more here: » Meaning of life: Encyclopedia II - Meaning of life - Scientific approaches and theories

"What is the meaning of life?" is a question many people ask themselves at some point during their lifetime. Some people believe that the meaning of life is one or more of the following: Survival and temporal success ...to accumulate wealth and increase social status ...to compete or co-operate with others ...to destroy others who harm you, or to practice nonviolence and nonresistance ...to gain and exercise power ...to leave a legacy, such as a work of art or a book ... ...

See also:

Meaning of life, Meaning of life - Popular beliefs, Meaning of life - Scientific approaches and theories, Meaning of life - Philosophical views, Meaning of life - Value as meaning, Meaning of life - Atheist views, Meaning of life - Existentialist views, Meaning of life - Humanist views, Meaning of life - Nihilist views, Meaning of life - Positivist views, Meaning of life - Pragmatist views, Meaning of life - Transhumanist views, Meaning of life - Religious beliefs, Meaning of life - Spiritual views, Meaning of life - Humorous treatments, Meaning of life - General philosophy topics, Meaning of life - General philosophy lists

Read more here: » Meaning of life: Encyclopedia II - Meaning of life - Popular beliefs

The very concept "the meaning of life" has become such a cliché that it has often been parodied, such as in the radio series The Hitchhiker's Guide to the Galaxy by Douglas Adams, later released as a novel, a television series, a film, and a computer game. His answer was 42. As the story goes, an advanced race of hyper-intelligent pan-dimensional beings (mice) builds a gigantic computer called Deep Thought to find the Answer to Life, the Universe, and Everything. Seven and a half million years later, the computer gives the answer: "4 ...

See also:

Meaning of life, Meaning of life - Popular beliefs, Meaning of life - Scientific approaches and theories, Meaning of life - Philosophical views, Meaning of life - Value as meaning, Meaning of life - Atheist views, Meaning of life - Existentialist views, Meaning of life - Humanist views, Meaning of life - Nihilist views, Meaning of life - Positivist views, Meaning of life - Pragmatist views, Meaning of life - Transhumanist views, Meaning of life - Religious beliefs, Meaning of life - Spiritual views, Meaning of life - Humorous treatments, Meaning of life - General philosophy topics, Meaning of life - General philosophy lists

Read more here: » Meaning of life: Encyclopedia II - Meaning of life - Humorous treatments

Body image is often measured by asking the subject to rate his or her current and ideal body shape using a series of drawings of increasing size. The difference between these two values is the amount of body satisfaction. Unfortunately, this method does not take into account the fact that a person might be aware of being under- or overweight and also satisfied with that circumstance. Consequently, an obese person, whose obesity causes him or her no psychological distress, might be rated as "having a poor ...

See also:

Body image, Body image - Research: Measuring Body Image, Body image - Causes and Influences, Body image - Relationship to Psychological Disorders, Body image - Attractiveness and Social Issues, Body image - Information on Specific Minority Populations, Body image - Native Americans, Body image - Work cited

Read more here: » Body image: Encyclopedia II - Body image - Research: Measuring Body Image

In the United States, members of the Christian Left come from a spectrum of denominations: Peace churches, elements of the Protestant mainline churches, and Roman Catholicism. The Christian Left has sometimes been viewed as a counterpart to the Christian Right, but in fact it is very different. While the Christian Right is almost uniquely American, the Christian Left is more global and diverse. However, the Christian Left does not seem to be as well-organized or publicized as its right-wing counterpart. Opponents state that thi ...

See also:

Christian left, Christian left - History, Christian left - Early Christianity as anti-establishment movement, Christian left - Movements, Christian left - Early antagonism between the Left and Christianity, Christian left - Alliance of the Left and Christianity, Christian left - Christian Left and campaigns for peace and human rights, Christian left - Christian Left in the United States, Christian left - Liberation Theology, Christian left - Christian Left and Homosexuality, Christian left - The Consistent Life Ethic, Christian left - Notable Christian leftists, Christian left - Australia, Christian left - Canada, Christian left - Colombia, Christian left - El Salvador, Christian left - Germany, Christian left - Italy, Christian left - Netherlands, Christian left - Nicaragua, Christian left - New Zealand, Christian left - Russia, Christian left - Slovenia, Christian left - Switzerland, Christian left - United States, Christian left - United Kingdom, Christian left - Venezuela, Christian left - Parties of the Christian left

Read more here: » Christian left: Encyclopedia II - Christian left - Christian Left in the United States

Some researchers also found that men judge the female figure they found most attractive as heavier than women's ratings of the ideal body shape. In contrast, that most women, including overweight women, desire men with a very low percentage of body fat, whether they be thin or muscular. This suggests that, contrary to the media focus, men are far more likely to be attracted to larger woman than women are to be attracted to larger men. Additionally, men are also more likely to be unsatisfied with their height, due to a perceived prefer ...

See also:

Body image, Body image - Research: Measuring Body Image, Body image - Causes and Influences, Body image - Relationship to Psychological Disorders, Body image - Attractiveness and Social Issues, Body image - Information on Specific Minority Populations, Body image - Native Americans, Body image - Work cited

Read more here: » Body image: Encyclopedia II - Body image - Attractiveness and Social Issues

Poor body images can often contribute to the onset of a variety of eating disorders, including anorexia nervosa, bulimia, and binge eating disorder. Other possible effects of the cultural obsession with looking slender include excessive exercising, fad diets, and lawsuits involving fast food chains. Concerns with body image have been linked to a decrease in self esteem and an increase in dieting among young women. This latter trend has been identified as an indicator of the onset of eating disorders such as anorexia nervosa and bulimi ...

See also:

Body image, Body image - Research: Measuring Body Image, Body image - Causes and Influences, Body image - Relationship to Psychological Disorders, Body image - Attractiveness and Social Issues, Body image - Information on Specific Minority Populations, Body image - Native Americans, Body image - Work cited

Read more here: » Body image: Encyclopedia II - Body image - Relationship to Psychological Disorders

As clearly evidenced by literature around the world, poor body images have existed at least since the widespread availability of mirrors, but one of the reasons most often cited for this continuing body dissatisfaction among young women is modern media influence, including that from movies, television, and magazines. Media representatives often reply that they are merely reflecting the ideals of the current generation or using whatever image best sells their products. However, research has shown that the media play a large role in reinforcin ...

See also:

Body image, Body image - Research: Measuring Body Image, Body image - Causes and Influences, Body image - Relationship to Psychological Disorders, Body image - Attractiveness and Social Issues, Body image - Information on Specific Minority Populations, Body image - Native Americans, Body image - Work cited

Read more here: » Body image: Encyclopedia II - Body image - Causes and Influences

A sketch of the proof of Zorn's lemma follows. Suppose the lemma is false. Then there exists a partially ordered set, or poset, P such that every totally ordered subset has an upper bound, and every element has a bigger one. For every totally ordered subset T we may then define a bigger element b(T), because T has an upper bound, and that upper bound has a bigger element. To actually ...

See also:

Zorn's lemma, Zorn's lemma - An example application, Zorn's lemma - Sketch of the proof of Zorn's lemma, Zorn's lemma - History

Read more here: » Zorn's lemma: Encyclopedia II - Zorn's lemma - Sketch of the proof of Zorn's lemma

Buddhism began to spread throughout Japan during the Heian period, primarily through two major esoteric sects, Tendai (Heavenly Terrace) and Shingon (True Word). Tendai originated in China and is based on the Lotus Sutra, one of the most important sutras of Mahayana Buddhism. Shingon is an indigenous sect with close affiliations to original Indian, Tibetan, and Chinese Buddhist thought founded by Kukai (also called Kobo Daishi). Kukai greatly impressed the emperors who succeeded Emperor Kammu (782-806), and also generations of Japanese, not ...

See also:

Heian period, Heian period - History, Heian period - Developments in Buddhism, Heian period - Heian period literature, Heian period - Heian period economics, Heian period - The Fujiwara Regency, Heian period - The Rise of the military class

Read more here: » Heian period: Encyclopedia II - Heian period - Developments in Buddhism

The function is an example of a cubic function with leading coefficient −7 and constant coefficient 3. ...

See also:

Polynomial, Polynomial - Elementary properties of polynomials, Polynomial - More advanced examples of polynomials, Polynomial - History, Polynomial - Polynomial functions, Polynomial - Graphs, Polynomial - End behavior, Polynomial - Number of x-intercepts, Polynomial - Number of turning points, Polynomial - Examples, Polynomial - Notes, Polynomial - Roots, Polynomial - Numerical analysis, Polynomial - Polynomials and calculus, Polynomial - Evaluation of polynomials, Polynomial - Finding roots, Polynomial - Several variables, Polynomial - Abstract algebra, Polynomial - Divisibility, Polynomial - More variables

Read more here: » Polynomial: Encyclopedia II - Polynomial - Examples

is a horizontal line with y-intercept a0. The graph of a degree 1 polynomial function (or linear function) f(x) = a0 + a1x, where is an oblique line with y-intercept a0 and slope See also:

Polynomial, Polynomial - Elementary properties of polynomials, Polynomial - More advanced examples of polynomials, Polynomial - History, Polynomial - Polynomial functions, Polynomial - Graphs, Polynomial - End behavior, Polynomial - Number of x-intercepts, Polynomial - Number of turning points, Polynomial - Examples, Polynomial - Notes, Polynomial - Roots, Polynomial - Numerical analysis, Polynomial - Polynomials and calculus, Polynomial - Evaluation of polynomials, Polynomial - Finding roots, Polynomial - Several variables, Polynomial - Abstract algebra, Polynomial - Divisibility, Polynomial - More variables

Read more here: » Polynomial: Encyclopedia II - Polynomial - Graphs

For given constants (i.e., numbers) a0, …, an in some field (possibly but not limited to R or C) with an non-zero, for n > 0, then a polynomial (function) of degree n is a function of the form More concisely, a polynomial function can be written in sigma notation as The constants a0, …, an are called the co ...

See also:

Polynomial, Polynomial - Elementary properties of polynomials, Polynomial - More advanced examples of polynomials, Polynomial - History, Polynomial - Polynomial functions, Polynomial - Graphs, Polynomial - End behavior, Polynomial - Number of x-intercepts, Polynomial - Number of turning points, Polynomial - Examples, Polynomial - Notes, Polynomial - Roots, Polynomial - Numerical analysis, Polynomial - Polynomials and calculus, Polynomial - Evaluation of polynomials, Polynomial - Finding roots, Polynomial - Several variables, Polynomial - Abstract algebra, Polynomial - Divisibility, Polynomial - More variables

Read more here: » Polynomial: Encyclopedia II - Polynomial - Polynomial functions

Also in linear algebra, the characteristic polynomial of a square matrix encodes several important properties of the matrix. In graph theory the chromatic polynomial of a graph encodes the different ways to vertex color the graph using x colors. In abstract algebra, one may define polynomials with coefficients in any ring. In knot theory the Alexander polynomial, the Jones ...

See also:

Polynomial, Polynomial - Elementary properties of polynomials, Polynomial - More advanced examples of polynomials, Polynomial - History, Polynomial - Polynomial functions, Polynomial - Graphs, Polynomial - End behavior, Polynomial - Number of x-intercepts, Polynomial - Number of turning points, Polynomial - Examples, Polynomial - Notes, Polynomial - Roots, Polynomial - Numerical analysis, Polynomial - Polynomials and calculus, Polynomial - Evaluation of polynomials, Polynomial - Finding roots, Polynomial - Several variables, Polynomial - Abstract algebra, Polynomial - Divisibility, Polynomial - More variables

Read more here: » Polynomial: Encyclopedia II - Polynomial - More advanced examples of polynomials

A root or zero of a polynomial f is a number ζ so that f(ζ) = 0. The fundamental theorem of algebra states that a polynomial of degree n over the complex numbers has exactly n complex roots (not necessarily distinct ones). Therefore a polynomial can be factorized as where each ζi i ...

See also:

Polynomial, Polynomial - Elementary properties of polynomials, Polynomial - More advanced examples of polynomials, Polynomial - History, Polynomial - Polynomial functions, Polynomial - Graphs, Polynomial - End behavior, Polynomial - Number of x-intercepts, Polynomial - Number of turning points, Polynomial - Examples, Polynomial - Notes, Polynomial - Roots, Polynomial - Numerical analysis, Polynomial - Polynomials and calculus, Polynomial - Evaluation of polynomials, Polynomial - Finding roots, Polynomial - Several variables, Polynomial - Abstract algebra, Polynomial - Divisibility, Polynomial - More variables

Read more here: » Polynomial: Encyclopedia II - Polynomial - Roots

Polynomial - Polynomials and calculus. One important aspect of calculus is the project of analyzing complicated functions by means of approximating them with polynomials. The culmination of these efforts is Taylor's theorem, which roughly states that every differentiable function locally looks like a polynomial, and the Stone-Weierstrass theorem, which states that every continuous function defined on a compact interval of the real axis can be approximated on the whole interval as closely as desired by a polynomial. Polynomials a ...

See also:

Polynomial, Polynomial - Elementary properties of polynomials, Polynomial - More advanced examples of polynomials, Polynomial - History, Polynomial - Polynomial functions, Polynomial - Graphs, Polynomial - End behavior, Polynomial - Number of x-intercepts, Polynomial - Number of turning points, Polynomial - Examples, Polynomial - Notes, Polynomial - Roots, Polynomial - Numerical analysis, Polynomial - Polynomials and calculus, Polynomial - Evaluation of polynomials, Polynomial - Finding roots, Polynomial - Several variables, Polynomial - Abstract algebra, Polynomial - Divisibility, Polynomial - More variables

Read more here: » Polynomial: Encyclopedia II - Polynomial - Numerical analysis