In psychology, compensation is a way people hide something that they cannot do well, by doing something else really well. They may not even be aware that they are doing it. Compensation can hide real or imagined problems and personal or physical inferiority. However it does not really solve the cause of the problem. Positive compensations may help a person overcome difficulties. But negative compensations do not, and can even make the problems worse.
There are two kinds of negative compensation:
A well-known example of failing overcompensation, can be seen in people going through a mid life crisis. Approaching midlife many people (especially men) lack the energy to maintain their psychological defenses, including their compensatory acts.
Alfred Adler, founder of individual psychology, used the word compensation in relation inferiority feelings. In his book "Study of Organ Inferiority and Its Physical Compensation" (1907) he wrote if a person feels inferior (weak) then they will (usually) try to compensate for it somewhere else.
Adler's reason to explore this was from his own experience. He was shy and yet he pushed himself to give lectures.
Adler also "transferred" this idea of compensation to psychic training.
Narcissistic people, by compensation theory, hide their feelings of low self-esteem by:
Narcissistic children try to compensate for their jealousy and anger by fantasizing about:
Christopher Lasch, an American historian and social critic wrote in his book "The Culture of Narcissism" (1979) that North American society in the 1970s was narcissistic. The narcissistic society:
Therefore, it is "fascinated" with fame.
Consumption can be an example of compensation (see study by Allison J. Pugh: "From compensation to ‘childhood wonder’"). Examples:
Shingle beach
A shingle beach (or pebble beach) is a beach made of pebbles, not sand. The stones may be from diameter. These beaches strongly resist erosion by waves.
This beach landform is quite common in Western Europe, because the pebbles come from the lines of flint in the chalk. Chesil Bank is an example. When the chalk erodes, the tough flint stays as pebbles.
Other examples are found in Bahrain, North America and in a number of other places. On the east coast of New Zealand's South Island, there are mixed shingle and sand fans in the braided rivers.
Puddingstone
Puddingstone is a kind of sedimentary rock. It is formed when pebbles, perhaps from a shingle beach, are cemented together by silica coming out of solution.
The rock which forms is extremely hard and resistant to wear. It is used for grindstones and building. The pebbles are usually made of flint, and both they and the matrix are made of silica.
Eponym
An eponym is a person's name that is used to name something else. This is common in science, such as asteroid, politics and medicine.
Injective function
In mathematics, a injective function is a function "f" : "A" → "B" with the following property. For every element "b" in the codomain "B", there is at most one element "a" in the domain "A" such that "f"("a")="b", or equivalently, distinct elements in the domain map to distinct elements in the codomain.
The term "injection" and the related terms "surjection" and "bijection" were introduced by Nicholas Bourbaki. In the 1930s, he and a group of other mathematicians published a series of books on modern advanced mathematics.
An injective function is often called a 1-1 (read "one-to-one") function. However, this is to be distinguish from a 1-1 correspondence, which is a bijective function (both injective and surjective).
Formally:
The element formula_5 is called a pre-image of the element formula_6 if formula_7 . Injections have one or none pre-images for every element "b" in "B".
Cardinality is the "number" of elements in a set. The cardinality of "A"={X,Y,Z,W} is 4. This is written as #"A"=4.
If the cardinality of the codomain is less than the cardinality of the domain, then the function cannot be an injection. For example, there is no injection from 6 elements to 5 elements, since it is impossible to map 6 elements to 5 elements without a duplicate.
Let "f"("x"):ℝ→ℝ be a real-valued function "y"="f"("x") of a real-valued argument "x". (This means both the input and output are real numbers.)
Example: The linear function of a slanted line is 1-1. That is, "y"="ax"+"b" where "a"≠0 is an injection. (It is also a surjection and thus a bijection.)
Example: The polynomial function of third degree: "f"("x")="x" is an injection. However, the polynomial function of third degree: "f"("x")="x" –3x is not an injection.
Example: The quadratic function "f"("x") = "x" is not an injection.
Note: One can make a non-injective function into an injective function by eliminating part of the domain. We call this restricting the domain. For example, restrict the domain of "f"("x")="x"² to non-negative numbers (positive numbers and zero). Define
This function is now an injection. (See also restriction of a function.)
Example: The exponential function "f"("x") = 10 is an injection. (However, it is not a surjection.)
Note: The fact that an exponential function is injective can be used in calculations.
Example: The logarithmic function base 10 "f"("x"):<nowiki>(0,+∞)</nowiki>→ℝ defined by "f"("x")=log("x") or "y"=log("x") is an injection (and a surjection). (This is the inverse function of 10.)
Example: The function "f":ℕ→ℕ that maps every natural number "n" to "2n" is an injection. Every even number has exactly one pre-image. Every odd number has no pre-image.
Lorient
Lorient is a commune and seaport in the Morbihan department in Brittany.
Lorient is twinned with:
Bongo
Bongo may mean:
In nature:
In geography:
In entertainment:
In vehicles:
In fiction:
In other uses:
Surjective function
In mathematics, a surjective or onto function is a function "f" : "A" → "B" with the following property. For every element "b" in the codomain "B", there is at least one element "a" in the domain "A" such that "f"("a")="b". This means that no element in the codomain is unmapped, and that the range and codomain of "f" are the same set.
The term "surjection" and the related terms "injection" and "bijection" were introduced by the group of mathematicians that called itself . In the 1930s, this group of mathematicians published a series of books on modern advanced mathematics. The French prefix "sur" means "above" or "onto" and was chosen since a surjective function maps its domain on to its codomain.
Formally:
where the element formula_4 is called the image of the element formula_5, and the element formula_5 a pre-image of the element formula_4.
The formal definition can also be interpreted in two ways:
A pre-image does "not" have to be unique. In the top image, both {X} and {Y} are pre-images of the element {1}. It is only important that there be at least one pre-image.
Let "f"("x"):ℝ→ℝ be a real-valued function "y"="f"("x") of a real-valued argument "x". (This means both the input and output are numbers.)
Finding a pre-image "x" for a given "y" is equivalent to either question:
In mathematics, we can find exact (analytic) roots only of polynomials of first, second (and third) degree. We find roots of all other functions approximately (numerically). This means a formal proof of surjectivity is rarely direct. So the discussions below are informal.
Example: The linear function of a slanted line is "onto". That is, "y"="ax"+"b" where "a"≠0 is a surjection. (It is also an injection and thus a bijection.)
Example: The cubic polynomial (of third degree) "f"("x")="x"-3"x" is a surjection.
Example: The quadratic function "f"("x") = "x" is not a surjection. There is no "x" such that "x" = −1. The range of "x"² is <nowiki>[0,+∞) </nowiki>, that is, the set of non-negative numbers. (Also, this function is not an injection.)