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Superadditive Function

Superadditive function

A sequence , n ≥ 1, is called superadditive if it satisfies the inequality ::

(1) \qquad a_ \geq a_n+a_m

for all m and n. The major reason for the use of superadditive sequences is the following lemma due to Fekete. :Lemma: For every superadditive sequence , n ≥ 1, the limit lim a_n/n exists and equal to sup a_n/n. Similarly, a function f(x) is superadditive if ::

f(x+y) \geq f(x)+f(y)

for all x and y in the domain of f. The analogue of Fekete lemma holds for superadditive functions as well. There are extensions of Fekete's lemma that do not require equation (1) to hold for all m and n. There are also results that allow one to deduce the rate of convergence to the limit whose existence is stated in Fekete's lemma if some kind of both superadditivity and subadditivity is present. A good exposition of this topic may be found in [2].

References

# # ---- Category:Mathematical analysis Category:Sequences and series

Sequence

:This is a page about mathematics. For other usages of "sequence", see: sequence (non-mathematical). In mathematics, a sequence is a list of objects (or events) arranged in a "linear" fashion, such that the order of the members is well defined and significant. For example, (C,Y,R) is a sequence of letters that differs from (Y,C,R), as the ordering matters. Sequences can be finite, as in this example, or infinite, such as the sequence of all even positive integers (2,4,6,...). The members of a sequence are also called its elements or terms, and the number of terms (possibly infinite) is called the length of the sequence.

Examples and notation

There are various and quite different notions of sequences in mathematics, some of which (e.g., exact sequence) are not covered by the notations introduced below. A sequence may be denoted (a₁,a₂, ...). For shortness, the notation (a_n) is also used. A more formal definition of a finite sequence with terms in a set S is a function from to S for some n≥0. An infinite sequence in S is a function from (the set of natural numbers) to S. A finite sequence is also called an n-tuple. A function from all integers into a set is sometimes called a bi-infinite sequence, since it may be thought of as a sequence indexed by negative integers grafted onto a sequence indexed by positive integers. Finite sequences include the null sequence ( ) that has no elements.

Types and properties of sequences

A subsequence of a given sequence is a sequence formed from the given sequence by deleting some of the elements without disturbing the relative positions of the remaining elements. If the terms of the sequence are a subset of a ordered set, then a monotonically increasing sequence is one for which each term is greater than or equal to the term before it; if each term is strictly greater than the one preceding it, the sequence is called strictly monotonically increasing. A monotonically decreasing sequence is defined similarly. Any sequence fulfilling the monotonicity property is called monotonic or monotone. This is a special case of the more general notion of monotonic function. The terms non-decreasing and non-increasing avoid any possible confusion with strictly increasing and strictly decreasing, respectively, see also strict. If the terms of a sequence are integers, then the sequence is an integer sequence. If the terms of a sequence are polynomials, then the sequence is a polynomial sequence. If S is endowed with a topology, then it is possible to talk about convergence of an infinite sequence in S. This is discussed in detail in the article about limits.

Sequences in analysis

In analysis, when talking about sequences, one usually understands sequences of the form :

(x_1, x_2, x_3, ...)

(x_0, x_1, x_2, ...)

i.e. infinite sequences of elements indexed by natural numbers. (It may be convenient to have the sequence start with an index different from 1 or 0. For example, the sequence defined by x_n = 1/log(n) would be defined only for

n\ge2

. When talking about such infinite sequences, it is usually sufficient (and does not change much for most considerations) to assume that the members of the sequence are defined at least for all indices large enough, that is, greater than some given

N

.) The most elementary type of sequences are numerical ones, that is, sequences of real or complex numbers. This type can be generalized to sequences of elements of some vector space. In analysis, the vector spaces considered are often function spaces. Even more generally, one can study sequences with elements in some topological space.

Series

The sum of a sequence is a series. More precisely, if

(x_1, x_2, x_3, ...)

is a sequence, one considers the sequence of partial sums

(S_1, S_2, S_3, ...)

with :

S_n=x_1+x_2+\dots + x_n=\sum\limits_^x_i.

This new sequence is called a series with the terms

x_1, x_2, x_3, ...

and is denoted as :

\sum\limits_^x_i.

If the sequence of partial sums is convergent, one also uses the infinite sum notation for its limit. For more details, see series.

External links

[http://www.research.att.com/~njas/sequences/index.html The On-Line Encyclopedia of Integer Sequences] Category:Elementary mathematics
- ko:수열 ja:列 (数学)

Inequality

:For the socioeconomic sense, see social inequality. social inequality In mathematics, an inequality is a statement about the relative size or order of two objects. (See also: equality) The notation a b means that a is greater than b. These relations are known as strict inequality; in contrast a ≤ b means that a is less than or equal to b and a ≥ b means that a is greater than or equal to b. If the sense of the inequality is the same for all values of the variables for which its members are defined, then the inequality is called an "absolute" or "unconditonal" inequality. If the sense of an inequality holds only for certain values of the variables involved, but is reversed or destroyed for other values of the variables, it is called a conditional inequality. The sense of an inequality is not changed if both sides are increased or decreased by the same number, or if both sides are multiplied or divided by a positive number; the sense of an inequality is reversed if both members are multiplied or divided by a negative number. Young students sometimes confuse the less-than and greater-than signs, which are mirror images of one another. A commonly taught mnemonic is that the sign represents a hungry alligator that is trying to eat the larger number; thus, it opens towards 8 in both 3 < 8 and 8 > 3.[http://mathforum.org/library/drmath/view/58428.html] The notation a >> b means that a is "much greater than" b. What this means exactly can vary, meaning anything from a factor of 100 difference to a ten order of magnitude difference. It is used in relation to equations in which a much greater value will cause the output of the equation to converge on a certain result.

Properties

Inequalities are governed by the following properties:

Trichotomy

The trichotomy property states:
- For any real numbers, "a" and "b", only one of the following is true:
- a b

Transitivity

The transitivity of inequalities states:
- For any real numbers, "a", "b", "c":
- If a > b and b > c; then a > c
- If a < b and b < c; then a < c

Reversal

The inequality relations are mirror images in the sense that:
- For any real numbers, "a" and "b":
- If a > b then b < a
- If a a

Addition and subtraction

The properties which deal with addition and subtraction states:
- For any real numbers, "a", "b", "c":
- If a > b; then a + c > b + c and a − c > b − c
- If a < b; then a + c < b + c and a − c < b − c

Multiplication and division

The properties which deal with multiplication and division state:
- For any real numbers, "a", "b", and "c":
- If c is positive and a > b; then a × c > b × c and a / c > b / c
- If c is positive and a < b; then a × c b; then a × c b × c and a / c > b / c

Applying a function to both sides

Any strictly monotonically increasing function may be applied to both sides of an inequality and it will still hold.

Chained notation

The notation a < b < c stands for a < b and b < c (from the transitivity above follows also a < c). Obviously, by the above laws, one can add/subtract the same number to all three terms, or multiply/divide all three terms by same nonzero number and reverse all inequalities according to sign. But care must be taken so that you really use the same number in all cases, eg. a < b + e < c is equivalent to a - e < b < c - e. This notation can be generalized to any number of terms, for instance a₁ ≤ a₂ ≤ ... ≤ a_n means that a_i ≤ a_j for any 1 ≤ i ≤ j ≤ n.

Well-known inequalities

See also list of inequalities. Mathematicians often use inequalities to bound quantities for which exact formulas cannot be computed easily. Some inequalities are used so often that they have names:
- Azuma's inequality
- Bernoulli inequality
- Boole's inequality
- Cauchy-Schwarz inequality
- Chebyshev's inequality
- Chernoff's inequality
- Cramér-Rao inequality
- Hoeffding's inequality
- Hölder's inequality
- Inequality of arithmetic and geometric means
- Jensen's inequality
- Markov's inequality
- Minkowski's inequality
- Nesbitt's inequality
- Pedoe's inequality
- Triangle inequality

References

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- Category:Inequalities Category:Elementary algebra ko:부등식 ja:不等式

Lemma (mathematics)

In mathematics, a lemma is a proven proposition which is used as a stepping stone to a larger result rather than an independent statement, in and of itself. A good stepping stone leads to many others, so some of the most powerful results in mathematics are known as lemmas: Zorn's lemma, Bézout's lemma, Gauss lemma, Fatou's lemma, Nakayama lemma, etc. There is no inherent distinction between a lemma and a theorem. The Greek word "lemma" (λημμα) itself means "anything which is received, such as a gift, profit, or a bribe." According to [1], the plural "lemmas" is commonly used. The correct Greek plural of lemma, however, is lemmata (λημματα). Both forms are used in English, although users of lemmas should be aware that Classical purists will consider their usage wrong, and users of lemmata should be aware that many readers may be unfamiliar with the term or consider its use unnecessarily pedantic.

References

# N. Higham, Handbook of writing for the mathematical sciences, Society for Industrial and Applied Mathematics, 1998 (p. 16) Category:Mathematical terminology
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Function (mathematics)

In mathematics, a function is a relation, such that each element of a set (the domain) is associated with a unique element of another (possibly the same) set (the codomain, not to be confused with the range). The concept of a function is fundamental to virtually every branch of mathematics and every quantitative science. The terms function, mapping, map and transformation are usually used synonymously. The term operation is frequently used for binary functions; functions whose domain is a set of functions, or a vector space, are often called operators (see also operator (programming)).

Intuitive introduction

Essentially, a function is a "rule" or procedure that assigns an "output" value to each given "input" value. The following are examples of functions:
- In a group of people, each person has a favorite colour—from the set of red, orange, yellow, green, cyan, blue, indigo, or violet. Here, the input is the person, and the output is one of the 8 colours. The favorite colour is a function of the person. For example, John has favorite colour red, while Kim has favorite colour violet. Note that more than one person may be associated with a given colour (e.g., John and Kim may both like red), but one person cannot have more or less than one favorite color.
- A stone is dropped from different stories of a tall building. The dropped stone may take 2 seconds to fall from the second story, and 4 seconds to fall from the 8th story. Here, the input is the story, and the output is the number of seconds. The relevant function describes the relationship between the time it takes the stone to reach the ground and the story. (See acceleration) The "rule" defining a function can be specified by a formula, a relationship, or simply a table listing the outputs against inputs. The most important feature of a function is that it is consistent, or deterministic, always producing the same output from a given input. In this way, a function may be thought of as a mechanism or "machine" (a "black box") consistently converting a given valid input into its unique associated output. In certain technical contexts, the input is often called the argument of the function, and the output the value of the function. A very common type of function occurs when the argument (input) and the value (output) are both numbers, the functional relationship is expressed by a formula, and the value (output) of the function is obtained by direct substitution of the argument into the formula. Consider for example :

f(x)=x^

which for any number x, assigns to x the associated value the square of x. A straightforward generalization is to allow functions depending on several arguments. For instance, :

g(x,y) = xy

is a function which takes the input, two expressions x and y, and assigns to it its product (output), xy. It might seem that this is not really a function as we described above, because this "rule" depends on two inputs. However, if we think of the two inputs together as a single pair (x, y), then we can interpret g as a function -- the argument (unified single input) is the ordered pair (x, y), and the function value (output) is xy. Such functions whose input consists of ordered pairs are called "binary" or "2-ary". In the sciences, we often encounter functions that are not given by (known) formulas. Consider for instance the temperature distribution on earth over time: this is a function which takes location and time as arguments and gives as output value the temperature at the indicated location at the indicated moment in time. We have seen that the intuitive notion of function is not limited to computations using single numbers and not even limited to computations; the mathematical notion of function is still more general and is not limited to situations involving numbers. Rather, a function links a "domain" (set of inputs) to a "codomain" (set of possible outputs) in such a way that every element of the domain is associated to precisely one element of the codomain. Functions are abstractly defined as certain relations, as will be seen below. Because of this generality, the function concept is fundamental to virtually every branch of mathematics and the quantitative sciences.

History

As a mathematical term, "function" was coined by Leibniz in 1694, to describe a quantity related to a curve, such as a curve's slope or a specific point of a curve. The functions Leibniz considered are today called differentiable functions, and they are the type of function most frequently encountered by nonmathematicians. For this type of function, one can talk about limits and derivatives; both are measurements of the change of output values associated to a change of input values, and these measurements are the basis of calculus. The word function was later used by Euler during the mid-18th century to describe an expression or formula involving various arguments, e.g. f(x) = sin(x) + x³. During the 19th century, mathematicians started to formalize all the different branches of mathematics. Weierstrass advocated building calculus on arithmetic rather than on geometry, which favoured Euler's definition over Leibniz's (see arithmetization of analysis). By broadening the definition of functions, mathematicians were then able to study "strange" mathematical objects such as continuous functions that are nowhere differentiable. These functions were first thought to be only theoretical curiosities, and they were collectively called "monsters" as late as the turn of the 20th century. However, powerful techniques from functional analysis have shown that these functions are in some sense "more common" than differentiable functions. Such functions have since been applied to the modeling of physical phenomena such as Brownian motion. Towards the end of the 19th century, mathematicians started trying to formalize all of mathematics using set theory, and they sought to define every mathematical object as a set. Dirichlet and Lobachevsky independently and almost simultaneously gave the modern "formal" definition of function (see formal definition below). In this definition, a function is a special case of a relation. In most cases of practical interest, however, the differences between the modern definition and Euler's definition are negligible. The notion of function as a rule for computing, rather than a special kind of relation, has been formalized in mathematical logic and theoretical computer science by means of several systems, including the lambda calculus, the theory of recursive functions and the Turing machine.

Formal definition

Formally a function f from a set X to a set Y, written f : X → Y, is an ordered triple (X, Y, G(f)), where G(f) is a subset of the cartesian product X × Y, such that for each x in X, there is a unique y in Y such that the ordered pair (x, y) is in G(f). X is called the domain of f, Y is called the codomain of F, and G(f) is called the graph of f. For each "input value" x in the domain, the corresponding unique "output value" y in the codomain is denoted by f(x). Equivalently a function f can be defined as a relation between X and Y which satisfies: # f is total, or entire: for all x in X, there exists a y in Y such that x f y (x is f-related to y), i.e. for each input value, there is at least one output value in Y. # f is many-to-one, or functional: if x f y and x f z, then y = z. i.e., many input values can be related to one output value, but one input value cannot be related to many output values. A relation between X and Y that satisfies condition (1) is a multivalued function. Every function is a multivalued function, but not every multivalued function is a function. A relation between X and Y that satisfies condition (2) is a partial function. Every function is a partial function, but not every partial function is a function. In this encyclopedia, the term "function" will mean a relation satisfying both conditions (1) and (2), unless otherwise stated. Consider the following three examples:

image:notMap1.png

This relation is total but not many-to-one; the element 3 in X is related to two elements b and c in Y. Therefore, this is a multivalued function, but not a function.

image:notMap2.png

This relation is many-to-one but not total; the element 1 in X is not related to any element of Y. Therefore, this is a partial function, but not a function.

image:mathmap2.png

This relation is both total and many-to-one, and so it is a function from X to Y. Note that the emphasis is on "-to-one" as "many" may actually mean "one". The function can be given explicitly by specifying its graph G(f) = or as :

f(x)=\left\

Domain (mathematics)

In mathematics, the domain of a function is the set of all input values to the function. X, the set of input values, is called the domain of f, and Y, the set of possible output values, is called the codomain. The range of f is the set of all actual outputs . Beware that sometimes the codomain is incorrectly called the range because of a failure to distinguish between possible and actual values. Given a function f : A → B, the set A is called the domain, or domain of definition of f. The set of all values in the codomain that f maps to is called the range of f, written f(A). A well-defined function must map every element of the domain to an element of its codomain. For example, the function f defined by : f(x) = 1/x has no value for f(0). Thus, the set R of real numbers cannot be its domain. In cases like this, the function is usually either defined on R\, or the "gap" is plugged by specifically defining f(0). If we extend the definition of f to : f(x) = 1/x, for x ≠ 0 : f(0) = 0, then f is defined for all real numbers and we can choose its domain to be R. Any function can be restricted to a subset of its domain. The restriction of g : A → B to S, where S ⊆ A, is written g |_S : S → B. Some well-known domains are as follows (note that each successive domain includes those above it):

Category theory

In category theory, instead of functions, one deals with morphisms, which are simply arrows from one object to another. The domain of any morphism is then simply the object where the arrow starts. In this context, many set theoretic ideas about domains have to be abandoned, or at least formulated more abstractly. For example, the notion of restricting a morphism to a subset of its domain must be modified. See subobject for more.

Complex analysis

In complex analysis, a domain is an open connected subset of the complex numbers.

Subadditive

A function f(x) is subadditive if ::

f(x+y)\leq f(x)+f(y)

for all x and y in the domain of f. A sequence , n ≥ 1, is called subadditive if it satisfies the inequality ::

(1) \qquad a_\leq a_n+a_m

for all m and n. The major reason for use of subadditive sequences is the following lemma due to M. Fekete. :Lemma: For every subadditive sequence , n ≥ 1, the limit lim a_n/n exists and is equal to inf a_n/n. The analogue of Fekete's lemma holds for subadditive functions as well. There are extensions of Fekete's lemma that do not require equation (1) to hold for all m and n. There are also results that allow one to deduce the rate of convergence to the limit whose existence is stated in Fekete's lemma if some kind of both superadditivity and subadditivity is present. See also: triangle inequality

References

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Note

A good exposition of this topic may be found in Steele's Probability theory and combinatorial optimization given in the references. Category:Mathematical analysis Category:Sequences and series

Category:Sequences and series

Category:Calculus Category:Mathematical analysis In mathematics, a sequence is a list of objects (or events) which have been ordered in a sequential fashion; such that each member either comes before, or after, every other member. More formally, a sequence is a function with a domain equal to the set of positive integers. A series is a sum of a sequence of terms. That is, a series is a list of numbers with addition operations between them.

Vejle-Vandel Jernbane

Vejle-Vandel-Grindsted Jernbane VVGJ var en lokal jernbane, der gik fra Vejle til Vandel og Grindsted. Vejle-Vandel Jernbane, VVJ, åbnet den 10. september 1897 var et af resultaterne af "jernbanekrigen" mellem Vejle og Kolding, der kæmpede om det vestlige opland og havde planer om en jernbane til Randbøl/Vandel området. Det blev Vejle der i jernbaneloven af 1894 fik banen til Vandel. Kolding fik i stedet en bane til Egtved. Kort efter Vejle-Vandel Jernbanes åbning fremkom planer om at forlænge banen mod Grindsted, og den 21. maj 1914 åbnedes forlængelsen, og VVJ ændrede navn til VVGJ, Vejle-Vandel-Grindsted Jernbane. VVGJ var en typisk oplandsbane, med en god godstrafik i de første år, særligt til De Forenede Papirfabrikkers to fabrikker i Haraldskær Fabrik og Vingsted Mølle samt mergel og kalk til jordforbedring af den flade hede mellem Vandel og Grindsted. Under 2. verdenskrig var der især store transporter for den tyske besættelsesmagt som prægede banens trafik, særligt til skydebanerne i Vingsted Mølle og i forbindelse med anlægget af flyvepladsen i Vandel. Hertil kom transport af store mængder tørv og brunkul til de østjyske byer. Efter krigen var banen stærkt nedslidt, og trods der foretoges moderniseringer med to skinnebusser fabrikeret på Scandia-fabrikkerne i Randers faldt persontrafikken og godstrafikken, og den nedlagdes helt 31. marts 1957. Selvom banen gik mod vest - set fra Vejle - førtes banen ind på Vejle Nord (nedrevet i 1984) og var beliggende tæt på Grejsdalsvej, hvor banen fra Vejle mod Herning også løber. Her havde banen driftfællesskab med Vejle-Give Jernbane indtil 1914 og herefter stod DSB for driften. Så tog den ellers et kraftigt sving igennem den indre by og krydsede adskillige travle gader, før den løb mod vest. Herefter fulgte følgende standsningssteder: Svanholm trinbræt, Trædballe trinbræt, Skibet station, Kvak Mølle trinbræt, Haraldskær Fabrik station, Teglgård trinbræt, Vingsted Mølle station, Kjeldkær trinbræt, Ravning station, Søgård trinbræt, Lihmskov station, Bindeballe station, Randbøl station, Vandel station, Østerby station, Billund station, Krog trinbræt, Løvlund station, Hinnum trinbræt og endelig Grindsted station. Vejle-Vandel 10. september 1897 - 31. marts 1957
- Længde: 28,3 km
- Sporvidde: 1435 mm
- Anlægslov af 8. maj 1894
- Koncession givet 6. november 1894 Vandel-Grindsted 21. maj 1914 - 31. marts 1957
- længde: 18,9 km
- Sporvidde: 1435 mm
- Anlægslov af 27. maj 1908
- Koncession givet 20. september 1912

Se også:

- Danske jernbaner
- Jernbane
- Jernbaneulykker

Eksterne henvisninger/Kilder

- Vandelbanen 1897-1957 / Viggo F. Hejlesen og Vigand Rasmussen
- http://www.signalpost.dk/vvgj.htm Kategori:Danske jernbaneselskaber Kategori:Danske jernbanelinjer Kategori:DK5 65.84

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Cañete la real
La villa malagueña de Cañete la Real se encuentra situada en el límite noroccidental de la provincia. Es un municipio fronterizo entre la Serranía de Ronda y la comarca de Antequera, y se eleva 742 metros sobre el nivel del mar. El principal río del municipio es el Guadalteba o río de Ortegícar, que atraviesa el sur del término municipal y abastece el embalse del mismo nombre. El

Superadditive function

References

Sequence

Examples and notation

Types and properties of sequences

Sequences in analysis

Series

See also

External links

Inequality

Properties

Trichotomy

Transitivity

Reversal

Addition and subtraction

Multiplication and division

Applying a function to both sides

Chained notation

Well-known inequalities

See also

References

Lemma (mathematics)

See also

References

Function (mathematics)

Intuitive introduction

History

Formal definition

Domain (mathematics)

Category theory

Complex analysis

See also

Subadditive

References

Note

Category:Sequences and series

Vejle-Vandel Jernbane

Se også:

Eksterne henvisninger/Kilder

Historia