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So when is an integer, then and is a mode. In the case that , then only is a mode.
Median
In general, there is no single formula to find the median for a binomial distribution, and it may even be non-unique. However, several special results have been established:
If is an integer, then the mean, median, and mode coincide and equal .
Any median must lie within the interval .
A median cannot lie too far away from the mean: .
The median is unique and equal to when (except for the case when and is odd).
When is a rational number (with the exception of \ and odd) the median is unique.
When and is odd, any number in the interval is a median of the binomial distribution. If and is even, then is the unique median.
Tail bounds
For , upper bounds can be derived for the lower tail of the cumulative distribution function , the probability that there are at most successes. Since , these bounds can also be seen as bounds for the upper tail of the cumulative distribution function for .
Hoeffding's inequality yields the simple bound
which is however not very tight. In particular, for , we have that (for fixed , with ), but Hoeffding's bound evaluates to a positive constant.
A sharper bound can be obtained from the Chernoff bound:
where is the relative entropy (or Kullback-Leibler divergence) between an -coin and a -coin (i.e. between the and distribution):
Asymptotically, this bound is reasonably tight; see for details.
One can also obtain lower bounds on the tail , known as anti-concentration bounds. By approximating the binomial coefficient with Stirling's formula it can be shown that
which implies the simpler but looser bound
For and for even , it is possible to make the denominator constant:
Statistical inference
Estimation of parameters | Binomial distribution | Wikipedia | 393 | 3876 | https://en.wikipedia.org/wiki/Binomial%20distribution | Mathematics | Statistics and probability | null |
When is known, the parameter can be estimated using the proportion of successes:
This estimator is found using maximum likelihood estimator and also the method of moments. This estimator is unbiased and uniformly with minimum variance, proven using Lehmann–Scheffé theorem, since it is based on a minimal sufficient and complete statistic (i.e.: ). It is also consistent both in probability and in MSE. This statistic is asymptotically normal thanks to the central limit theorem, because it is the same as taking the mean over Bernoulli samples. It has a variance of , a property which is used in various ways, such as in Wald's confidence intervals.
A closed form Bayes estimator for also exists when using the Beta distribution as a conjugate prior distribution. When using a general as a prior, the posterior mean estimator is:
The Bayes estimator is asymptotically efficient and as the sample size approaches infinity (), it approaches the MLE solution. The Bayes estimator is biased (how much depends on the priors), admissible and consistent in probability. Using the Bayesian estimator with the Beta distribution can be used with Thompson sampling.
For the special case of using the standard uniform distribution as a non-informative prior, , the posterior mean estimator becomes:
(A posterior mode should just lead to the standard estimator.) This method is called the rule of succession, which was introduced in the 18th century by Pierre-Simon Laplace.
When relying on Jeffreys prior, the prior is , which leads to the estimator:
When estimating with very rare events and a small (e.g.: if ), then using the standard estimator leads to which sometimes is unrealistic and undesirable. In such cases there are various alternative estimators. One way is to use the Bayes estimator , leading to:
Another method is to use the upper bound of the confidence interval obtained using the rule of three:
Confidence intervals for the parameter p
Even for quite large values of n, the actual distribution of the mean is significantly nonnormal. Because of this problem several methods to estimate confidence intervals have been proposed. | Binomial distribution | Wikipedia | 469 | 3876 | https://en.wikipedia.org/wiki/Binomial%20distribution | Mathematics | Statistics and probability | null |
In the equations for confidence intervals below, the variables have the following meaning:
n1 is the number of successes out of n, the total number of trials
is the proportion of successes
is the quantile of a standard normal distribution (i.e., probit) corresponding to the target error rate . For example, for a 95% confidence level the error = 0.05, so = 0.975 and = 1.96.
Wald method
A continuity correction of may be added.
Agresti–Coull method
Here the estimate of is modified to
This method works well for and . See here for . For use the Wilson (score) method below.
Arcsine method
Wilson (score) method
The notation in the formula below differs from the previous formulas in two respects:
Firstly, has a slightly different interpretation in the formula below: it has its ordinary meaning of 'the th quantile of the standard normal distribution', rather than being a shorthand for 'the th quantile'.
Secondly, this formula does not use a plus-minus to define the two bounds. Instead, one may use to get the lower bound, or use to get the upper bound. For example: for a 95% confidence level the error = 0.05, so one gets the lower bound by using , and one gets the upper bound by using .
Comparison
The so-called "exact" (Clopper–Pearson) method is the most conservative. (Exact does not mean perfectly accurate; rather, it indicates that the estimates will not be less conservative than the true value.)
The Wald method, although commonly recommended in textbooks, is the most biased.
Related distributions
Sums of binomials
If and are independent binomial variables with the same probability , then is again a binomial variable; its distribution is :
A Binomial distributed random variable can be considered as the sum of Bernoulli distributed random variables. So the sum of two Binomial distributed random variables and is equivalent to the sum of Bernoulli distributed random variables, which means . This can also be proven directly using the addition rule.
However, if and do not have the same probability , then the variance of the sum will be smaller than the variance of a binomial variable distributed as .
Poisson binomial distribution
The binomial distribution is a special case of the Poisson binomial distribution, which is the distribution of a sum of independent non-identical Bernoulli trials .
Ratio of two binomial distributions | Binomial distribution | Wikipedia | 508 | 3876 | https://en.wikipedia.org/wiki/Binomial%20distribution | Mathematics | Statistics and probability | null |
This result was first derived by Katz and coauthors in 1978.
Let and be independent. Let .
Then log(T) is approximately normally distributed with mean log(p1/p2) and variance .
Conditional binomials
If X ~ B(n, p) and Y | X ~ B(X, q) (the conditional distribution of Y, given X), then Y is a simple binomial random variable with distribution Y ~ B(n, pq).
For example, imagine throwing n balls to a basket UX and taking the balls that hit and throwing them to another basket UY. If p is the probability to hit UX then X ~ B(n, p) is the number of balls that hit UX. If q is the probability to hit UY then the number of balls that hit UY is Y ~ B(X, q) and therefore Y ~ B(n, pq).
Since and , by the law of total probability,
Since the equation above can be expressed as
Factoring and pulling all the terms that don't depend on out of the sum now yields
After substituting in the expression above, we get
Notice that the sum (in the parentheses) above equals by the binomial theorem. Substituting this in finally yields
and thus as desired.
Bernoulli distribution
The Bernoulli distribution is a special case of the binomial distribution, where . Symbolically, has the same meaning as . Conversely, any binomial distribution, , is the distribution of the sum of independent Bernoulli trials, , each with the same probability .
Normal approximation | Binomial distribution | Wikipedia | 332 | 3876 | https://en.wikipedia.org/wiki/Binomial%20distribution | Mathematics | Statistics and probability | null |
If is large enough, then the skew of the distribution is not too great. In this case a reasonable approximation to is given by the normal distribution
and this basic approximation can be improved in a simple way by using a suitable continuity correction.
The basic approximation generally improves as increases (at least 20) and is better when is not near to 0 or 1. Various rules of thumb may be used to decide whether is large enough, and is far enough from the extremes of zero or one:
One rule is that for the normal approximation is adequate if the absolute value of the skewness is strictly less than 0.3; that is, if
This can be made precise using the Berry–Esseen theorem.
A stronger rule states that the normal approximation is appropriate only if everything within 3 standard deviations of its mean is within the range of possible values; that is, only if
This 3-standard-deviation rule is equivalent to the following conditions, which also imply the first rule above.
The rule is totally equivalent to request that
Moving terms around yields:
Since , we can apply the square power and divide by the respective factors and , to obtain the desired conditions:
Notice that these conditions automatically imply that . On the other hand, apply again the square root and divide by 3,
Subtracting the second set of inequalities from the first one yields:
and so, the desired first rule is satisfied,
Another commonly used rule is that both values and must be greater than or equal to 5. However, the specific number varies from source to source, and depends on how good an approximation one wants. In particular, if one uses 9 instead of 5, the rule implies the results stated in the previous paragraphs.
Assume that both values and are greater than 9. Since , we easily have that
We only have to divide now by the respective factors and , to deduce the alternative form of the 3-standard-deviation rule:
The following is an example of applying a continuity correction. Suppose one wishes to calculate for a binomial random variable . If has a distribution given by the normal approximation, then is approximated by . The addition of 0.5 is the continuity correction; the uncorrected normal approximation gives considerably less accurate results. | Binomial distribution | Wikipedia | 454 | 3876 | https://en.wikipedia.org/wiki/Binomial%20distribution | Mathematics | Statistics and probability | null |
This approximation, known as de Moivre–Laplace theorem, is a huge time-saver when undertaking calculations by hand (exact calculations with large are very onerous); historically, it was the first use of the normal distribution, introduced in Abraham de Moivre's book The Doctrine of Chances in 1738. Nowadays, it can be seen as a consequence of the central limit theorem since is a sum of independent, identically distributed Bernoulli variables with parameter . This fact is the basis of a hypothesis test, a "proportion z-test", for the value of using , the sample proportion and estimator of , in a common test statistic.
For example, suppose one randomly samples people out of a large population and ask them whether they agree with a certain statement. The proportion of people who agree will of course depend on the sample. If groups of n people were sampled repeatedly and truly randomly, the proportions would follow an approximate normal distribution with mean equal to the true proportion p of agreement in the population and with standard deviation
Poisson approximation
The binomial distribution converges towards the Poisson distribution as the number of trials goes to infinity while the product converges to a finite limit. Therefore, the Poisson distribution with parameter can be used as an approximation to of the binomial distribution if is sufficiently large and is sufficiently small. According to rules of thumb, this approximation is good if and such that , or if and such that , or if and .
Concerning the accuracy of Poisson approximation, see Novak, ch. 4, and references therein.
Limiting distributions
Poisson limit theorem: As approaches and approaches 0 with the product held fixed, the distribution approaches the Poisson distribution with expected value .
de Moivre–Laplace theorem: As approaches while remains fixed, the distribution of
approaches the normal distribution with expected value 0 and variance 1. This result is sometimes loosely stated by saying that the distribution of is asymptotically normal with expected value 0 and variance 1. This result is a specific case of the central limit theorem.
Beta distribution
The binomial distribution and beta distribution are different views of the same model of repeated Bernoulli trials. The binomial distribution is the PMF of successes given independent events each with a probability of success.
Mathematically, when and , the beta distribution and the binomial distribution are related by a factor of : | Binomial distribution | Wikipedia | 482 | 3876 | https://en.wikipedia.org/wiki/Binomial%20distribution | Mathematics | Statistics and probability | null |
Beta distributions also provide a family of prior probability distributions for binomial distributions in Bayesian inference:
Given a uniform prior, the posterior distribution for the probability of success given independent events with observed successes is a beta distribution.
Computational methods
Random number generation
Methods for random number generation where the marginal distribution is a binomial distribution are well-established.
One way to generate random variates samples from a binomial distribution is to use an inversion algorithm. To do so, one must calculate the probability that for all values from through . (These probabilities should sum to a value close to one, in order to encompass the entire sample space.) Then by using a pseudorandom number generator to generate samples uniformly between 0 and 1, one can transform the calculated samples into discrete numbers by using the probabilities calculated in the first step.
History
This distribution was derived by Jacob Bernoulli. He considered the case where where is the probability of success and and are positive integers. Blaise Pascal had earlier considered the case where , tabulating the corresponding binomial coefficients in what is now recognized as Pascal's triangle. | Binomial distribution | Wikipedia | 227 | 3876 | https://en.wikipedia.org/wiki/Binomial%20distribution | Mathematics | Statistics and probability | null |
In mathematics, a binary relation associates elements of one set called the domain with elements of another set called the codomain. Precisely, a binary relation over sets and is a set of ordered pairs where is in and is in . It encodes the common concept of relation: an element is related to an element , if and only if the pair belongs to the set of ordered pairs that defines the binary relation.
An example of a binary relation is the "divides" relation over the set of prime numbers and the set of integers , in which each prime is related to each integer that is a multiple of , but not to an integer that is not a multiple of . In this relation, for instance, the prime number is related to numbers such as , , , , but not to or , just as the prime number is related to , , and , but not to or .
Binary relations, and especially homogeneous relations, are used in many branches of mathematics to model a wide variety of concepts. These include, among others:
the "is greater than", "is equal to", and "divides" relations in arithmetic;
the "is congruent to" relation in geometry;
the "is adjacent to" relation in graph theory;
the "is orthogonal to" relation in linear algebra.
A function may be defined as a binary relation that meets additional constraints. Binary relations are also heavily used in computer science.
A binary relation over sets and is an element of the power set of Since the latter set is ordered by inclusion (), each relation has a place in the lattice of subsets of A binary relation is called a homogeneous relation when . A binary relation is also called a heterogeneous relation when it is not necessary that .
Since relations are sets, they can be manipulated using set operations, including union, intersection, and complementation, and satisfying the laws of an algebra of sets. Beyond that, operations like the converse of a relation and the composition of relations are available, satisfying the laws of a calculus of relations, for which there are textbooks by Ernst Schröder, Clarence Lewis, and Gunther Schmidt. A deeper analysis of relations involves decomposing them into subsets called concepts, and placing them in a complete lattice. | Binary relation | Wikipedia | 450 | 3931 | https://en.wikipedia.org/wiki/Binary%20relation | Mathematics | Set theory | null |
In some systems of axiomatic set theory, relations are extended to classes, which are generalizations of sets. This extension is needed for, among other things, modeling the concepts of "is an element of" or "is a subset of" in set theory, without running into logical inconsistencies such as Russell's paradox.
A binary relation is the most studied special case of an -ary relation over sets , which is a subset of the Cartesian product
Definition
Given sets and , the Cartesian product is defined as and its elements are called ordered pairs.
A over sets and is a subset of The set is called the or of , and the set the or of . In order to specify the choices of the sets and , some authors define a or as an ordered triple , where is a subset of called the of the binary relation. The statement reads " is -related to " and is denoted by . The or of is the set of all such that for at least one . The codomain of definition, , or of is the set of all such that for at least one . The of is the union of its domain of definition and its codomain of definition.
When a binary relation is called a (or ). To emphasize the fact that and are allowed to be different, a binary relation is also called a heterogeneous relation. The prefix hetero is from the Greek ἕτερος (heteros, "other, another, different").
A heterogeneous relation has been called a rectangular relation, suggesting that it does not have the square-like symmetry of a homogeneous relation on a set where Commenting on the development of binary relations beyond homogeneous relations, researchers wrote, "... a variant of the theory has evolved that treats relations from the very beginning as or , i.e. as relations where the normal case is that they are relations between different sets."
The terms correspondence, dyadic relation and two-place relation are synonyms for binary relation, though some authors use the term "binary relation" for any subset of a Cartesian product without reference to and , and reserve the term "correspondence" for a binary relation with reference to and .
In a binary relation, the order of the elements is important; if then can be true or false independently of . For example, divides , but does not divide .
Operations
Union
If and are binary relations over sets and then is the of and over and . | Binary relation | Wikipedia | 500 | 3931 | https://en.wikipedia.org/wiki/Binary%20relation | Mathematics | Set theory | null |
The identity element is the empty relation. For example, is the union of < and =, and is the union of > and =.
Intersection
If and are binary relations over sets and then is the of and over and .
The identity element is the universal relation. For example, the relation "is divisible by 6" is the intersection of the relations "is divisible by 3" and "is divisible by 2".
Composition
If is a binary relation over sets and , and is a binary relation over sets and then (also denoted by ) is the of and over and .
The identity element is the identity relation. The order of and in the notation used here agrees with the standard notational order for composition of functions. For example, the composition (is parent of)(is mother of) yields (is maternal grandparent of), while the composition (is mother of)(is parent of) yields (is grandmother of). For the former case, if is the parent of and is the mother of , then is the maternal grandparent of .
Converse
If is a binary relation over sets and then is the , also called , of over and .
For example, is the converse of itself, as is , and and are each other's converse, as are and . A binary relation is equal to its converse if and only if it is symmetric.
Complement
If is a binary relation over sets and then (also denoted by ) is the of over and .
For example, and are each other's complement, as are and , and , and , and for total orders also and , and and .
The complement of the converse relation is the converse of the complement:
If the complement has the following properties:
If a relation is symmetric, then so is the complement.
The complement of a reflexive relation is irreflexive—and vice versa.
The complement of a strict weak order is a total preorder—and vice versa.
Restriction
If is a binary homogeneous relation over a set and is a subset of then is the of to over .
If is a binary relation over sets and and if is a subset of then is the of to over and .
If a relation is reflexive, irreflexive, symmetric, antisymmetric, asymmetric, transitive, total, trichotomous, a partial order, total order, strict weak order, total preorder (weak order), or an equivalence relation, then so too are its restrictions. | Binary relation | Wikipedia | 508 | 3931 | https://en.wikipedia.org/wiki/Binary%20relation | Mathematics | Set theory | null |
However, the transitive closure of a restriction is a subset of the restriction of the transitive closure, i.e., in general not equal. For example, restricting the relation " is parent of " to females yields the relation " is mother of the woman "; its transitive closure does not relate a woman with her paternal grandmother. On the other hand, the transitive closure of "is parent of" is "is ancestor of"; its restriction to females does relate a woman with her paternal grandmother.
Also, the various concepts of completeness (not to be confused with being "total") do not carry over to restrictions. For example, over the real numbers a property of the relation is that every non-empty subset with an upper bound in has a least upper bound (also called supremum) in However, for the rational numbers this supremum is not necessarily rational, so the same property does not hold on the restriction of the relation to the rational numbers.
A binary relation over sets and is said to be a relation over and , written if is a subset of , that is, for all and if , then . If is contained in and is contained in , then and are called written . If is contained in but is not contained in , then is said to be than , written For example, on the rational numbers, the relation is smaller than , and equal to the composition .
Matrix representation
Binary relations over sets and can be represented algebraically by logical matrices indexed by and with entries in the Boolean semiring (addition corresponds to OR and multiplication to AND) where matrix addition corresponds to union of relations, matrix multiplication corresponds to composition of relations (of a relation over and and a relation over and ), the Hadamard product corresponds to intersection of relations, the zero matrix corresponds to the empty relation, and the matrix of ones corresponds to the universal relation. Homogeneous relations (when ) form a matrix semiring (indeed, a matrix semialgebra over the Boolean semiring) where the identity matrix corresponds to the identity relation.
Examples
Types of binary relations
Some important types of binary relations over sets and are listed below. | Binary relation | Wikipedia | 435 | 3931 | https://en.wikipedia.org/wiki/Binary%20relation | Mathematics | Set theory | null |
Uniqueness properties:
Injective (also called left-unique): for all and all if and then . In other words, every element of the codomain has at most one preimage element. For such a relation, is called a primary key of . For example, the green and blue binary relations in the diagram are injective, but the red one is not (as it relates both and to ), nor the black one (as it relates both and to ).
Functional (also called right-unique or univalent): for all and all if and then . In other words, every element of the domain has at most one image element. Such a binary relation is called a or . For such a relation, is called of . For example, the red and green binary relations in the diagram are functional, but the blue one is not (as it relates to both and ), nor the black one (as it relates to both and ).
One-to-one: injective and functional. For example, the green binary relation in the diagram is one-to-one, but the red, blue and black ones are not.
One-to-many: injective and not functional. For example, the blue binary relation in the diagram is one-to-many, but the red, green and black ones are not.
Many-to-one: functional and not injective. For example, the red binary relation in the diagram is many-to-one, but the green, blue and black ones are not.
Many-to-many: not injective nor functional. For example, the black binary relation in the diagram is many-to-many, but the red, green and blue ones are not. | Binary relation | Wikipedia | 361 | 3931 | https://en.wikipedia.org/wiki/Binary%20relation | Mathematics | Set theory | null |
Totality properties (only definable if the domain and codomain are specified):
Total (also called left-total): for all there exists a such that . In other words, every element of the domain has at least one image element. In other words, the domain of definition of is equal to . This property, is different from the definition of (also called by some authors) in Properties. Such a binary relation is called a . For example, the red and green binary relations in the diagram are total, but the blue one is not (as it does not relate to any real number), nor the black one (as it does not relate to any real number). As another example, is a total relation over the integers. But it is not a total relation over the positive integers, because there is no in the positive integers such that . However, is a total relation over the positive integers, the rational numbers and the real numbers. Every reflexive relation is total: for a given , choose .
Surjective (also called right-total): for all , there exists an such that . In other words, every element of the codomain has at least one preimage element. In other words, the codomain of definition of is equal to . For example, the green and blue binary relations in the diagram are surjective, but the red one is not (as it does not relate any real number to ), nor the black one (as it does not relate any real number to ). | Binary relation | Wikipedia | 314 | 3931 | https://en.wikipedia.org/wiki/Binary%20relation | Mathematics | Set theory | null |
Uniqueness and totality properties (only definable if the domain and codomain are specified):
A function (also called mapping): a binary relation that is functional and total. In other words, every element of the domain has exactly one image element. For example, the red and green binary relations in the diagram are functions, but the blue and black ones are not.
An injection: a function that is injective. For example, the green relation in the diagram is an injection, but the red one is not; the black and the blue relation is not even a function.
A surjection: a function that is surjective. For example, the green relation in the diagram is a surjection, but the red one is not.
A bijection: a function that is injective and surjective. In other words, every element of the domain has exactly one image element and every element of the codomain has exactly one preimage element. For example, the green binary relation in the diagram is a bijection, but the red one is not.
If relations over proper classes are allowed:
Set-like (also called local): for all , the class of all such that , i.e. , is a set. For example, the relation is set-like, and every relation on two sets is set-like. The usual ordering < over the class of ordinal numbers is a set-like relation, while its inverse > is not.
Sets versus classes
Certain mathematical "relations", such as "equal to", "subset of", and "member of", cannot be understood to be binary relations as defined above, because their domains and codomains cannot be taken to be sets in the usual systems of axiomatic set theory. For example, to model the general concept of "equality" as a binary relation , take the domain and codomain to be the "class of all sets", which is not a set in the usual set theory. | Binary relation | Wikipedia | 416 | 3931 | https://en.wikipedia.org/wiki/Binary%20relation | Mathematics | Set theory | null |
In most mathematical contexts, references to the relations of equality, membership and subset are harmless because they can be understood implicitly to be restricted to some set in the context. The usual work-around to this problem is to select a "large enough" set , that contains all the objects of interest, and work with the restriction instead of . Similarly, the "subset of" relation needs to be restricted to have domain and codomain (the power set of a specific set ): the resulting set relation can be denoted by Also, the "member of" relation needs to be restricted to have domain and codomain to obtain a binary relation that is a set. Bertrand Russell has shown that assuming to be defined over all sets leads to a contradiction in naive set theory, see Russell's paradox.
Another solution to this problem is to use a set theory with proper classes, such as NBG or Morse–Kelley set theory, and allow the domain and codomain (and so the graph) to be proper classes: in such a theory, equality, membership, and subset are binary relations without special comment. (A minor modification needs to be made to the concept of the ordered triple , as normally a proper class cannot be a member of an ordered tuple; or of course one can identify the binary relation with its graph in this context.) With this definition one can for instance define a binary relation over every set and its power set.
Homogeneous relation
A homogeneous relation over a set is a binary relation over and itself, i.e. it is a subset of the Cartesian product It is also simply called a (binary) relation over .
A homogeneous relation over a set may be identified with a directed simple graph permitting loops, where is the vertex set and is the edge set (there is an edge from a vertex to a vertex if and only if ).
The set of all homogeneous relations over a set is the power set which is a Boolean algebra augmented with the involution of mapping of a relation to its converse relation. Considering composition of relations as a binary operation on , it forms a semigroup with involution. | Binary relation | Wikipedia | 434 | 3931 | https://en.wikipedia.org/wiki/Binary%20relation | Mathematics | Set theory | null |
Some important properties that a homogeneous relation over a set may have are:
: for all . For example, is a reflexive relation but > is not.
: for all not . For example, is an irreflexive relation, but is not.
: for all if then . For example, "is a blood relative of" is a symmetric relation.
: for all if and then For example, is an antisymmetric relation.
: for all if then not . A relation is asymmetric if and only if it is both antisymmetric and irreflexive. For example, > is an asymmetric relation, but is not.
: for all if and then . A transitive relation is irreflexive if and only if it is asymmetric. For example, "is ancestor of" is a transitive relation, while "is parent of" is not.
: for all if then or .
: for all or .
: for all if then some exists such that and .
A is a relation that is reflexive, antisymmetric, and transitive. A is a relation that is irreflexive, asymmetric, and transitive. A is a relation that is reflexive, antisymmetric, transitive and connected. A is a relation that is irreflexive, asymmetric, transitive and connected.
An is a relation that is reflexive, symmetric, and transitive.
For example, " divides " is a partial, but not a total order on natural numbers "" is a strict total order on and " is parallel to " is an equivalence relation on the set of all lines in the Euclidean plane.
All operations defined in section also apply to homogeneous relations.
Beyond that, a homogeneous relation over a set may be subjected to closure operations like:
the smallest reflexive relation over containing ,
the smallest transitive relation over containing ,
the smallest equivalence relation over containing .
Calculus of relations
Developments in algebraic logic have facilitated usage of binary relations. The calculus of relations includes the algebra of sets, extended by composition of relations and the use of converse relations. The inclusion meaning that implies , sets the scene in a lattice of relations. But since the inclusion symbol is superfluous. Nevertheless, composition of relations and manipulation of the operators according to Schröder rules, provides a calculus to work in the power set of | Binary relation | Wikipedia | 488 | 3931 | https://en.wikipedia.org/wiki/Binary%20relation | Mathematics | Set theory | null |
In contrast to homogeneous relations, the composition of relations operation is only a partial function. The necessity of matching target to source of composed relations has led to the suggestion that the study of heterogeneous relations is a chapter of category theory as in the category of sets, except that the morphisms of this category are relations. The of the category Rel are sets, and the relation-morphisms compose as required in a category.
Induced concept lattice
Binary relations have been described through their induced concept lattices:
A concept satisfies two properties:
The logical matrix of is the outer product of logical vectors logical vectors.
is maximal, not contained in any other outer product. Thus is described as a non-enlargeable rectangle.
For a given relation the set of concepts, enlarged by their joins and meets, forms an "induced lattice of concepts", with inclusion forming a preorder.
The MacNeille completion theorem (1937) (that any partial order may be embedded in a complete lattice) is cited in a 2013 survey article "Decomposition of relations on concept lattices". The decomposition is
, where and are functions, called or left-total, functional relations in this context. The "induced concept lattice is isomorphic to the cut completion of the partial order that belongs to the minimal decomposition of the relation ."
Particular cases are considered below: total order corresponds to Ferrers type, and identity corresponds to difunctional, a generalization of equivalence relation on a set.
Relations may be ranked by the Schein rank which counts the number of concepts necessary to cover a relation. Structural analysis of relations with concepts provides an approach for data mining.
Particular relations
Proposition: If is a surjective relation and is its transpose, then where is the identity relation.
Proposition: If is a serial relation, then where is the identity relation.
Difunctional
The idea of a difunctional relation is to partition objects by distinguishing attributes, as a generalization of the concept of an equivalence relation. One way this can be done is with an intervening set of indicators. The partitioning relation is a composition of relations using relations Jacques Riguet named these relations difunctional since the composition involves functional relations, commonly called partial functions.
In 1950 Riguet showed that such relations satisfy the inclusion: | Binary relation | Wikipedia | 469 | 3931 | https://en.wikipedia.org/wiki/Binary%20relation | Mathematics | Set theory | null |
In automata theory, the term rectangular relation has also been used to denote a difunctional relation. This terminology recalls the fact that, when represented as a logical matrix, the columns and rows of a difunctional relation can be arranged as a block matrix with rectangular blocks of ones on the (asymmetric) main diagonal. More formally, a relation on is difunctional if and only if it can be written as the union of Cartesian products , where the are a partition of a subset of and the likewise a partition of a subset of .
Using the notation , a difunctional relation can also be characterized as a relation such that wherever and have a non-empty intersection, then these two sets coincide; formally implies
In 1997 researchers found "utility of binary decomposition based on difunctional dependencies in database management." Furthermore, difunctional relations are fundamental in the study of bisimulations.
In the context of homogeneous relations, a partial equivalence relation is difunctional.
Ferrers type
A strict order on a set is a homogeneous relation arising in order theory.
In 1951 Jacques Riguet adopted the ordering of an integer partition, called a Ferrers diagram, to extend ordering to binary relations in general.
The corresponding logical matrix of a general binary relation has rows which finish with a sequence of ones. Thus the dots of a Ferrer's diagram are changed to ones and aligned on the right in the matrix.
An algebraic statement required for a Ferrers type relation R is
If any one of the relations is of Ferrers type, then all of them are.
Contact
Suppose is the power set of , the set of all subsets of . Then a relation is a contact relation if it satisfies three properties:
The set membership relation, "is an element of", satisfies these properties so is a contact relation. The notion of a general contact relation was introduced by Georg Aumann in 1970.
In terms of the calculus of relations, sufficient conditions for a contact relation include
where is the converse of set membership ().
Preorder R\R
Every relation generates a preorder which is the left residual. In terms of converse and complements, Forming the diagonal of , the corresponding row of and column of will be of opposite logical values, so the diagonal is all zeros. Then
, so that is a reflexive relation. | Binary relation | Wikipedia | 484 | 3931 | https://en.wikipedia.org/wiki/Binary%20relation | Mathematics | Set theory | null |
To show transitivity, one requires that Recall that is the largest relation such that Then
(repeat)
(Schröder's rule)
(complementation)
(definition)
The inclusion relation Ω on the power set of can be obtained in this way from the membership relation on subsets of :
Fringe of a relation
Given a relation , its fringe is the sub-relation defined as
When is a partial identity relation, difunctional, or a block diagonal relation, then . Otherwise the operator selects a boundary sub-relation described in terms of its logical matrix: is the side diagonal if is an upper right triangular linear order or strict order. is the block fringe if is irreflexive () or upper right block triangular. is a sequence of boundary rectangles when is of Ferrers type.
On the other hand, when is a dense, linear, strict order.
Mathematical heaps
Given two sets and , the set of binary relations between them can be equipped with a ternary operation where denotes the converse relation of . In 1953 Viktor Wagner used properties of this ternary operation to define semiheaps, heaps, and generalized heaps. The contrast of heterogeneous and homogeneous relations is highlighted by these definitions: | Binary relation | Wikipedia | 249 | 3931 | https://en.wikipedia.org/wiki/Binary%20relation | Mathematics | Set theory | null |
A bijection, bijective function, or one-to-one correspondence between two mathematical sets is a function such that each element of the second set (the codomain) is the image of exactly one element of the first set (the domain). Equivalently, a bijection is a relation between two sets such that each element of either set is paired with exactly one element of the other set.
A function is bijective if and only if it is invertible; that is, a function is bijective if and only if there is a function the inverse of , such that each of the two ways for composing the two functions produces an identity function: for each in and for each in
For example, the multiplication by two defines a bijection from the integers to the even numbers, which has the division by two as its inverse function.
A function is bijective if and only if it is both injective (or one-to-one)—meaning that each element in the codomain is mapped from at most one element of the domain—and surjective (or onto)—meaning that each element of the codomain is mapped from at least one element of the domain. The term one-to-one correspondence must not be confused with one-to-one function, which means injective but not necessarily surjective.
The elementary operation of counting establishes a bijection from some finite set to the first natural numbers , up to the number of elements in the counted set. It results that two finite sets have the same number of elements if and only if there exists a bijection between them. More generally, two sets are said to have the same cardinal number if there exists a bijection between them.
A bijective function from a set to itself is also called a permutation, and the set of all permutations of a set forms its symmetric group.
Some bijections with further properties have received specific names, which include automorphisms, isomorphisms, homeomorphisms, diffeomorphisms, permutation groups, and most geometric transformations. Galois correspondences are bijections between sets of mathematical objects of apparently very different nature. | Bijection | Wikipedia | 460 | 3942 | https://en.wikipedia.org/wiki/Bijection | Mathematics | Functions: General | null |
Definition
For a binary relation pairing elements of set X with elements of set Y to be a bijection, four properties must hold:
each element of X must be paired with at least one element of Y,
no element of X may be paired with more than one element of Y,
each element of Y must be paired with at least one element of X, and
no element of Y may be paired with more than one element of X.
Satisfying properties (1) and (2) means that a pairing is a function with domain X. It is more common to see properties (1) and (2) written as a single statement: Every element of X is paired with exactly one element of Y. Functions which satisfy property (3) are said to be "onto Y " and are called surjections (or surjective functions). Functions which satisfy property (4) are said to be "one-to-one functions" and are called injections (or injective functions). With this terminology, a bijection is a function which is both a surjection and an injection, or using other words, a bijection is a function which is both "one-to-one" and "onto".
Examples
Batting line-up of a baseball or cricket team
Consider the batting line-up of a baseball or cricket team (or any list of all the players of any sports team where every player holds a specific spot in a line-up). The set X will be the players on the team (of size nine in the case of baseball) and the set Y will be the positions in the batting order (1st, 2nd, 3rd, etc.) The "pairing" is given by which player is in what position in this order. Property (1) is satisfied since each player is somewhere in the list. Property (2) is satisfied since no player bats in two (or more) positions in the order. Property (3) says that for each position in the order, there is some player batting in that position and property (4) states that two or more players are never batting in the same position in the list. | Bijection | Wikipedia | 438 | 3942 | https://en.wikipedia.org/wiki/Bijection | Mathematics | Functions: General | null |
Seats and students of a classroom
In a classroom there are a certain number of seats. A group of students enter the room and the instructor asks them to be seated. After a quick look around the room, the instructor declares that there is a bijection between the set of students and the set of seats, where each student is paired with the seat they are sitting in. What the instructor observed in order to reach this conclusion was that:
Every student was in a seat (there was no one standing),
No student was in more than one seat,
Every seat had someone sitting there (there were no empty seats), and
No seat had more than one student in it.
The instructor was able to conclude that there were just as many seats as there were students, without having to count either set.
More mathematical examples | Bijection | Wikipedia | 164 | 3942 | https://en.wikipedia.org/wiki/Bijection | Mathematics | Functions: General | null |
For any set X, the identity function 1X: X → X, 1X(x) = x is bijective.
The function f: R → R, f(x) = 2x + 1 is bijective, since for each y there is a unique x = (y − 1)/2 such that f(x) = y. More generally, any linear function over the reals, f: R → R, f(x) = ax + b (where a is non-zero) is a bijection. Each real number y is obtained from (or paired with) the real number x = (y − b)/a.
The function f: R → (−π/2, π/2), given by f(x) = arctan(x) is bijective, since each real number x is paired with exactly one angle y in the interval (−π/2, π/2) so that tan(y) = x (that is, y = arctan(x)). If the codomain (−π/2, π/2) was made larger to include an integer multiple of π/2, then this function would no longer be onto (surjective), since there is no real number which could be paired with the multiple of π/2 by this arctan function.
The exponential function, g: R → R, g(x) = ex, is not bijective: for instance, there is no x in R such that g(x) = −1, showing that g is not onto (surjective). However, if the codomain is restricted to the positive real numbers , then g would be bijective; its inverse (see below) is the natural logarithm function ln.
The function h: R → R+, h(x) = x2 is not bijective: for instance, h(−1) = h(1) = 1, showing that h is not one-to-one (injective). However, if the domain is restricted to , then h would be bijective; its inverse is the positive square root function.
By Schröder–Bernstein theorem, given any two sets X and Y, and two injective functions f: X → Y and g: Y → X, there exists a bijective function h: X → Y. | Bijection | Wikipedia | 505 | 3942 | https://en.wikipedia.org/wiki/Bijection | Mathematics | Functions: General | null |
Inverses
A bijection f with domain X (indicated by f: X → Y in functional notation) also defines a converse relation starting in Y and going to X (by turning the arrows around). The process of "turning the arrows around" for an arbitrary function does not, in general, yield a function, but properties (3) and (4) of a bijection say that this inverse relation is a function with domain Y. Moreover, properties (1) and (2) then say that this inverse function is a surjection and an injection, that is, the inverse function exists and is also a bijection. Functions that have inverse functions are said to be invertible. A function is invertible if and only if it is a bijection.
Stated in concise mathematical notation, a function f: X → Y is bijective if and only if it satisfies the condition
for every y in Y there is a unique x in X with y = f(x).
Continuing with the baseball batting line-up example, the function that is being defined takes as input the name of one of the players and outputs the position of that player in the batting order. Since this function is a bijection, it has an inverse function which takes as input a position in the batting order and outputs the player who will be batting in that position.
Composition
The composition of two bijections f: X → Y and g: Y → Z is a bijection, whose inverse is given by is .
Conversely, if the composition of two functions is bijective, it only follows that f is injective and g is surjective.
Cardinality
If X and Y are finite sets, then there exists a bijection between the two sets X and Y if and only if X and Y have the same number of elements. Indeed, in axiomatic set theory, this is taken as the definition of "same number of elements" (equinumerosity), and generalising this definition to infinite sets leads to the concept of cardinal number, a way to distinguish the various sizes of infinite sets. | Bijection | Wikipedia | 439 | 3942 | https://en.wikipedia.org/wiki/Bijection | Mathematics | Functions: General | null |
Properties
A function f: R → R is bijective if and only if its graph meets every horizontal and vertical line exactly once.
If X is a set, then the bijective functions from X to itself, together with the operation of functional composition (∘), form a group, the symmetric group of X, which is denoted variously by S(X), SX, or X! (X factorial).
Bijections preserve cardinalities of sets: for a subset A of the domain with cardinality |A| and subset B of the codomain with cardinality |B|, one has the following equalities:
|f(A)| = |A| and |f−1(B)| = |B|.
If X and Y are finite sets with the same cardinality, and f: X → Y, then the following are equivalent:
f is a bijection.
f is a surjection.
f is an injection.
For a finite set S, there is a bijection between the set of possible total orderings of the elements and the set of bijections from S to S. That is to say, the number of permutations of elements of S is the same as the number of total orderings of that set—namely, n!.
Category theory
Bijections are precisely the isomorphisms in the category Set of sets and set functions. However, the bijections are not always the isomorphisms for more complex categories. For example, in the category Grp of groups, the morphisms must be homomorphisms since they must preserve the group structure, so the isomorphisms are group isomorphisms which are bijective homomorphisms.
Generalization to partial functions
The notion of one-to-one correspondence generalizes to partial functions, where they are called partial bijections, although partial bijections are only required to be injective. The reason for this relaxation is that a (proper) partial function is already undefined for a portion of its domain; thus there is no compelling reason to constrain its inverse to be a total function, i.e. defined everywhere on its domain. The set of all partial bijections on a given base set is called the symmetric inverse semigroup. | Bijection | Wikipedia | 470 | 3942 | https://en.wikipedia.org/wiki/Bijection | Mathematics | Functions: General | null |
Another way of defining the same notion is to say that a partial bijection from A to B is any relation
R (which turns out to be a partial function) with the property that R is the graph of a bijection f:A′→B′, where A′ is a subset of A and B′ is a subset of B.
When the partial bijection is on the same set, it is sometimes called a one-to-one partial transformation. An example is the Möbius transformation simply defined on the complex plane, rather than its completion to the extended complex plane.
Gallery | Bijection | Wikipedia | 122 | 3942 | https://en.wikipedia.org/wiki/Bijection | Mathematics | Functions: General | null |
Biochemistry, or biological chemistry, is the study of chemical processes within and relating to living organisms. A sub-discipline of both chemistry and biology, biochemistry may be divided into three fields: structural biology, enzymology, and metabolism. Over the last decades of the 20th century, biochemistry has become successful at explaining living processes through these three disciplines. Almost all areas of the life sciences are being uncovered and developed through biochemical methodology and research. Biochemistry focuses on understanding the chemical basis that allows biological molecules to give rise to the processes that occur within living cells and between cells, in turn relating greatly to the understanding of tissues and organs as well as organism structure and function. Biochemistry is closely related to molecular biology, the study of the molecular mechanisms of biological phenomena.
Much of biochemistry deals with the structures, functions, and interactions of biological macromolecules such as proteins, nucleic acids, carbohydrates, and lipids. They provide the structure of cells and perform many of the functions associated with life. The chemistry of the cell also depends upon the reactions of small molecules and ions. These can be inorganic (for example, water and metal ions) or organic (for example, the amino acids, which are used to synthesize proteins). The mechanisms used by cells to harness energy from their environment via chemical reactions are known as metabolism. The findings of biochemistry are applied primarily in medicine, nutrition, and agriculture. In medicine, biochemists investigate the causes and cures of diseases. Nutrition studies how to maintain health and wellness and also the effects of nutritional deficiencies. In agriculture, biochemists investigate soil and fertilizers with the goal of improving crop cultivation, crop storage, and pest control. In recent decades, biochemical principles and methods have been combined with problem-solving approaches from engineering to manipulate living systems in order to produce useful tools for research, industrial processes, and diagnosis and control of diseasethe discipline of biotechnology.
History | Biochemistry | Wikipedia | 394 | 3954 | https://en.wikipedia.org/wiki/Biochemistry | Biology and health sciences | Chemistry | null |
At its most comprehensive definition, biochemistry can be seen as a study of the components and composition of living things and how they come together to become life. In this sense, the history of biochemistry may therefore go back as far as the ancient Greeks. However, biochemistry as a specific scientific discipline began sometime in the 19th century, or a little earlier, depending on which aspect of biochemistry is being focused on. Some argued that the beginning of biochemistry may have been the discovery of the first enzyme, diastase (now called amylase), in 1833 by Anselme Payen, while others considered Eduard Buchner's first demonstration of a complex biochemical process alcoholic fermentation in cell-free extracts in 1897 to be the birth of biochemistry. Some might also point as its beginning to the influential 1842 work by Justus von Liebig, Animal chemistry, or, Organic chemistry in its applications to physiology and pathology, which presented a chemical theory of metabolism, or even earlier to the 18th century studies on fermentation and respiration by Antoine Lavoisier. Many other pioneers in the field who helped to uncover the layers of complexity of biochemistry have been proclaimed founders of modern biochemistry. Emil Fischer, who studied the chemistry of proteins, and F. Gowland Hopkins, who studied enzymes and the dynamic nature of biochemistry, represent two examples of early biochemists.
The term "biochemistry" was first used when Vinzenz Kletzinsky (1826–1882) had his "Compendium der Biochemie" printed in Vienna in 1858; it derived from a combination of biology and chemistry. In 1877, Felix Hoppe-Seyler used the term ( in German) as a synonym for physiological chemistry in the foreword to the first issue of Zeitschrift für Physiologische Chemie (Journal of Physiological Chemistry) where he argued for the setting up of institutes dedicated to this field of study. The German chemist Carl Neuberg however is often cited to have coined the word in 1903, while some credited it to Franz Hofmeister. | Biochemistry | Wikipedia | 418 | 3954 | https://en.wikipedia.org/wiki/Biochemistry | Biology and health sciences | Chemistry | null |
It was once generally believed that life and its materials had some essential property or substance (often referred to as the "vital principle") distinct from any found in non-living matter, and it was thought that only living beings could produce the molecules of life. In 1828, Friedrich Wöhler published a paper on his serendipitous urea synthesis from potassium cyanate and ammonium sulfate; some regarded that as a direct overthrow of vitalism and the establishment of organic chemistry. However, the Wöhler synthesis has sparked controversy as some reject the death of vitalism at his hands. Since then, biochemistry has advanced, especially since the mid-20th century, with the development of new techniques such as chromatography, X-ray diffraction, dual polarisation interferometry, NMR spectroscopy, radioisotopic labeling, electron microscopy and molecular dynamics simulations. These techniques allowed for the discovery and detailed analysis of many molecules and metabolic pathways of the cell, such as glycolysis and the Krebs cycle (citric acid cycle), and led to an understanding of biochemistry on a molecular level.
Another significant historic event in biochemistry is the discovery of the gene, and its role in the transfer of information in the cell. In the 1950s, James D. Watson, Francis Crick, Rosalind Franklin and Maurice Wilkins were instrumental in solving DNA structure and suggesting its relationship with the genetic transfer of information. In 1958, George Beadle and Edward Tatum received the Nobel Prize for work in fungi showing that one gene produces one enzyme. In 1988, Colin Pitchfork was the first person convicted of murder with DNA evidence, which led to the growth of forensic science. More recently, Andrew Z. Fire and Craig C. Mello received the 2006 Nobel Prize for discovering the role of RNA interference (RNAi) in the silencing of gene expression.
Starting materials: the chemical elements of life
Around two dozen chemical elements are essential to various kinds of biological life. Most rare elements on Earth are not needed by life (exceptions being selenium and iodine), while a few common ones (aluminum and titanium) are not used. Most organisms share element needs, but there are a few differences between plants and animals. For example, ocean algae use bromine, but land plants and animals do not seem to need any. All animals require sodium, but is not an essential element for plants. Plants need boron and silicon, but animals may not (or may need ultra-small amounts). | Biochemistry | Wikipedia | 508 | 3954 | https://en.wikipedia.org/wiki/Biochemistry | Biology and health sciences | Chemistry | null |
Just six elements—carbon, hydrogen, nitrogen, oxygen, calcium and phosphorus—make up almost 99% of the mass of living cells, including those in the human body (see composition of the human body for a complete list). In addition to the six major elements that compose most of the human body, humans require smaller amounts of possibly 18 more.
Biomolecules
The 4 main classes of molecules in biochemistry (often called biomolecules) are carbohydrates, lipids, proteins, and nucleic acids. Many biological molecules are polymers: in this terminology, monomers are relatively small macromolecules that are linked together to create large macromolecules known as polymers. When monomers are linked together to synthesize a biological polymer, they undergo a process called dehydration synthesis. Different macromolecules can assemble in larger complexes, often needed for biological activity.
Carbohydrates
Two of the main functions of carbohydrates are energy storage and providing structure. One of the common sugars known as glucose is a carbohydrate, but not all carbohydrates are sugars. There are more carbohydrates on Earth than any other known type of biomolecule; they are used to store energy and genetic information, as well as play important roles in cell to cell interactions and communications.
The simplest type of carbohydrate is a monosaccharide, which among other properties contains carbon, hydrogen, and oxygen, mostly in a ratio of 1:2:1 (generalized formula CnH2nOn, where n is at least 3). Glucose (C6H12O6) is one of the most important carbohydrates; others include fructose (C6H12O6), the sugar commonly associated with the sweet taste of fruits, and deoxyribose (C5H10O4), a component of DNA. A monosaccharide can switch between acyclic (open-chain) form and a cyclic form. The open-chain form can be turned into a ring of carbon atoms bridged by an oxygen atom created from the carbonyl group of one end and the hydroxyl group of another. The cyclic molecule has a hemiacetal or hemiketal group, depending on whether the linear form was an aldose or a ketose. | Biochemistry | Wikipedia | 499 | 3954 | https://en.wikipedia.org/wiki/Biochemistry | Biology and health sciences | Chemistry | null |
In these cyclic forms, the ring usually has 5 or 6 atoms. These forms are called furanoses and pyranoses, respectively—by analogy with furan and pyran, the simplest compounds with the same carbon-oxygen ring (although they lack the carbon-carbon double bonds of these two molecules). For example, the aldohexose glucose may form a hemiacetal linkage between the hydroxyl on carbon 1 and the oxygen on carbon 4, yielding a molecule with a 5-membered ring, called glucofuranose. The same reaction can take place between carbons 1 and 5 to form a molecule with a 6-membered ring, called glucopyranose. Cyclic forms with a 7-atom ring called heptoses are rare.
Two monosaccharides can be joined by a glycosidic or ester bond into a disaccharide through a dehydration reaction during which a molecule of water is released. The reverse reaction in which the glycosidic bond of a disaccharide is broken into two monosaccharides is termed hydrolysis. The best-known disaccharide is sucrose or ordinary sugar, which consists of a glucose molecule and a fructose molecule joined. Another important disaccharide is lactose found in milk, consisting of a glucose molecule and a galactose molecule. Lactose may be hydrolysed by lactase, and deficiency in this enzyme results in lactose intolerance.
When a few (around three to six) monosaccharides are joined, it is called an oligosaccharide (oligo- meaning "few"). These molecules tend to be used as markers and signals, as well as having some other uses. Many monosaccharides joined form a polysaccharide. They can be joined in one long linear chain, or they may be branched. Two of the most common polysaccharides are cellulose and glycogen, both consisting of repeating glucose monomers. Cellulose is an important structural component of plant's cell walls and glycogen is used as a form of energy storage in animals. | Biochemistry | Wikipedia | 466 | 3954 | https://en.wikipedia.org/wiki/Biochemistry | Biology and health sciences | Chemistry | null |
Sugar can be characterized by having reducing or non-reducing ends. A reducing end of a carbohydrate is a carbon atom that can be in equilibrium with the open-chain aldehyde (aldose) or keto form (ketose). If the joining of monomers takes place at such a carbon atom, the free hydroxy group of the pyranose or furanose form is exchanged with an OH-side-chain of another sugar, yielding a full acetal. This prevents opening of the chain to the aldehyde or keto form and renders the modified residue non-reducing. Lactose contains a reducing end at its glucose moiety, whereas the galactose moiety forms a full acetal with the C4-OH group of glucose. Saccharose does not have a reducing end because of full acetal formation between the aldehyde carbon of glucose (C1) and the keto carbon of fructose (C2).
Lipids
Lipids comprise a diverse range of molecules and to some extent is a catchall for relatively water-insoluble or nonpolar compounds of biological origin, including waxes, fatty acids, fatty-acid derived phospholipids, sphingolipids, glycolipids, and terpenoids (e.g., retinoids and steroids). Some lipids are linear, open-chain aliphatic molecules, while others have ring structures. Some are aromatic (with a cyclic [ring] and planar [flat] structure) while others are not. Some are flexible, while others are rigid.
Lipids are usually made from one molecule of glycerol combined with other molecules. In triglycerides, the main group of bulk lipids, there is one molecule of glycerol and three fatty acids. Fatty acids are considered the monomer in that case, and may be saturated (no double bonds in the carbon chain) or unsaturated (one or more double bonds in the carbon chain). | Biochemistry | Wikipedia | 422 | 3954 | https://en.wikipedia.org/wiki/Biochemistry | Biology and health sciences | Chemistry | null |
Most lipids have some polar character and are largely nonpolar. In general, the bulk of their structure is nonpolar or hydrophobic ("water-fearing"), meaning that it does not interact well with polar solvents like water. Another part of their structure is polar or hydrophilic ("water-loving") and will tend to associate with polar solvents like water. This makes them amphiphilic molecules (having both hydrophobic and hydrophilic portions). In the case of cholesterol, the polar group is a mere –OH (hydroxyl or alcohol).
In the case of phospholipids, the polar groups are considerably larger and more polar, as described below.
Lipids are an integral part of our daily diet. Most oils and milk products that we use for cooking and eating like butter, cheese, ghee etc. are composed of fats. Vegetable oils are rich in various polyunsaturated fatty acids (PUFA). Lipid-containing foods undergo digestion within the body and are broken into fatty acids and glycerol, the final degradation products of fats and lipids. Lipids, especially phospholipids, are also used in various pharmaceutical products, either as co-solubilizers (e.g. in parenteral infusions) or else as drug carrier components (e.g. in a liposome or transfersome).
Proteins | Biochemistry | Wikipedia | 301 | 3954 | https://en.wikipedia.org/wiki/Biochemistry | Biology and health sciences | Chemistry | null |
Proteins are very large molecules—macro-biopolymers—made from monomers called amino acids. An amino acid consists of an alpha carbon atom attached to an amino group, –NH2, a carboxylic acid group, –COOH (although these exist as –NH3+ and –COO− under physiologic conditions), a simple hydrogen atom, and a side chain commonly denoted as "–R". The side chain "R" is different for each amino acid of which there are 20 standard ones. It is this "R" group that makes each amino acid different, and the properties of the side chains greatly influence the overall three-dimensional conformation of a protein. Some amino acids have functions by themselves or in a modified form; for instance, glutamate functions as an important neurotransmitter. Amino acids can be joined via a peptide bond. In this dehydration synthesis, a water molecule is removed and the peptide bond connects the nitrogen of one amino acid's amino group to the carbon of the other's carboxylic acid group. The resulting molecule is called a dipeptide, and short stretches of amino acids (usually, fewer than thirty) are called peptides or polypeptides. Longer stretches merit the title proteins. As an example, the important blood serum protein albumin contains 585 amino acid residues.
Proteins can have structural and/or functional roles. For instance, movements of the proteins actin and myosin ultimately are responsible for the contraction of skeletal muscle. One property many proteins have is that they specifically bind to a certain molecule or class of molecules—they may be extremely selective in what they bind. Antibodies are an example of proteins that attach to one specific type of molecule. Antibodies are composed of heavy and light chains. Two heavy chains would be linked to two light chains through disulfide linkages between their amino acids. Antibodies are specific through variation based on differences in the N-terminal domain. | Biochemistry | Wikipedia | 409 | 3954 | https://en.wikipedia.org/wiki/Biochemistry | Biology and health sciences | Chemistry | null |
The enzyme-linked immunosorbent assay (ELISA), which uses antibodies, is one of the most sensitive tests modern medicine uses to detect various biomolecules. Probably the most important proteins, however, are the enzymes. Virtually every reaction in a living cell requires an enzyme to lower the activation energy of the reaction. These molecules recognize specific reactant molecules called substrates; they then catalyze the reaction between them. By lowering the activation energy, the enzyme speeds up that reaction by a rate of 1011 or more; a reaction that would normally take over 3,000 years to complete spontaneously might take less than a second with an enzyme. The enzyme itself is not used up in the process and is free to catalyze the same reaction with a new set of substrates. Using various modifiers, the activity of the enzyme can be regulated, enabling control of the biochemistry of the cell as a whole.
The structure of proteins is traditionally described in a hierarchy of four levels. The primary structure of a protein consists of its linear sequence of amino acids; for instance, "alanine-glycine-tryptophan-serine-glutamate-asparagine-glycine-lysine-...". Secondary structure is concerned with local morphology (morphology being the study of structure). Some combinations of amino acids will tend to curl up in a coil called an α-helix or into a sheet called a β-sheet; some α-helixes can be seen in the hemoglobin schematic above. Tertiary structure is the entire three-dimensional shape of the protein. This shape is determined by the sequence of amino acids. In fact, a single change can change the entire structure. The alpha chain of hemoglobin contains 146 amino acid residues; substitution of the glutamate residue at position 6 with a valine residue changes the behavior of hemoglobin so much that it results in sickle-cell disease. Finally, quaternary structure is concerned with the structure of a protein with multiple peptide subunits, like hemoglobin with its four subunits. Not all proteins have more than one subunit. | Biochemistry | Wikipedia | 447 | 3954 | https://en.wikipedia.org/wiki/Biochemistry | Biology and health sciences | Chemistry | null |
Ingested proteins are usually broken up into single amino acids or dipeptides in the small intestine and then absorbed. They can then be joined to form new proteins. Intermediate products of glycolysis, the citric acid cycle, and the pentose phosphate pathway can be used to form all twenty amino acids, and most bacteria and plants possess all the necessary enzymes to synthesize them. Humans and other mammals, however, can synthesize only half of them. They cannot synthesize isoleucine, leucine, lysine, methionine, phenylalanine, threonine, tryptophan, and valine. Because they must be ingested, these are the essential amino acids. Mammals do possess the enzymes to synthesize alanine, asparagine, aspartate, cysteine, glutamate, glutamine, glycine, proline, serine, and tyrosine, the nonessential amino acids. While they can synthesize arginine and histidine, they cannot produce it in sufficient amounts for young, growing animals, and so these are often considered essential amino acids.
If the amino group is removed from an amino acid, it leaves behind a carbon skeleton called an α-keto acid. Enzymes called transaminases can easily transfer the amino group from one amino acid (making it an α-keto acid) to another α-keto acid (making it an amino acid). This is important in the biosynthesis of amino acids, as for many of the pathways, intermediates from other biochemical pathways are converted to the α-keto acid skeleton, and then an amino group is added, often via transamination. The amino acids may then be linked together to form a protein.
A similar process is used to break down proteins. It is first hydrolyzed into its component amino acids. Free ammonia (NH3), existing as the ammonium ion (NH4+) in blood, is toxic to life forms. A suitable method for excreting it must therefore exist. Different tactics have evolved in different animals, depending on the animals' needs. Unicellular organisms release the ammonia into the environment. Likewise, bony fish can release ammonia into the water where it is quickly diluted. In general, mammals convert ammonia into urea, via the urea cycle. | Biochemistry | Wikipedia | 499 | 3954 | https://en.wikipedia.org/wiki/Biochemistry | Biology and health sciences | Chemistry | null |
In order to determine whether two proteins are related, or in other words to decide whether they are homologous or not, scientists use sequence-comparison methods. Methods like sequence alignments and structural alignments are powerful tools that help scientists identify homologies between related molecules. The relevance of finding homologies among proteins goes beyond forming an evolutionary pattern of protein families. By finding how similar two protein sequences are, we acquire knowledge about their structure and therefore their function.
Nucleic acids
Nucleic acids, so-called because of their prevalence in cellular nuclei, is the generic name of the family of biopolymers. They are complex, high-molecular-weight biochemical macromolecules that can convey genetic information in all living cells and viruses. The monomers are called nucleotides, and each consists of three components: a nitrogenous heterocyclic base (either a purine or a pyrimidine), a pentose sugar, and a phosphate group.
The most common nucleic acids are deoxyribonucleic acid (DNA) and ribonucleic acid (RNA). The phosphate group and the sugar of each nucleotide bond with each other to form the backbone of the nucleic acid, while the sequence of nitrogenous bases stores the information. The most common nitrogenous bases are adenine, cytosine, guanine, thymine, and uracil. The nitrogenous bases of each strand of a nucleic acid will form hydrogen bonds with certain other nitrogenous bases in a complementary strand of nucleic acid. Adenine binds with thymine and uracil, thymine binds only with adenine, and cytosine and guanine can bind only with one another. Adenine, thymine, and uracil contain two hydrogen bonds, while hydrogen bonds formed between cytosine and guanine are three.
Aside from the genetic material of the cell, nucleic acids often play a role as second messengers, as well as forming the base molecule for adenosine triphosphate (ATP), the primary energy-carrier molecule found in all living organisms. Also, the nitrogenous bases possible in the two nucleic acids are different: adenine, cytosine, and guanine occur in both RNA and DNA, while thymine occurs only in DNA and uracil occurs in RNA.
Metabolism
Carbohydrates as energy source | Biochemistry | Wikipedia | 495 | 3954 | https://en.wikipedia.org/wiki/Biochemistry | Biology and health sciences | Chemistry | null |
Glucose is an energy source in most life forms. For instance, polysaccharides are broken down into their monomers by enzymes (glycogen phosphorylase removes glucose residues from glycogen, a polysaccharide). Disaccharides like lactose or sucrose are cleaved into their two component monosaccharides.
Glycolysis (anaerobic)
Glucose is mainly metabolized by a very important ten-step pathway called glycolysis, the net result of which is to break down one molecule of glucose into two molecules of pyruvate. This also produces a net two molecules of ATP, the energy currency of cells, along with two reducing equivalents of converting NAD+ (nicotinamide adenine dinucleotide: oxidized form) to NADH (nicotinamide adenine dinucleotide: reduced form). This does not require oxygen; if no oxygen is available (or the cell cannot use oxygen), the NAD is restored by converting the pyruvate to lactate (lactic acid) (e.g. in humans) or to ethanol plus carbon dioxide (e.g. in yeast). Other monosaccharides like galactose and fructose can be converted into intermediates of the glycolytic pathway. | Biochemistry | Wikipedia | 281 | 3954 | https://en.wikipedia.org/wiki/Biochemistry | Biology and health sciences | Chemistry | null |
Aerobic
In aerobic cells with sufficient oxygen, as in most human cells, the pyruvate is further metabolized. It is irreversibly converted to acetyl-CoA, giving off one carbon atom as the waste product carbon dioxide, generating another reducing equivalent as NADH. The two molecules acetyl-CoA (from one molecule of glucose) then enter the citric acid cycle, producing two molecules of ATP, six more NADH molecules and two reduced (ubi)quinones (via FADH2 as enzyme-bound cofactor), and releasing the remaining carbon atoms as carbon dioxide. The produced NADH and quinol molecules then feed into the enzyme complexes of the respiratory chain, an electron transport system transferring the electrons ultimately to oxygen and conserving the released energy in the form of a proton gradient over a membrane (inner mitochondrial membrane in eukaryotes). Thus, oxygen is reduced to water and the original electron acceptors NAD+ and quinone are regenerated. This is why humans breathe in oxygen and breathe out carbon dioxide. The energy released from transferring the electrons from high-energy states in NADH and quinol is conserved first as proton gradient and converted to ATP via ATP synthase. This generates an additional 28 molecules of ATP (24 from the 8 NADH + 4 from the 2 quinols), totaling to 32 molecules of ATP conserved per degraded glucose (two from glycolysis + two from the citrate cycle). It is clear that using oxygen to completely oxidize glucose provides an organism with far more energy than any oxygen-independent metabolic feature, and this is thought to be the reason why complex life appeared only after Earth's atmosphere accumulated large amounts of oxygen.
Gluconeogenesis | Biochemistry | Wikipedia | 366 | 3954 | https://en.wikipedia.org/wiki/Biochemistry | Biology and health sciences | Chemistry | null |
In vertebrates, vigorously contracting skeletal muscles (during weightlifting or sprinting, for example) do not receive enough oxygen to meet the energy demand, and so they shift to anaerobic metabolism, converting glucose to lactate.
The combination of glucose from noncarbohydrates origin, such as fat and proteins. This only happens when glycogen supplies in the liver are worn out. The pathway is a crucial reversal of glycolysis from pyruvate to glucose and can use many sources like amino acids, glycerol and Krebs Cycle. Large scale protein and fat catabolism usually occur when those suffer from starvation or certain endocrine disorders. The liver regenerates the glucose, using a process called gluconeogenesis. This process is not quite the opposite of glycolysis, and actually requires three times the amount of energy gained from glycolysis (six molecules of ATP are used, compared to the two gained in glycolysis). Analogous to the above reactions, the glucose produced can then undergo glycolysis in tissues that need energy, be stored as glycogen (or starch in plants), or be converted to other monosaccharides or joined into di- or oligosaccharides. The combined pathways of glycolysis during exercise, lactate's crossing via the bloodstream to the liver, subsequent gluconeogenesis and release of glucose into the bloodstream is called the Cori cycle.
Relationship to other "molecular-scale" biological sciences | Biochemistry | Wikipedia | 322 | 3954 | https://en.wikipedia.org/wiki/Biochemistry | Biology and health sciences | Chemistry | null |
Researchers in biochemistry use specific techniques native to biochemistry, but increasingly combine these with techniques and ideas developed in the fields of genetics, molecular biology, and biophysics. There is not a defined line between these disciplines. Biochemistry studies the chemistry required for biological activity of molecules, molecular biology studies their biological activity, genetics studies their heredity, which happens to be carried by their genome. This is shown in the following schematic that depicts one possible view of the relationships between the fields:
Biochemistry is the study of the chemical substances and vital processes occurring in live organisms. Biochemists focus heavily on the role, function, and structure of biomolecules. The study of the chemistry behind biological processes and the synthesis of biologically active molecules are applications of biochemistry. Biochemistry studies life at the atomic and molecular level.
Genetics is the study of the effect of genetic differences in organisms. This can often be inferred by the absence of a normal component (e.g. one gene). The study of "mutants" – organisms that lack one or more functional components with respect to the so-called "wild type" or normal phenotype. Genetic interactions (epistasis) can often confound simple interpretations of such "knockout" studies.
Molecular biology is the study of molecular underpinnings of the biological phenomena, focusing on molecular synthesis, modification, mechanisms and interactions. The central dogma of molecular biology, where genetic material is transcribed into RNA and then translated into protein, despite being oversimplified, still provides a good starting point for understanding the field. This concept has been revised in light of emerging novel roles for RNA.
Chemical biology seeks to develop new tools based on small molecules that allow minimal perturbation of biological systems while providing detailed information about their function. Further, chemical biology employs biological systems to create non-natural hybrids between biomolecules and synthetic devices (for example emptied viral capsids that can deliver gene therapy or drug molecules). | Biochemistry | Wikipedia | 399 | 3954 | https://en.wikipedia.org/wiki/Biochemistry | Biology and health sciences | Chemistry | null |
In abstract algebra, a Boolean algebra or Boolean lattice is a complemented distributive lattice. This type of algebraic structure captures essential properties of both set operations and logic operations. A Boolean algebra can be seen as a generalization of a power set algebra or a field of sets, or its elements can be viewed as generalized truth values. It is also a special case of a De Morgan algebra and a Kleene algebra (with involution).
Every Boolean algebra gives rise to a Boolean ring, and vice versa, with ring multiplication corresponding to conjunction or meet ∧, and ring addition to exclusive disjunction or symmetric difference (not disjunction ∨). However, the theory of Boolean rings has an inherent asymmetry between the two operators, while the axioms and theorems of Boolean algebra express the symmetry of the theory described by the duality principle.
History
The term "Boolean algebra" honors George Boole (1815–1864), a self-educated English mathematician. He introduced the algebraic system initially in a small pamphlet, The Mathematical Analysis of Logic, published in 1847 in response to an ongoing public controversy between Augustus De Morgan and William Hamilton, and later as a more substantial book, The Laws of Thought, published in 1854. Boole's formulation differs from that described above in some important respects. For example, conjunction and disjunction in Boole were not a dual pair of operations. Boolean algebra emerged in the 1860s, in papers written by William Jevons and Charles Sanders Peirce. The first systematic presentation of Boolean algebra and distributive lattices is owed to the 1890 Vorlesungen of Ernst Schröder. The first extensive treatment of Boolean algebra in English is A. N. Whitehead's 1898 Universal Algebra. Boolean algebra as an axiomatic algebraic structure in the modern axiomatic sense begins with a 1904 paper by Edward V. Huntington. Boolean algebra came of age as serious mathematics with the work of Marshall Stone in the 1930s, and with Garrett Birkhoff's 1940 Lattice Theory. In the 1960s, Paul Cohen, Dana Scott, and others found deep new results in mathematical logic and axiomatic set theory using offshoots of Boolean algebra, namely forcing and Boolean-valued models.
Definition | Boolean algebra (structure) | Wikipedia | 477 | 3959 | https://en.wikipedia.org/wiki/Boolean%20algebra%20%28structure%29 | Mathematics | Order theory | null |
A Boolean algebra is a set , equipped with two binary operations (called "meet" or "and"), (called "join" or "or"), a unary operation (called "complement" or "not") and two elements and in (called "bottom" and "top", or "least" and "greatest" element, also denoted by the symbols and , respectively), such that for all elements , and of , the following axioms hold:
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Note, however, that the absorption law and even the associativity law can be excluded from the set of axioms as they can be derived from the other axioms (see Proven properties).
A Boolean algebra with only one element is called a trivial Boolean algebra or a degenerate Boolean algebra. (In older works, some authors required and to be distinct elements in order to exclude this case.)
It follows from the last three pairs of axioms above (identity, distributivity and complements), or from the absorption axiom, that
if and only if .
The relation defined by if these equivalent conditions hold, is a partial order with least element 0 and greatest element 1. The meet and the join of two elements coincide with their infimum and supremum, respectively, with respect to ≤.
The first four pairs of axioms constitute a definition of a bounded lattice.
It follows from the first five pairs of axioms that any complement is unique.
The set of axioms is self-dual in the sense that if one exchanges with and with in an axiom, the result is again an axiom. Therefore, by applying this operation to a Boolean algebra (or Boolean lattice), one obtains another Boolean algebra with the same elements; it is called its dual.
Examples
The simplest non-trivial Boolean algebra, the two-element Boolean algebra, has only two elements, and , and is defined by the rules: | Boolean algebra (structure) | Wikipedia | 460 | 3959 | https://en.wikipedia.org/wiki/Boolean%20algebra%20%28structure%29 | Mathematics | Order theory | null |
It has applications in logic, interpreting as false, as true, as and, as or, and as not. Expressions involving variables and the Boolean operations represent statement forms, and two such expressions can be shown to be equal using the above axioms if and only if the corresponding statement forms are logically equivalent.
The two-element Boolean algebra is also used for circuit design in electrical engineering; here 0 and 1 represent the two different states of one bit in a digital circuit, typically high and low voltage. Circuits are described by expressions containing variables, and two such expressions are equal for all values of the variables if and only if the corresponding circuits have the same input–output behavior. Furthermore, every possible input–output behavior can be modeled by a suitable Boolean expression.
The two-element Boolean algebra is also important in the general theory of Boolean algebras, because an equation involving several variables is generally true in all Boolean algebras if and only if it is true in the two-element Boolean algebra (which can be checked by a trivial brute force algorithm for small numbers of variables). This can for example be used to show that the following laws (Consensus theorems) are generally valid in all Boolean algebras:
The power set (set of all subsets) of any given nonempty set forms a Boolean algebra, an algebra of sets, with the two operations (union) and (intersection). The smallest element 0 is the empty set and the largest element is the set itself.
After the two-element Boolean algebra, the simplest Boolean algebra is that defined by the power set of two atoms: | Boolean algebra (structure) | Wikipedia | 333 | 3959 | https://en.wikipedia.org/wiki/Boolean%20algebra%20%28structure%29 | Mathematics | Order theory | null |
The set of all subsets of that are either finite or cofinite is a Boolean algebra and an algebra of sets called the finite–cofinite algebra. If is infinite then the set of all cofinite subsets of , which is called the Fréchet filter, is a free ultrafilter on . However, the Fréchet filter is not an ultrafilter on the power set of .
Starting with the propositional calculus with sentence symbols, form the Lindenbaum algebra (that is, the set of sentences in the propositional calculus modulo logical equivalence). This construction yields a Boolean algebra. It is in fact the free Boolean algebra on generators. A truth assignment in propositional calculus is then a Boolean algebra homomorphism from this algebra to the two-element Boolean algebra.
Given any linearly ordered set with a least element, the interval algebra is the smallest Boolean algebra of subsets of containing all of the half-open intervals such that is in and is either in or equal to . Interval algebras are useful in the study of Lindenbaum–Tarski algebras; every countable Boolean algebra is isomorphic to an interval algebra.
For any natural number , the set of all positive divisors of , defining if divides , forms a distributive lattice. This lattice is a Boolean algebra if and only if is square-free. The bottom and the top elements of this Boolean algebra are the natural numbers and , respectively. The complement of is given by . The meet and the join of and are given by the greatest common divisor () and the least common multiple () of and , respectively. The ring addition is given by . The picture shows an example for . As a counter-example, considering the non-square-free , the greatest common divisor of 30 and its complement 2 would be 2, while it should be the bottom element 1.
Other examples of Boolean algebras arise from topological spaces: if is a topological space, then the collection of all subsets of that are both open and closed forms a Boolean algebra with the operations (union) and (intersection).
If is an arbitrary ring then its set of central idempotents, which is the set
becomes a Boolean algebra when its operations are defined by and .
Homomorphisms and isomorphisms
A homomorphism between two Boolean algebras and is a function such that for all , in :
,
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It then follows that for all in . The class of all Boolean algebras, together with this notion of morphism, forms a full subcategory of the category of lattices.
An isomorphism between two Boolean algebras and is a homomorphism with an inverse homomorphism, that is, a homomorphism such that the composition is the identity function on , and the composition is the identity function on . A homomorphism of Boolean algebras is an isomorphism if and only if it is bijective.
Boolean rings
Every Boolean algebra gives rise to a ring by defining (this operation is called symmetric difference in the case of sets and XOR in the case of logic) and . The zero element of this ring coincides with the 0 of the Boolean algebra; the multiplicative identity element of the ring is the of the Boolean algebra. This ring has the property that for all in ; rings with this property are called Boolean rings.
Conversely, if a Boolean ring is given, we can turn it into a Boolean algebra by defining and .
Since these two constructions are inverses of each other, we can say that every Boolean ring arises from a Boolean algebra, and vice versa. Furthermore, a map is a homomorphism of Boolean algebras if and only if it is a homomorphism of Boolean rings. The categories of Boolean rings and Boolean algebras are equivalent; in fact the categories are isomorphic.
Hsiang (1985) gave a rule-based algorithm to check whether two arbitrary expressions denote the same value in every Boolean ring.
More generally, Boudet, Jouannaud, and Schmidt-Schauß (1989) gave an algorithm to solve equations between arbitrary Boolean-ring expressions.
Employing the similarity of Boolean rings and Boolean algebras, both algorithms have applications in automated theorem proving.
Ideals and filters | Boolean algebra (structure) | Wikipedia | 388 | 3959 | https://en.wikipedia.org/wiki/Boolean%20algebra%20%28structure%29 | Mathematics | Order theory | null |
An ideal of the Boolean algebra is a nonempty subset such that for all , in we have in and for all in we have in . This notion of ideal coincides with the notion of ring ideal in the Boolean ring . An ideal of is called prime if and if in always implies in or in . Furthermore, for every we have that , and then if is prime we have or for every . An ideal of is called maximal if and if the only ideal properly containing is itself. For an ideal , if and , then or is contained in another proper ideal . Hence, such an is not maximal, and therefore the notions of prime ideal and maximal ideal are equivalent in Boolean algebras. Moreover, these notions coincide with ring theoretic ones of prime ideal and maximal ideal in the Boolean ring .
The dual of an ideal is a filter. A filter of the Boolean algebra is a nonempty subset such that for all , in we have in and for all in we have in . The dual of a maximal (or prime) ideal in a Boolean algebra is ultrafilter. Ultrafilters can alternatively be described as 2-valued morphisms from to the two-element Boolean algebra. The statement every filter in a Boolean algebra can be extended to an ultrafilter is called the ultrafilter lemma and cannot be proven in Zermelo–Fraenkel set theory (ZF), if ZF is consistent. Within ZF, the ultrafilter lemma is strictly weaker than the axiom of choice.
The ultrafilter lemma has many equivalent formulations: every Boolean algebra has an ultrafilter, every ideal in a Boolean algebra can be extended to a prime ideal, etc.
Representations
It can be shown that every finite Boolean algebra is isomorphic to the Boolean algebra of all subsets of a finite set. Therefore, the number of elements of every finite Boolean algebra is a power of two.
Stone's celebrated representation theorem for Boolean algebras states that every Boolean algebra is isomorphic to the Boolean algebra of all clopen sets in some (compact totally disconnected Hausdorff) topological space.
Axiomatics | Boolean algebra (structure) | Wikipedia | 450 | 3959 | https://en.wikipedia.org/wiki/Boolean%20algebra%20%28structure%29 | Mathematics | Order theory | null |
The first axiomatization of Boolean lattices/algebras in general was given by the English philosopher and mathematician Alfred North Whitehead in 1898.
It included the above axioms and additionally and .
In 1904, the American mathematician Edward V. Huntington (1874–1952) gave probably the most parsimonious axiomatization based on , , , even proving the associativity laws (see box).
He also proved that these axioms are independent of each other.
In 1933, Huntington set out the following elegant axiomatization for Boolean algebra. It requires just one binary operation and a unary functional symbol , to be read as 'complement', which satisfy the following laws:
Herbert Robbins immediately asked: If the Huntington equation is replaced with its dual, to wit:
do (1), (2), and (4) form a basis for Boolean algebra? Calling (1), (2), and (4) a Robbins algebra, the question then becomes: Is every Robbins algebra a Boolean algebra? This question (which came to be known as the Robbins conjecture) remained open for decades, and became a favorite question of Alfred Tarski and his students. In 1996, William McCune at Argonne National Laboratory, building on earlier work by Larry Wos, Steve Winker, and Bob Veroff, answered Robbins's question in the affirmative: Every Robbins algebra is a Boolean algebra. Crucial to McCune's proof was the computer program EQP he designed. For a simplification of McCune's proof, see Dahn (1998).
Further work has been done for reducing the number of axioms; see Minimal axioms for Boolean algebra.
Generalizations
Removing the requirement of existence of a unit from the axioms of Boolean algebra yields "generalized Boolean algebras". Formally, a distributive lattice is a generalized Boolean lattice, if it has a smallest element and for any elements and in such that , there exists an element such that and . Defining as the unique such that and , we say that the structure is a generalized Boolean algebra, while is a generalized Boolean semilattice. Generalized Boolean lattices are exactly the ideals of Boolean lattices. | Boolean algebra (structure) | Wikipedia | 465 | 3959 | https://en.wikipedia.org/wiki/Boolean%20algebra%20%28structure%29 | Mathematics | Order theory | null |
A structure that satisfies all axioms for Boolean algebras except the two distributivity axioms is called an orthocomplemented lattice. Orthocomplemented lattices arise naturally in quantum logic as lattices of closed linear subspaces for separable Hilbert spaces. | Boolean algebra (structure) | Wikipedia | 64 | 3959 | https://en.wikipedia.org/wiki/Boolean%20algebra%20%28structure%29 | Mathematics | Order theory | null |
A bicycle, also called a pedal cycle, bike, push-bike or cycle, is a human-powered or motor-assisted, pedal-driven, single-track vehicle, with two wheels attached to a frame, one behind the other. A is called a cyclist, or bicyclist.
Bicycles were introduced in the 19th century in Europe. By the early 21st century there were more than 1 billion bicycles. There are many more bicycles than cars. Bicycles are the principal means of transport in many regions. They also provide a popular form of recreation, and have been adapted for use as children's toys. Bicycles are used for fitness, military and police applications, courier services, bicycle racing, and artistic cycling.
The basic shape and configuration of a typical upright or "safety" bicycle, has changed little since the first chain-driven model was developed around 1885. However, many details have been improved, especially since the advent of modern materials and computer-aided design. These have allowed for a proliferation of specialized designs for many types of cycling. In the 21st century, electric bicycles have become popular.
The bicycle's invention has had an enormous effect on society, both in terms of culture and of advancing modern industrial methods. Several components that played a key role in the development of the automobile were initially invented for use in the bicycle, including ball bearings, pneumatic tires, chain-driven sprockets, and tension-spoked wheels.
Etymology
The word bicycle first appeared in English print in The Daily News in 1868, to describe "Bysicles and trysicles" on the "Champs Elysées and Bois de Boulogne". The word was first used in 1847 in a French publication to describe an unidentified two-wheeled vehicle, possibly a carriage. The design of the bicycle was an advance on the velocipede, although the words were used with some degree of overlap for a time.
Other words for bicycle include "bike", "pushbike", "pedal cycle", or "cycle". In Unicode, the code point for "bicycle" is 0x1F6B2. The entity 🚲 in HTML produces 🚲. | Bicycle | Wikipedia | 448 | 3973 | https://en.wikipedia.org/wiki/Bicycle | Technology | Transportation | null |
Although bike and cycle are used interchangeably to refer mostly to two types of two-wheelers, the terms still vary across the world. In India, for example, a cycle refers only to a two-wheeler using pedal power whereas the term bike is used to describe a two-wheeler using internal combustion engine or electric motors as a source of motive power instead of motorcycle/motorbike.
History
The "dandy horse", also called Draisienne or Laufmaschine ("running machine"), was the first human means of transport to use only two wheels in tandem and was invented by the German Baron Karl von Drais. It is regarded as the first bicycle and von Drais is seen as the "father of the bicycle", but it did not have pedals. Von Drais introduced it to the public in Mannheim in 1817 and in Paris in 1818. Its rider sat astride a wooden frame supported by two in-line wheels and pushed the vehicle along with his or her feet while steering the front wheel.
The first mechanically propelled, two-wheeled vehicle may have been built by Kirkpatrick MacMillan, a Scottish blacksmith, in 1839, although the claim is often disputed. He is also associated with the first recorded instance of a cycling traffic offense, when a Glasgow newspaper in 1842 reported an accident in which an anonymous "gentleman from Dumfries-shire... bestride a velocipede... of ingenious design" knocked over a little girl in Glasgow and was fined five shillings (). | Bicycle | Wikipedia | 316 | 3973 | https://en.wikipedia.org/wiki/Bicycle | Technology | Transportation | null |
In the early 1860s, Frenchmen Pierre Michaux and Pierre Lallement took bicycle design in a new direction by adding a mechanical crank drive with pedals on an enlarged front wheel (the velocipede). This was the first in mass production. Another French inventor named Douglas Grasso had a failed prototype of Pierre Lallement's bicycle several years earlier. Several inventions followed using rear-wheel drive, the best known being the rod-driven velocipede by Scotsman Thomas McCall in 1869. In that same year, bicycle wheels with wire spokes were patented by Eugène Meyer of Paris. The French vélocipède, made of iron and wood, developed into the "penny-farthing" (historically known as an "ordinary bicycle", a retronym, since there was then no other kind). It featured a tubular steel frame on which were mounted wire-spoked wheels with solid rubber tires. These bicycles were difficult to ride due to their high seat and poor weight distribution. In 1868 Rowley Turner, a sales agent of the Coventry Sewing Machine Company (which soon became the Coventry Machinists Company), brought a Michaux cycle to Coventry, England. His uncle, Josiah Turner, and business partner James Starley, used this as a basis for the 'Coventry Model' in what became Britain's first cycle factory.
The dwarf ordinary addressed some of these faults by reducing the front wheel diameter and setting the seat further back. This, in turn, required gearing—effected in a variety of ways—to efficiently use pedal power. Having to both pedal and steer via the front wheel remained a problem. Englishman J.K. Starley (nephew of James Starley), J.H. Lawson, and Shergold solved this problem by introducing the chain drive (originated by the unsuccessful "bicyclette" of Englishman Henry Lawson), connecting the frame-mounted cranks to the rear wheel. These models were known as safety bicycles, dwarf safeties, or upright bicycles for their lower seat height and better weight distribution, although without pneumatic tires the ride of the smaller-wheeled bicycle would be much rougher than that of the larger-wheeled variety. Starley's 1885 Rover, manufactured in Coventry is usually described as the first recognizably modern bicycle. Soon the seat tube was added which created the modern bike's double-triangle diamond frame. | Bicycle | Wikipedia | 495 | 3973 | https://en.wikipedia.org/wiki/Bicycle | Technology | Transportation | null |
Further innovations increased comfort and ushered in a second bicycle craze, the 1890s Golden Age of Bicycles. In 1888, Scotsman John Boyd Dunlop introduced the first practical pneumatic tire, which soon became universal. Willie Hume demonstrated the supremacy of Dunlop's tyres in 1889, winning the tyre's first-ever races in Ireland and then England. Soon after, the rear freewheel was developed, enabling the rider to coast. This refinement led to the 1890s invention of coaster brakes. Dérailleur gears and hand-operated Bowden cable-pull brakes were also developed during these years, but were only slowly adopted by casual riders.
The Svea Velocipede with vertical pedal arrangement and locking hubs was introduced in 1892 by the Swedish engineers Fredrik Ljungström and Birger Ljungström. It attracted attention at the World Fair and was produced in a few thousand units.
In the 1870s many cycling clubs flourished. They were popular in a time when there were no cars on the market and the principal mode of transportation was horse-drawn vehicles, such the horse and buggy or the horsecar. Among the earliest clubs was The Bicycle Touring Club, which has operated since 1878. By the turn of the century, cycling clubs flourished on both sides of the Atlantic, and touring and racing became widely popular. The Raleigh Bicycle Company was founded in Nottingham, England in 1888. It became the biggest bicycle manufacturing company in the world, making over two million bikes per year.
Bicycles and horse buggies were the two mainstays of private transportation just prior to the automobile, and the grading of smooth roads in the late 19th century was stimulated by the widespread advertising, production, and use of these devices. More than 1 billion bicycles have been manufactured worldwide as of the early 21st century. Bicycles are the most common vehicle of any kind in the world, and the most numerous model of any kind of vehicle, whether human-powered or motor vehicle, is the Chinese Flying Pigeon, with numbers exceeding 500 million. The next most numerous vehicle, the Honda Super Cub motorcycle, has more than 100 million units made, while most produced car, the Toyota Corolla, has reached 44 million and counting.
Uses
Bicycles are used for transportation, bicycle commuting, and utility cycling. They are also used professionally by mail carriers, paramedics, police, messengers, and general delivery services. Military uses of bicycles include communications, reconnaissance, troop movement, supply of provisions, and patrol, such as in bicycle infantries. | Bicycle | Wikipedia | 510 | 3973 | https://en.wikipedia.org/wiki/Bicycle | Technology | Transportation | null |
They are also used for recreational purposes, including bicycle touring, mountain biking, physical fitness, and play. Bicycle sports include racing, BMX racing, track racing, criterium, roller racing, sportives and time trials. Major multi-stage professional events are the Giro d'Italia, the Tour de France, the Vuelta a España, the Tour de Pologne, and the Volta a Portugal. They are also used for entertainment and pleasure in other ways, such as in organised mass rides, artistic cycling and freestyle BMX.
Technical aspects
The bicycle has undergone continual adaptation and improvement since its inception. These innovations have continued with the advent of modern materials and computer-aided design, allowing for a proliferation of specialized bicycle types, improved bicycle safety, and riding comfort.
Types
Bicycles can be categorized in many different ways: by function, by number of riders, by general construction, by gearing or by means of propulsion. The more common types include utility bicycles, mountain bicycles, racing bicycles, touring bicycles, hybrid bicycles, cruiser bicycles, and BMX bikes. Less common are tandems, low riders, tall bikes, fixed gear, folding models, amphibious bicycles, cargo bikes, recumbents and electric bicycles.
Unicycles, tricycles and quadracycles are not strictly bicycles, as they have respectively one, three and four wheels, but are often referred to informally as "bikes" or "cycles".
Dynamics
A bicycle stays upright while moving forward by being steered so as to keep its center of mass over the wheels. This steering is usually provided by the rider, but under certain conditions may be provided by the bicycle itself.
The combined center of mass of a bicycle and its rider must lean into a turn to successfully navigate it. This lean is induced by a method known as countersteering, which can be performed by the rider turning the handlebars directly with the hands or indirectly by leaning the bicycle.
Short-wheelbase or tall bicycles, when braking, can generate enough stopping force at the front wheel to flip longitudinally. The act of purposefully using this force to lift the rear wheel and balance on the front without tipping over is a trick known as a stoppie, endo, or front wheelie.
Performance | Bicycle | Wikipedia | 452 | 3973 | https://en.wikipedia.org/wiki/Bicycle | Technology | Transportation | null |
The bicycle is extraordinarily efficient in both biological and mechanical terms. The bicycle is the most efficient human-powered means of transportation in terms of energy a person must expend to travel a given distance. From a mechanical viewpoint, up to 99% of the energy delivered by the rider into the pedals is transmitted to the wheels, although the use of gearing mechanisms may reduce this by 10–15%. In terms of the ratio of cargo weight a bicycle can carry to total weight, it is also an efficient means of cargo transportation.
A human traveling on a bicycle at low to medium speeds of around uses only the power required to walk. Air drag, which is proportional to the square of speed, requires dramatically higher power outputs as speeds increase. If the rider is sitting upright, the rider's body creates about 75% of the total drag of the bicycle/rider combination. Drag can be reduced by seating the rider in a more aerodynamically streamlined position. Drag can also be reduced by covering the bicycle with an aerodynamic fairing. The fastest recorded unpaced speed on a flat surface is .
In addition, the carbon dioxide generated in the production and transportation of the food required by the bicyclist, per mile traveled, is less than that generated by energy efficient motorcars.
Parts
Frame
The great majority of modern bicycles have a frame with upright seating that looks much like the first chain-driven bike. These upright bicycles almost always feature the diamond frame, a truss consisting of two triangles: the front triangle and the rear triangle. The front triangle consists of the head tube, top tube, down tube, and seat tube. The head tube contains the headset, the set of bearings that allows the fork to turn smoothly for steering and balance. The top tube connects the head tube to the seat tube at the top, and the down tube connects the head tube to the bottom bracket. The rear triangle consists of the seat tube and paired chain stays and seat stays. The chain stays run parallel to the chain, connecting the bottom bracket to the rear dropout, where the axle for the rear wheel is held. The seat stays connect the top of the seat tube (at or near the same point as the top tube) to the rear fork ends. | Bicycle | Wikipedia | 456 | 3973 | https://en.wikipedia.org/wiki/Bicycle | Technology | Transportation | null |
Historically, women's bicycle frames had a top tube that connected in the middle of the seat tube instead of the top, resulting in a lower standover height at the expense of compromised structural integrity, since this places a strong bending load in the seat tube, and bicycle frame members are typically weak in bending. This design, referred to as a step-through frame or as an open frame, allows the rider to mount and dismount in a dignified way while wearing a skirt or dress. While some women's bicycles continue to use this frame style, there is also a variation, the mixte, which splits the top tube laterally into two thinner top tubes that bypass the seat tube on each side and connect to the rear fork ends. The ease of stepping through is also appreciated by those with limited flexibility or other joint problems. Because of its persistent image as a "women's" bicycle, step-through frames are not common for larger frames.
Step-throughs were popular partly for practical reasons and partly for social mores of the day. For most of the history of bicycles' popularity women have worn long skirts, and the lower frame accommodated these better than the top-tube. Furthermore, it was considered "unladylike" for women to open their legs to mount and dismount—in more conservative times women who rode bicycles at all were vilified as immoral or immodest. These practices were akin to the older practice of riding horse sidesaddle.
Another style is the recumbent bicycle. These are inherently more aerodynamic than upright versions, as the rider may lean back onto a support and operate pedals that are on about the same level as the seat. The world's fastest bicycle is a recumbent bicycle but this type was banned from competition in 1934 by the Union Cycliste Internationale. | Bicycle | Wikipedia | 374 | 3973 | https://en.wikipedia.org/wiki/Bicycle | Technology | Transportation | null |
Historically, materials used in bicycles have followed a similar pattern as in aircraft, the goal being high strength and low weight. Since the late 1930s alloy steels have been used for frame and fork tubes in higher quality machines. By the 1980s aluminum welding techniques had improved to the point that aluminum tube could safely be used in place of steel. Since then aluminum alloy frames and other components have become popular due to their light weight, and most mid-range bikes are now principally aluminum alloy of some kind. More expensive bikes use carbon fibre due to its significantly lighter weight and profiling ability, allowing designers to make a bike both stiff and compliant by manipulating the lay-up. Virtually all professional racing bicycles now use carbon fibre frames, as they have the best strength to weight ratio. A typical modern carbon fiber frame can weigh less than .
Other exotic frame materials include titanium and advanced alloys. Bamboo, a natural composite material with high strength-to-weight ratio and stiffness has been used for bicycles since 1894. Recent versions use bamboo for the primary frame with glued metal connections and parts, priced as exotic models.
Drivetrain and gearing
The drivetrain begins with pedals which rotate the cranks, which are held in axis by the bottom bracket. Most bicycles use a chain to transmit power to the rear wheel. A very small number of bicycles use a shaft drive to transmit power, or special belts. Hydraulic bicycle transmissions have been built, but they are currently inefficient and complex.
Since cyclists' legs are most efficient over a narrow range of pedaling speeds, or cadence, a variable gear ratio helps a cyclist to maintain an optimum pedalling speed while covering varied terrain. Some, mainly utility, bicycles use hub gears with between 3 and 14 ratios, but most use the generally more efficient dérailleur system, by which the chain is moved between different cogs called chainrings and sprockets to select a ratio. A dérailleur system normally has two dérailleurs, or mechs, one at the front to select the chainring and another at the back to select the sprocket. Most bikes have two or three chainrings, and from 5 to 11 sprockets on the back, with the number of theoretical gears calculated by multiplying front by back. In reality, many gears overlap or require the chain to run diagonally, so the number of usable gears is fewer. | Bicycle | Wikipedia | 489 | 3973 | https://en.wikipedia.org/wiki/Bicycle | Technology | Transportation | null |
An alternative to chaindrive is to use a synchronous belt. These are toothed and work much the same as a chain—popular with commuters and long distance cyclists they require little maintenance. They cannot be shifted across a cassette of sprockets, and are used either as single speed or with a hub gear.
Different gears and ranges of gears are appropriate for different people and styles of cycling. Multi-speed bicycles allow gear selection to suit the circumstances: a cyclist could use a high gear when cycling downhill, a medium gear when cycling on a flat road, and a low gear when cycling uphill. In a lower gear every turn of the pedals leads to fewer rotations of the rear wheel. This allows the energy required to move the same distance to be distributed over more pedal turns, reducing fatigue when riding uphill, with a heavy load, or against strong winds. A higher gear allows a cyclist to make fewer pedal turns to maintain a given speed, but with more effort per turn of the pedals.
With a chain drive transmission, a chainring attached to a crank drives the chain, which in turn rotates the rear wheel via the rear sprocket(s) (cassette or freewheel). There are four gearing options: two-speed hub gear integrated with chain ring, up to 3 chain rings, up to 12 sprockets, hub gear built into rear wheel (3-speed to 14-speed). The most common options are either a rear hub or multiple chain rings combined with multiple sprockets (other combinations of options are possible but less common).
Steering
The handlebars connect to the stem that connects to the fork that connects to the front wheel, and the whole assembly connects to the bike and rotates about the steering axis via the headset bearings. Three styles of handlebar are common. Upright handlebars, the norm in Europe and elsewhere until the 1970s, curve gently back toward the rider, offering a natural grip and comfortable upright position. Drop handlebars "drop" as they curve forward and down, offering the cyclist best braking power from a more aerodynamic "crouched" position, as well as more upright positions in which the hands grip the brake lever mounts, the forward curves, or the upper flat sections for increasingly upright postures. Mountain bikes generally feature a 'straight handlebar' or 'riser bar' with varying degrees of sweep backward and centimeters rise upwards, as well as wider widths which can provide better handling due to increased leverage against the wheel. | Bicycle | Wikipedia | 512 | 3973 | https://en.wikipedia.org/wiki/Bicycle | Technology | Transportation | null |
Seating
Saddles also vary with rider preference, from the cushioned ones favored by short-distance riders to narrower saddles which allow more room for leg swings. Comfort depends on riding position. With comfort bikes and hybrids, cyclists sit high over the seat, their weight directed down onto the saddle, such that a wider and more cushioned saddle is preferable. For racing bikes where the rider is bent over, weight is more evenly distributed between the handlebars and saddle, the hips are flexed, and a narrower and harder saddle is more efficient. Differing saddle designs exist for male and female cyclists, accommodating the genders' differing anatomies and sit bone width measurements, although bikes typically are sold with saddles most appropriate for men. Suspension seat posts and seat springs provide comfort by absorbing shock but can add to the overall weight of the bicycle.
A recumbent bicycle has a reclined chair-like seat that some riders find more comfortable than a saddle, especially riders who suffer from certain types of seat, back, neck, shoulder, or wrist pain. Recumbent bicycles may have either under-seat or over-seat steering.
Brakes
Bicycle brakes may be rim brakes, in which friction pads are compressed against the wheel rims; hub brakes, where the mechanism is contained within the wheel hub, or disc brakes, where pads act on a rotor attached to the hub. Most road bicycles use rim brakes, but some use disc brakes. Disc brakes are more common for mountain bikes, tandems and recumbent bicycles than on other types of bicycles, due to their increased power, coupled with an increased weight and complexity.
With hand-operated brakes, force is applied to brake levers mounted on the handlebars and transmitted via Bowden cables or hydraulic lines to the friction pads, which apply pressure to the braking surface, causing friction which slows the bicycle down. A rear hub brake may be either hand-operated or pedal-actuated, as in the back pedal coaster brakes which were popular in North America until the 1960s. | Bicycle | Wikipedia | 413 | 3973 | https://en.wikipedia.org/wiki/Bicycle | Technology | Transportation | null |
Track bicycles do not have brakes, because all riders ride in the same direction around a track which does not necessitate sharp deceleration. Track riders are still able to slow down because all track bicycles are fixed-gear, meaning that there is no freewheel. Without a freewheel, coasting is impossible, so when the rear wheel is moving, the cranks are moving. To slow down, the rider applies resistance to the pedals, acting as a braking system which can be as effective as a conventional rear wheel brake, but not as effective as a front wheel brake.
Suspension
Bicycle suspension refers to the system or systems used to suspend the rider and all or part of the bicycle. This serves two purposes: to keep the wheels in continuous contact with the ground, improving control, and to isolate the rider and luggage from jarring due to rough surfaces, improving comfort.
Bicycle suspensions are used primarily on mountain bicycles, but are also common on hybrid bicycles, as they can help deal with problematic vibration from poor surfaces. Suspension is especially important on recumbent bicycles, since while an upright bicycle rider can stand on the pedals to achieve some of the benefits of suspension, a recumbent rider cannot.
Basic mountain bicycles and hybrids usually have front suspension only, whilst more sophisticated ones also have rear suspension. Road bicycles tend to have no suspension.
Wheels and tires
The wheel axle fits into fork ends in the frame and fork. A pair of wheels may be called a wheelset, especially in the context of ready-built "off the shelf", performance-oriented wheels.
Tires vary enormously depending on their intended purpose. Road bicycles use tires 18 to 25 millimeters wide, most often completely smooth, or slick, and inflated to high pressure to roll fast on smooth surfaces. Off-road tires are usually between wide, and have treads for gripping in muddy conditions or metal studs for ice.
Groupset
Groupset generally refers to all of the components that make up a bicycle excluding the bicycle frame, fork, stem, wheels, tires, and rider contact points, such as the saddle and handlebars.
Accessories | Bicycle | Wikipedia | 429 | 3973 | https://en.wikipedia.org/wiki/Bicycle | Technology | Transportation | null |
Some components, which are often optional accessories on sports bicycles, are standard features on utility bicycles to enhance their usefulness, comfort, safety and visibility. Fenders with spoilers (mudflaps) protect the cyclist and moving parts from spray when riding through wet areas. In some countries (e.g. Germany, UK), fenders are called mudguards. The chainguards protect clothes from oil on the chain while preventing clothing from being caught between the chain and crankset teeth. Kick stands keep bicycles upright when parked, and bike locks deter theft. Front-mounted baskets, front or rear luggage carriers or racks, and panniers mounted above either or both wheels can be used to carry equipment or cargo. Pegs can be fastened to one, or both of the wheel hubs to either help the rider perform certain tricks, or allow a place for extra riders to stand, or rest. Parents sometimes add rear-mounted child seats, an auxiliary saddle fitted to the crossbar, or both to transport children. Bicycles can also be fitted with a hitch to tow a trailer for carrying cargo, a child, or both.
Toe-clips and toestraps and clipless pedals help keep the foot locked in the proper pedal position and enable cyclists to pull and push the pedals. Technical accessories include cyclocomputers for measuring speed, distance, heart rate, GPS data etc. Other accessories include lights, reflectors, mirrors, racks, trailers, bags, water bottles and cages, and bell. Bicycle lights, reflectors, and helmets are required by law in some geographic regions depending on the legal code. It is more common to see bicycles with bottle generators, dynamos, lights, fenders, racks and bells in Europe. Bicyclists also have specialized form fitting and high visibility clothing.
Children's bicycles may be outfitted with cosmetic enhancements such as bike horns, streamers, and spoke beads. Training wheels are sometimes used when learning to ride, but a dedicated balance bike teaches independent riding more effectively.
Bicycle helmets can reduce injury in the event of a collision or accident, and a suitable helmet is legally required of riders in many jurisdictions. Helmets may be classified as an accessory or as an item of clothing.
Bike trainers are used to enable cyclists to cycle while the bike remains stationary. They are frequently used to warm up before races or indoors when riding conditions are unfavorable. | Bicycle | Wikipedia | 491 | 3973 | https://en.wikipedia.org/wiki/Bicycle | Technology | Transportation | null |
Standards
A number of formal and industry standards exist for bicycle components to help make spare parts exchangeable and to maintain a minimum product safety.
The International Organization for Standardization (ISO) has a special technical committee for cycles, TC149, that has the scope of "Standardization in the field of cycles, their components and accessories with particular reference to terminology, testing methods and requirements for performance and safety, and interchangeability".
The European Committee for Standardization (CEN) also has a specific Technical Committee, TC333, that defines European standards for cycles. Their mandate states that EN cycle standards shall harmonize with ISO standards. Some CEN cycle standards were developed before ISO published their standards, leading to strong European influences in this area. European cycle standards tend to describe minimum safety requirements, while ISO standards have historically harmonized parts geometry.
Maintenance and repair
Like all devices with mechanical moving parts, bicycles require a certain amount of regular maintenance and replacement of worn parts. A bicycle is relatively simple compared with a car, so some cyclists choose to do at least part of the maintenance themselves. Some components are easy to handle using relatively simple tools, while other components may require specialist manufacturer-dependent tools.
Many bicycle components are available at several different price/quality points; manufacturers generally try to keep all components on any particular bike at about the same quality level, though at the very cheap end of the market there may be some skimping on less obvious components (e.g. bottom bracket).
There are several hundred assisted-service Community Bicycle Organizations worldwide. At a Community Bicycle Organization, laypeople bring in bicycles needing repair or maintenance; volunteers teach them how to do the required steps.
Full service is available from bicycle mechanics at a local bike shop.
In areas where it is available, some cyclists purchase roadside assistance from companies such as the Better World Club or the American Automobile Association.
Maintenance
The most basic maintenance item is keeping the tires correctly inflated; this can make a noticeable difference as to how the bike feels to ride. Bicycle tires usually have a marking on the sidewall indicating the pressure appropriate for that tire. Bicycles use much higher pressures than cars: car tires are normally in the range of , whereas bicycle tires are normally in the range of . | Bicycle | Wikipedia | 449 | 3973 | https://en.wikipedia.org/wiki/Bicycle | Technology | Transportation | null |
Another basic maintenance item is regular lubrication of the chain and pivot points for derailleurs and brake components. Most of the bearings on a modern bike are sealed and grease-filled and require little or no attention; such bearings will usually last for or more. The crank bearings require periodic maintenance, which involves removing, cleaning and repacking with the correct grease.
The chain and the brake blocks are the components which wear out most quickly, so these need to be checked from time to time, typically every or so. Most local bike shops will do such checks for free. Note that when a chain becomes badly worn it will also wear out the rear cogs/cassette and eventually the chain ring(s), so replacing a chain when only moderately worn will prolong the life of other components.
Over the longer term, tires do wear out, after ; a rash of punctures is often the most visible sign of a worn tire.
Repair
Very few bicycle components can actually be repaired; replacement of the failing component is the normal practice.
The most common roadside problem is a puncture of the tire's inner tube. A patch kit may be employed to fix the puncture or the tube can be replaced, though the latter solution comes at a greater cost and waste of material. Some brands of tires are much more puncture-resistant than others, often incorporating one or more layers of Kevlar; the downside of such tires is that they may be heavier and/or more difficult to fit and remove.
Tools
There are specialized bicycle tools for use both in the shop and at the roadside. Many cyclists carry tool kits. These may include a tire patch kit (which, in turn, may contain any combination of a hand pump or CO2 pump, tire levers, spare tubes, self-adhesive patches, or tube-patching material, an adhesive, a piece of sandpaper or a metal grater (for roughening the tube surface to be patched) and sometimes even a block of French chalk), wrenches, hex keys, screwdrivers, and a chain tool. Special, thin wrenches are often required for maintaining various screw-fastened parts, specifically, the frequently lubricated ball-bearing "cones". There are also cycling-specific multi-tools that combine many of these implements into a single compact device. More specialized bicycle components may require more complex tools, including proprietary tools specific for a given manufacturer. | Bicycle | Wikipedia | 499 | 3973 | https://en.wikipedia.org/wiki/Bicycle | Technology | Transportation | null |
Social and historical aspects
The bicycle has had a considerable effect on human society, in both the cultural and industrial realms.
In daily life
Around the turn of the 20th century, bicycles reduced crowding in inner-city tenements by allowing workers to commute from more spacious dwellings in the suburbs. They also reduced dependence on horses. Bicycles allowed people to travel for leisure into the country, since bicycles were three times as energy efficient as walking and three to four times as fast.
In built-up cities around the world, urban planning uses cycling infrastructure like bikeways to reduce traffic congestion and air pollution. A number of cities around the world have implemented schemes known as bicycle sharing systems or community bicycle programs. The first of these was the White Bicycle plan in Amsterdam in 1965. It was followed by yellow bicycles in La Rochelle and green bicycles in Cambridge. These initiatives complement public transport systems and offer an alternative to motorized traffic to help reduce congestion and pollution. In Europe, especially in the Netherlands and parts of Germany and Denmark, bicycle commuting is common. In Copenhagen, a cyclists' organization runs a Cycling Embassy that promotes biking for commuting and sightseeing. The United Kingdom has a tax break scheme (IR 176) that allows employees to buy a new bicycle tax free to use for commuting.
In the Netherlands all train stations offer free bicycle parking, or a more secure parking place for a small fee, with the larger stations also offering bicycle repair shops. Cycling is so popular that the parking capacity may be exceeded, while in some places such as Delft the capacity is usually exceeded. In Trondheim in Norway, the Trampe bicycle lift has been developed to encourage cyclists by giving assistance on a steep hill. Buses in many cities have bicycle carriers mounted on the front.
There are towns in some countries where bicycle culture has been an integral part of the landscape for generations, even without much official support. That is the case of Ílhavo, in Portugal.
In cities where bicycles are not integrated into the public transportation system, commuters often use bicycles as elements of a mixed-mode commute, where the bike is used to travel to and from train stations or other forms of rapid transit. Some students who commute several miles drive a car from home to a campus parking lot, then ride a bicycle to class. Folding bicycles are useful in these scenarios, as they are less cumbersome when carried aboard. Los Angeles removed a small amount of seating on some trains to make more room for bicycles and wheel chairs. | Bicycle | Wikipedia | 506 | 3973 | https://en.wikipedia.org/wiki/Bicycle | Technology | Transportation | null |
Some US companies, notably in the tech sector, are developing both innovative cycle designs and cycle-friendliness in the workplace. Foursquare, whose CEO Dennis Crowley "pedaled to pitch meetings ... [when he] was raising money from venture capitalists" on a two-wheeler, chose a new location for its New York headquarters "based on where biking would be easy". Parking in the office was also integral to HQ planning. Mitchell Moss, who runs the Rudin Center for Transportation Policy & Management at New York University, said in 2012: "Biking has become the mode of choice for the educated high tech worker".
Bicycles offer an important mode of transport in many developing countries. Until recently, bicycles have been a staple of everyday life throughout Asian countries. They are the most frequently used method of transport for commuting to work, school, shopping, and life in general. In Europe, bicycles are commonly used. They also offer a degree of exercise to keep individuals healthy.
Bicycles are also celebrated in the visual arts. An example of this is the Bicycle Film Festival, a film festival hosted all around the world.
Poverty alleviation
Female emancipation
The safety bicycle gave women unprecedented mobility, contributing to their emancipation in Western nations. As bicycles became safer and cheaper, more women had access to the personal freedom that bicycles embodied, and so the bicycle came to symbolize the New Woman of the late 19th century, especially in Britain and the United States. The bicycle craze in the 1890s also led to a movement for so-called rational dress, which helped liberate women from corsets and ankle-length skirts and other restrictive garments, substituting the then-shocking bloomers. | Bicycle | Wikipedia | 344 | 3973 | https://en.wikipedia.org/wiki/Bicycle | Technology | Transportation | null |
The bicycle was recognized by 19th-century feminists and suffragists as a "freedom machine" for women. American Susan B. Anthony said in a New York World interview on 2 February 1896: "I think it has done more to emancipate woman than any one thing in the world. I rejoice every time I see a woman ride by on a wheel. It gives her a feeling of self-reliance and independence the moment she takes her seat; and away she goes, the picture of untrammelled womanhood." In 1895 Frances Willard, the tightly laced president of the Woman's Christian Temperance Union, wrote A Wheel Within a Wheel: How I Learned to Ride the Bicycle, with Some Reflections by the Way, a 75-page illustrated memoir praising "Gladys", her bicycle, for its "gladdening effect" on her health and political optimism. Willard used a cycling metaphor to urge other suffragists to action.
In 1985, Georgena Terry started the first women-specific bicycle company. Her designs featured frame geometry and wheel sizes chosen to better fit women, with shorter top tubes and more suitable reach.
Economic implications
Bicycle manufacturing proved to be a training ground for other industries and led to the development of advanced metalworking techniques, both for the frames themselves and for special components such as ball bearings, washers, and sprockets. These techniques later enabled skilled metalworkers and mechanics to develop the components used in early automobiles and aircraft.
Wilbur and Orville Wright, a pair of businessmen, ran the Wright Cycle Company which designed, manufactured and sold their bicycles during the bike boom of the 1890s.
They also served to teach the industrial models later adopted, including mechanization and mass production (later copied and adopted by Ford and General Motors), vertical integration (also later copied and adopted by Ford), aggressive advertising (as much as 10% of all advertising in U.S. periodicals in 1898 was by bicycle makers), lobbying for better roads (which had the side benefit of acting as advertising, and of improving sales by providing more places to ride), all first practiced by Pope. In addition, bicycle makers adopted the annual model change (later derided as planned obsolescence, and usually credited to General Motors), which proved very successful. | Bicycle | Wikipedia | 466 | 3973 | https://en.wikipedia.org/wiki/Bicycle | Technology | Transportation | null |
Early bicycles were an example of conspicuous consumption, being adopted by the fashionable elites. In addition, by serving as a platform for accessories, which could ultimately cost more than the bicycle itself, it paved the way for the likes of the Barbie doll.
Bicycles helped create, or enhance, new kinds of businesses, such as bicycle messengers, traveling seamstresses, riding academies, and racing rinks. Their board tracks were later adapted to early motorcycle and automobile racing. There were a variety of new inventions, such as spoke tighteners, and specialized lights, socks and shoes, and even cameras, such as the Eastman Company's Poco. Probably the best known and most widely used of these inventions, adopted well beyond cycling, is Charles Bennett's Bike Web, which came to be called the jock strap.
They also presaged a move away from public transit that would explode with the introduction of the automobile.
J. K. Starley's company became the Rover Cycle Company Ltd. in the late 1890s, and then renamed the Rover Company when it started making cars. Morris Motors Limited (in Oxford) and Škoda also began in the bicycle business, as did the Wright brothers. Alistair Craig, whose company eventually emerged to become the engine manufacturers Ailsa Craig, also started from manufacturing bicycles, in Glasgow in March 1885.
In general, U.S. and European cycle manufacturers used to assemble cycles from their own frames and components made by other companies, although very large companies (such as Raleigh) used to make almost every part of a bicycle (including bottom brackets, axles, etc.) In recent years, those bicycle makers have greatly changed their methods of production. Now, almost none of them produce their own frames.
Many newer or smaller companies only design and market their products; the actual production is done by Asian companies. For example, some 60% of the world's bicycles are now being made in China. Despite this shift in production, as nations such as China and India become more wealthy, their own use of bicycles has declined due to the increasing affordability of cars and motorcycles. One of the major reasons for the proliferation of Chinese-made bicycles in foreign markets is the lower cost of labor in China.
In line with the European financial crisis of that time, in 2011 the number of bicycle sales in Italy (1.75 million) passed the number of new car sales.
Environmental impact | Bicycle | Wikipedia | 485 | 3973 | https://en.wikipedia.org/wiki/Bicycle | Technology | Transportation | null |
One of the profound economic implications of bicycle use is that it liberates the user from motor fuel consumption. (Ballantine, 1972) The bicycle is an inexpensive, fast, healthy and environmentally friendly mode of transport. Ivan Illich stated that bicycle use extended the usable physical environment for people, while alternatives such as cars and motorways degraded and confined people's environment and mobility. Currently, two billion bicycles are in use around the world. Children, students, professionals, laborers, civil servants and seniors are pedaling around their communities. They all experience the freedom and the natural opportunity for exercise that the bicycle easily provides. Bicycle also has lowest carbon intensity of travel.
Manufacturing
The global bicycle market is $61 billion in 2011. , 130 million bicycles were sold every year globally and 66% of them were made in China.
Legal requirements
Early in its development, as with automobiles, there were restrictions on the operation of bicycles. Along with advertising, and to gain free publicity, Albert A. Pope litigated on behalf of cyclists.
The 1968 Vienna Convention on Road Traffic of the United Nations considers a bicycle to be a vehicle, and a person controlling a bicycle (whether actually riding or not) is considered an operator or driver. The traffic codes of many countries reflect these definitions and demand that a bicycle satisfy certain legal requirements before it can be used on public roads. In many jurisdictions, it is an offense to use a bicycle that is not in a roadworthy condition.
In some countries, bicycles must have functioning front and rear lights when ridden after dark.
Some countries require child and/or adult cyclists to wear helmets, as this may protect riders from head trauma. Countries which require adult cyclists to wear helmets include Spain, New Zealand and Australia. Mandatory helmet wearing is one of the most controversial topics in the cycling world, with proponents arguing that it reduces head injuries and thus is an acceptable requirement, while opponents argue that by making cycling seem more dangerous and cumbersome, it reduces cyclist numbers on the streets, creating an overall negative health effect (fewer people cycling for their own health, and the remaining cyclists being more exposed through a reversed safety in numbers effect).
Theft | Bicycle | Wikipedia | 435 | 3973 | https://en.wikipedia.org/wiki/Bicycle | Technology | Transportation | null |
Bicycles are popular targets for theft, due to their value and ease of resale. The number of bicycles stolen annually is difficult to quantify as a large number of crimes are not reported. Around 50% of the participants in the Montreal International Journal of Sustainable Transportation survey were subjected to a bicycle theft in their lifetime as active cyclists. Most bicycles have serial numbers that can be recorded to verify identity in case of theft. | Bicycle | Wikipedia | 83 | 3973 | https://en.wikipedia.org/wiki/Bicycle | Technology | Transportation | null |
In inorganic chemistry, bicarbonate (IUPAC-recommended nomenclature: hydrogencarbonate) is an intermediate form in the deprotonation of carbonic acid. It is a polyatomic anion with the chemical formula .
Bicarbonate serves a crucial biochemical role in the physiological pH buffering system.
The term "bicarbonate" was coined in 1814 by the English chemist William Hyde Wollaston. The name lives on as a trivial name.
Chemical properties
The bicarbonate ion (hydrogencarbonate ion) is an anion with the empirical formula and a molecular mass of 61.01 daltons; it consists of one central carbon atom surrounded by three oxygen atoms in a trigonal planar arrangement, with a hydrogen atom attached to one of the oxygens. It is isoelectronic with nitric acid . The bicarbonate ion carries a negative one formal charge and is an amphiprotic species which has both acidic and basic properties. It is both the conjugate base of carbonic acid ; and the conjugate acid of , the carbonate ion, as shown by these equilibrium reactions:
+ 2 H2O + H2O + OH− H2CO3 + 2 OH−
H2CO3 + 2 H2O + H3O+ + H2O + 2 H3O+.
A bicarbonate salt forms when a positively charged ion attaches to the negatively charged oxygen atoms of the ion, forming an ionic compound. Many bicarbonates are soluble in water at standard temperature and pressure; in particular, sodium bicarbonate contributes to total dissolved solids, a common parameter for assessing water quality.
Physiological role
Bicarbonate () is a vital component of the pH buffering system of the human body (maintaining acid–base homeostasis). 70%–75% of CO2 in the body is converted into carbonic acid (H2CO3), which is the conjugate acid of and can quickly turn into it. | Bicarbonate | Wikipedia | 407 | 3982 | https://en.wikipedia.org/wiki/Bicarbonate | Physical sciences | Carbonic oxyanions | Chemistry |
With carbonic acid as the central intermediate species, bicarbonate – in conjunction with water, hydrogen ions, and carbon dioxide – forms this buffering system, which is maintained at the volatile equilibrium required to provide prompt resistance to pH changes in both the acidic and basic directions. This is especially important for protecting tissues of the central nervous system, where pH changes too far outside of the normal range in either direction could prove disastrous (see acidosis or alkalosis). Recently it has been also demonstrated that cellular bicarbonate metabolism can be regulated by mTORC1 signaling.
Additionally, bicarbonate plays a key role in the digestive system. It raises the internal pH of the stomach, after highly acidic digestive juices have finished in their digestion of food. Bicarbonate also acts to regulate pH in the small intestine. It is released from the pancreas in response to the hormone secretin to neutralize the acidic chyme entering the duodenum from the stomach.
Bicarbonate in the environment
Bicarbonate is the dominant form of dissolved inorganic carbon in sea water, and in most fresh waters. As such it is an important sink in the carbon cycle.
Some plants like Chara utilize carbonate and produce calcium carbonate (CaCO3) as result of biological metabolism.
In freshwater ecology, strong photosynthetic activity by freshwater plants in daylight releases gaseous oxygen into the water and at the same time produces bicarbonate ions. These shift the pH upward until in certain circumstances the degree of alkalinity can become toxic to some organisms or can make other chemical constituents such as ammonia toxic. In darkness, when no photosynthesis occurs, respiration processes release carbon dioxide, and no new bicarbonate ions are produced, resulting in a rapid fall in pH.
The flow of bicarbonate ions from rocks weathered by the carbonic acid in rainwater is an important part of the carbon cycle.
Other uses
The most common salt of the bicarbonate ion is sodium bicarbonate, NaHCO3, which is commonly known as baking soda. When heated or exposed to an acid such as acetic acid (vinegar), sodium bicarbonate releases carbon dioxide. This is used as a leavening agent in baking.
Ammonium bicarbonate is used in the manufacturing of some cookies, crackers, and biscuits. | Bicarbonate | Wikipedia | 475 | 3982 | https://en.wikipedia.org/wiki/Bicarbonate | Physical sciences | Carbonic oxyanions | Chemistry |
Diagnostics
In diagnostic medicine, the blood value of bicarbonate is one of several indicators of the state of acid–base physiology in the body. It is measured, along with chloride, potassium, and sodium, to assess electrolyte levels in an electrolyte panel test (which has Current Procedural Terminology, CPT, code 80051).
The parameter standard bicarbonate concentration (SBCe) is the bicarbonate concentration in the blood at a PaCO2 of , full oxygen saturation and 36 °C.
Bicarbonate compounds
Sodium bicarbonate
Potassium bicarbonate
Caesium bicarbonate
Magnesium bicarbonate
Calcium bicarbonate
Ammonium bicarbonate
Carbonic acid | Bicarbonate | Wikipedia | 140 | 3982 | https://en.wikipedia.org/wiki/Bicarbonate | Physical sciences | Carbonic oxyanions | Chemistry |
In mathematics, more specifically in functional analysis, a Banach space (pronounced ) is a complete normed vector space. Thus, a Banach space is a vector space with a metric that allows the computation of vector length and distance between vectors and is complete in the sense that a Cauchy sequence of vectors always converges to a well-defined limit that is within the space.
Banach spaces are named after the Polish mathematician Stefan Banach, who introduced this concept and studied it systematically in 1920–1922 along with Hans Hahn and Eduard Helly.
Maurice René Fréchet was the first to use the term "Banach space" and Banach in turn then coined the term "Fréchet space".
Banach spaces originally grew out of the study of function spaces by Hilbert, Fréchet, and Riesz earlier in the century. Banach spaces play a central role in functional analysis. In other areas of analysis, the spaces under study are often Banach spaces.
Definition
A Banach space is a complete normed space
A normed space is a pair
consisting of a vector space over a scalar field (where is commonly or ) together with a distinguished
norm Like all norms, this norm induces a translation invariant
distance function, called the canonical or (norm) induced metric, defined for all vectors by
This makes into a metric space
A sequence is called or or if for every real there exists some index such that
whenever and are greater than
The normed space is called a and the canonical metric is called a if is a , which by definition means for every Cauchy sequence in there exists some such that
where because this sequence's convergence to can equivalently be expressed as:
The norm of a normed space is called a if is a Banach space.
L-semi-inner product
For any normed space there exists an L-semi-inner product on such that for all in general, there may be infinitely many L-semi-inner products that satisfy this condition. L-semi-inner products are a generalization of inner products, which are what fundamentally distinguish Hilbert spaces from all other Banach spaces. This shows that all normed spaces (and hence all Banach spaces) can be considered as being generalizations of (pre-)Hilbert spaces.
Characterization in terms of series | Banach space | Wikipedia | 467 | 3989 | https://en.wikipedia.org/wiki/Banach%20space | Mathematics | Linear algebra | null |
The vector space structure allows one to relate the behavior of Cauchy sequences to that of converging series of vectors.
A normed space is a Banach space if and only if each absolutely convergent series in converges to a value that lies within
Topology
The canonical metric of a normed space induces the usual metric topology on which is referred to as the canonical or norm induced topology.
Every normed space is automatically assumed to carry this Hausdorff topology, unless indicated otherwise.
With this topology, every Banach space is a Baire space, although there exist normed spaces that are Baire but not Banach. The norm is always a continuous function with respect to the topology that it induces.
The open and closed balls of radius centered at a point are, respectively, the sets
Any such ball is a convex and bounded subset of but a compact ball / neighborhood exists if and only if is a finite-dimensional vector space.
In particular, no infinite–dimensional normed space can be locally compact or have the Heine–Borel property.
If is a vector and is a scalar then
Using shows that this norm-induced topology is translation invariant, which means that for any and the subset is open (respectively, closed) in if and only if this is true of its translation
Consequently, the norm induced topology is completely determined by any neighbourhood basis at the origin. Some common neighborhood bases at the origin include:
where is a sequence in of positive real numbers that converges to in (such as or for instance).
So for example, every open subset of can be written as a union
indexed by some subset where every may be picked from the aforementioned sequence (the open balls can be replaced with closed balls, although then the indexing set and radii may also need to be replaced).
Additionally, can always be chosen to be countable if is a , which by definition means that contains some countable dense subset.
Homeomorphism classes of separable Banach spaces
All finite–dimensional normed spaces are separable Banach spaces and any two Banach spaces of the same finite dimension are linearly homeomorphic.
Every separable infinite–dimensional Hilbert space is linearly isometrically isomorphic to the separable Hilbert sequence space with its usual norm | Banach space | Wikipedia | 461 | 3989 | https://en.wikipedia.org/wiki/Banach%20space | Mathematics | Linear algebra | null |
The Anderson–Kadec theorem states that every infinite–dimensional separable Fréchet space is homeomorphic to the product space of countably many copies of (this homeomorphism need not be a linear map).
Thus all infinite–dimensional separable Fréchet spaces are homeomorphic to each other (or said differently, their topology is unique up to a homeomorphism).
Since every Banach space is a Fréchet space, this is also true of all infinite–dimensional separable Banach spaces, including
In fact, is even homeomorphic to its own unit which stands in sharp contrast to finite–dimensional spaces (the Euclidean plane is not homeomorphic to the unit circle, for instance).
This pattern in homeomorphism classes extends to generalizations of metrizable (locally Euclidean) topological manifolds known as , which are metric spaces that are around every point, locally homeomorphic to some open subset of a given Banach space (metric Hilbert manifolds and metric Fréchet manifolds are defined similarly).
For example, every open subset of a Banach space is canonically a metric Banach manifold modeled on since the inclusion map is an open local homeomorphism.
Using Hilbert space microbundles, David Henderson showed in 1969 that every metric manifold modeled on a separable infinite–dimensional Banach (or Fréchet) space can be topologically embedded as an subset of and, consequently, also admits a unique smooth structure making it into a Hilbert manifold.
Compact and convex subsets
There is a compact subset of whose convex hull is closed and thus also compact (see this footnote for an example).
However, like in all Banach spaces, the convex hull of this (and every other) compact subset will be compact. But if a normed space is not complete then it is in general guaranteed that will be compact whenever is; an example can even be found in a (non-complete) pre-Hilbert vector subspace of
As a topological vector space | Banach space | Wikipedia | 407 | 3989 | https://en.wikipedia.org/wiki/Banach%20space | Mathematics | Linear algebra | null |
This norm-induced topology also makes into what is known as a topological vector space (TVS), which by definition is a vector space endowed with a topology making the operations of addition and scalar multiplication continuous. It is emphasized that the TVS is a vector space together with a certain type of topology; that is to say, when considered as a TVS, it is associated with particular norm or metric (both of which are "forgotten"). This Hausdorff TVS is even locally convex because the set of all open balls centered at the origin forms a neighbourhood basis at the origin consisting of convex balanced open sets. This TVS is also , which by definition refers to any TVS whose topology is induced by some (possibly unknown) norm. Normable TVSs are characterized by being Hausdorff and having a bounded convex neighborhood of the origin.
All Banach spaces are barrelled spaces, which means that every barrel is neighborhood of the origin (all closed balls centered at the origin are barrels, for example) and guarantees that the Banach–Steinhaus theorem holds.
Comparison of complete metrizable vector topologies
The open mapping theorem implies that if and are topologies on that make both and into complete metrizable TVS (for example, Banach or Fréchet spaces) and if one topology is finer or coarser than the other then they must be equal (that is, if or then ).
So for example, if and are Banach spaces with topologies and and if one of these spaces has some open ball that is also an open subset of the other space (or equivalently, if one of or is continuous) then their topologies are identical and their norms are equivalent.
Completeness
Complete norms and equivalent norms
Two norms, and on a vector space are said to be if they induce the same topology; this happens if and only if there exist positive real numbers such that for all If and are two equivalent norms on a vector space then is a Banach space if and only if is a Banach space.
See this footnote for an example of a continuous norm on a Banach space that is equivalent to that Banach space's given norm.
All norms on a finite-dimensional vector space are equivalent and every finite-dimensional normed space is a Banach space.
Complete norms vs complete metrics | Banach space | Wikipedia | 474 | 3989 | https://en.wikipedia.org/wiki/Banach%20space | Mathematics | Linear algebra | null |
A metric on a vector space is induced by a norm on if and only if is translation invariant and , which means that for all scalars and all in which case the function defines a norm on and the canonical metric induced by is equal to
Suppose that is a normed space and that is the norm topology induced on Suppose that is metric on such that the topology that induces on is equal to If is translation invariant then is a Banach space if and only if is a complete metric space.
If is translation invariant, then it may be possible for to be a Banach space but for to be a complete metric space (see this footnote for an example). In contrast, a theorem of Klee, which also applies to all metrizable topological vector spaces, implies that if there exists complete metric on that induces the norm topology on then is a Banach space.
A Fréchet space is a locally convex topological vector space whose topology is induced by some translation-invariant complete metric.
Every Banach space is a Fréchet space but not conversely; indeed, there even exist Fréchet spaces on which no norm is a continuous function (such as the space of real sequences with the product topology).
However, the topology of every Fréchet space is induced by some countable family of real-valued (necessarily continuous) maps called seminorms, which are generalizations of norms.
It is even possible for a Fréchet space to have a topology that is induced by a countable family of (such norms would necessarily be continuous)
but to not be a Banach/normable space because its topology can not be defined by any norm.
An example of such a space is the Fréchet space whose definition can be found in the article on spaces of test functions and distributions.
Complete norms vs complete topological vector spaces | Banach space | Wikipedia | 372 | 3989 | https://en.wikipedia.org/wiki/Banach%20space | Mathematics | Linear algebra | null |
There is another notion of completeness besides metric completeness and that is the notion of a complete topological vector space (TVS) or TVS-completeness, which uses the theory of uniform spaces.
Specifically, the notion of TVS-completeness uses a unique translation-invariant uniformity, called the canonical uniformity, that depends on vector subtraction and the topology that the vector space is endowed with, and so in particular, this notion of TVS completeness is independent of whatever norm induced the topology (and even applies to TVSs that are even metrizable).
Every Banach space is a complete TVS. Moreover, a normed space is a Banach space (that is, its norm-induced metric is complete) if and only if it is complete as a topological vector space.
If is a metrizable topological vector space (such as any norm induced topology, for example), then is a complete TVS if and only if it is a complete TVS, meaning that it is enough to check that every Cauchy in converges in to some point of (that is, there is no need to consider the more general notion of arbitrary Cauchy nets).
If is a topological vector space whose topology is induced by (possibly unknown) norm (such spaces are called ), then is a complete topological vector space if and only if may be assigned a norm that induces on the topology and also makes into a Banach space.
A Hausdorff locally convex topological vector space is normable if and only if its strong dual space is normable, in which case is a Banach space ( denotes the strong dual space of whose topology is a generalization of the dual norm-induced topology on the continuous dual space ; see this footnote for more details).
If is a metrizable locally convex TVS, then is normable if and only if is a Fréchet–Urysohn space.
This shows that in the category of locally convex TVSs, Banach spaces are exactly those complete spaces that are both metrizable and have metrizable strong dual spaces.
Completions
Every normed space can be isometrically embedded onto a dense vector subspace of Banach space, where this Banach space is called a of the normed space. This Hausdorff completion is unique up to isometric isomorphism. | Banach space | Wikipedia | 483 | 3989 | https://en.wikipedia.org/wiki/Banach%20space | Mathematics | Linear algebra | null |
More precisely, for every normed space there exist a Banach space and a mapping such that is an isometric mapping and is dense in If is another Banach space such that there is an isometric isomorphism from onto a dense subset of then is isometrically isomorphic to
This Banach space is the Hausdorff of the normed space The underlying metric space for is the same as the metric completion of with the vector space operations extended from to The completion of is sometimes denoted by
General theory
Linear operators, isomorphisms
If and are normed spaces over the same ground field the set of all continuous -linear maps is denoted by In infinite-dimensional spaces, not all linear maps are continuous. A linear mapping from a normed space to another normed space is continuous if and only if it is bounded on the closed unit ball of Thus, the vector space can be given the operator norm
For a Banach space, the space is a Banach space with respect to this norm. In categorical contexts, it is sometimes convenient to restrict the function space between two Banach spaces to only the short maps; in that case the space reappears as a natural bifunctor.
If is a Banach space, the space forms a unital Banach algebra; the multiplication operation is given by the composition of linear maps.
If and are normed spaces, they are isomorphic normed spaces if there exists a linear bijection such that and its inverse are continuous. If one of the two spaces or is complete (or reflexive, separable, etc.) then so is the other space. Two normed spaces and are isometrically isomorphic if in addition, is an isometry, that is, for every in The Banach–Mazur distance between two isomorphic but not isometric spaces and gives a measure of how much the two spaces and differ. | Banach space | Wikipedia | 383 | 3989 | https://en.wikipedia.org/wiki/Banach%20space | Mathematics | Linear algebra | null |
Continuous and bounded linear functions and seminorms
Every continuous linear operator is a bounded linear operator and if dealing only with normed spaces then the converse is also true. That is, a linear operator between two normed spaces is bounded if and only if it is a continuous function. So in particular, because the scalar field (which is or ) is a normed space, a linear functional on a normed space is a bounded linear functional if and only if it is a continuous linear functional. This allows for continuity-related results (like those below) to be applied to Banach spaces. Although boundedness is the same as continuity for linear maps between normed spaces, the term "bounded" is more commonly used when dealing primarily with Banach spaces.
If is a subadditive function (such as a norm, a sublinear function, or real linear functional), then is continuous at the origin if and only if is uniformly continuous on all of ; and if in addition then is continuous if and only if its absolute value is continuous, which happens if and only if is an open subset of
And very importantly for applying the Hahn–Banach theorem, a linear functional is continuous if and only if this is true of its real part and moreover, and the real part completely determines which is why the Hahn–Banach theorem is often stated only for real linear functionals.
Also, a linear functional on is continuous if and only if the seminorm is continuous, which happens if and only if there exists a continuous seminorm such that ; this last statement involving the linear functional and seminorm is encountered in many versions of the Hahn–Banach theorem.
Basic notions
The Cartesian product of two normed spaces is not canonically equipped with a norm. However, several equivalent norms are commonly used, such as
which correspond (respectively) to the coproduct and product in the category of Banach spaces and short maps (discussed above). For finite (co)products, these norms give rise to isomorphic normed spaces, and the product (or the direct sum ) is complete if and only if the two factors are complete.
If is a closed linear subspace of a normed space there is a natural norm on the quotient space
The quotient is a Banach space when is complete. The quotient map from onto sending to its class is linear, onto and has norm except when in which case the quotient is the null space. | Banach space | Wikipedia | 505 | 3989 | https://en.wikipedia.org/wiki/Banach%20space | Mathematics | Linear algebra | null |
The closed linear subspace of is said to be a complemented subspace of if is the range of a surjective bounded linear projection In this case, the space is isomorphic to the direct sum of and the kernel of the projection
Suppose that and are Banach spaces and that There exists a canonical factorization of as
where the first map is the quotient map, and the second map sends every class in the quotient to the image in This is well defined because all elements in the same class have the same image. The mapping is a linear bijection from onto the range whose inverse need not be bounded.
Classical spaces
Basic examples of Banach spaces include: the Lp spaces and their special cases, the sequence spaces that consist of scalar sequences indexed by natural numbers ; among them, the space of absolutely summable sequences and the space of square summable sequences; the space of sequences tending to zero and the space of bounded sequences; the space of continuous scalar functions on a compact Hausdorff space equipped with the max norm,
According to the Banach–Mazur theorem, every Banach space is isometrically isomorphic to a subspace of some For every separable Banach space there is a closed subspace of such that
Any Hilbert space serves as an example of a Banach space. A Hilbert space on is complete for a norm of the form
where
is the inner product, linear in its first argument that satisfies the following:
For example, the space is a Hilbert space.
The Hardy spaces, the Sobolev spaces are examples of Banach spaces that are related to spaces and have additional structure. They are important in different branches of analysis, Harmonic analysis and Partial differential equations among others.
Banach algebras
A Banach algebra is a Banach space over or together with a structure of algebra over , such that the product map is continuous. An equivalent norm on can be found so that for all
Examples | Banach space | Wikipedia | 394 | 3989 | https://en.wikipedia.org/wiki/Banach%20space | Mathematics | Linear algebra | null |
The Banach space with the pointwise product, is a Banach algebra.
The disk algebra consists of functions holomorphic in the open unit disk and continuous on its closure: Equipped with the max norm on the disk algebra is a closed subalgebra of
The Wiener algebra is the algebra of functions on the unit circle with absolutely convergent Fourier series. Via the map associating a function on to the sequence of its Fourier coefficients, this algebra is isomorphic to the Banach algebra where the product is the convolution of sequences.
For every Banach space the space of bounded linear operators on with the composition of maps as product, is a Banach algebra.
A C*-algebra is a complex Banach algebra with an antilinear involution such that The space of bounded linear operators on a Hilbert space is a fundamental example of C*-algebra. The Gelfand–Naimark theorem states that every C*-algebra is isometrically isomorphic to a C*-subalgebra of some The space of complex continuous functions on a compact Hausdorff space is an example of commutative C*-algebra, where the involution associates to every function its complex conjugate
Dual space
If is a normed space and the underlying field (either the real or the complex numbers), the continuous dual space is the space of continuous linear maps from into or continuous linear functionals.
The notation for the continuous dual is in this article.
Since is a Banach space (using the absolute value as norm), the dual is a Banach space, for every normed space The Dixmier–Ng theorem characterizes the dual spaces of Banach spaces.
The main tool for proving the existence of continuous linear functionals is the Hahn–Banach theorem.
In particular, every continuous linear functional on a subspace of a normed space can be continuously extended to the whole space, without increasing the norm of the functional.
An important special case is the following: for every vector in a normed space there exists a continuous linear functional on such that
When is not equal to the vector, the functional must have norm one, and is called a norming functional for | Banach space | Wikipedia | 452 | 3989 | https://en.wikipedia.org/wiki/Banach%20space | Mathematics | Linear algebra | null |
The Hahn–Banach separation theorem states that two disjoint non-empty convex sets in a real Banach space, one of them open, can be separated by a closed affine hyperplane.
The open convex set lies strictly on one side of the hyperplane, the second convex set lies on the other side but may touch the hyperplane.
A subset in a Banach space is total if the linear span of is dense in The subset is total in if and only if the only continuous linear functional that vanishes on is the functional: this equivalence follows from the Hahn–Banach theorem.
If is the direct sum of two closed linear subspaces and then the dual of is isomorphic to the direct sum of the duals of and
If is a closed linear subspace in one can associate the in the dual,
The orthogonal is a closed linear subspace of the dual. The dual of is isometrically isomorphic to
The dual of is isometrically isomorphic to
The dual of a separable Banach space need not be separable, but:
When is separable, the above criterion for totality can be used for proving the existence of a countable total subset in
Weak topologies
The weak topology on a Banach space is the coarsest topology on for which all elements in the continuous dual space are continuous.
The norm topology is therefore finer than the weak topology.
It follows from the Hahn–Banach separation theorem that the weak topology is Hausdorff, and that a norm-closed convex subset of a Banach space is also weakly closed.
A norm-continuous linear map between two Banach spaces and is also weakly continuous, that is, continuous from the weak topology of to that of
If is infinite-dimensional, there exist linear maps which are not continuous. The space of all linear maps from to the underlying field (this space is called the algebraic dual space, to distinguish it from also induces a topology on which is finer than the weak topology, and much less used in functional analysis.
On a dual space there is a topology weaker than the weak topology of called weak* topology.
It is the coarsest topology on for which all evaluation maps where ranges over are continuous.
Its importance comes from the Banach–Alaoglu theorem.
The Banach–Alaoglu theorem can be proved using Tychonoff's theorem about infinite products of compact Hausdorff spaces.
When is separable, the unit ball of the dual is a metrizable compact in the weak* topology. | Banach space | Wikipedia | 511 | 3989 | https://en.wikipedia.org/wiki/Banach%20space | Mathematics | Linear algebra | null |
Examples of dual spaces
The dual of is isometrically isomorphic to : for every bounded linear functional on there is a unique element such that
The dual of is isometrically isomorphic to .
The dual of Lebesgue space is isometrically isomorphic to when and
For every vector in a Hilbert space the mapping
defines a continuous linear functional on The Riesz representation theorem states that every continuous linear functional on is of the form for a uniquely defined vector in
The mapping is an antilinear isometric bijection from onto its dual
When the scalars are real, this map is an isometric isomorphism.
When is a compact Hausdorff topological space, the dual of is the space of Radon measures in the sense of Bourbaki.
The subset of consisting of non-negative measures of mass 1 (probability measures) is a convex w*-closed subset of the unit ball of
The extreme points of are the Dirac measures on
The set of Dirac measures on equipped with the w*-topology, is homeomorphic to
The result has been extended by Amir and Cambern to the case when the multiplicative Banach–Mazur distance between and is
The theorem is no longer true when the distance is
In the commutative Banach algebra the maximal ideals are precisely kernels of Dirac measures on
More generally, by the Gelfand–Mazur theorem, the maximal ideals of a unital commutative Banach algebra can be identified with its characters—not merely as sets but as topological spaces: the former with the hull-kernel topology and the latter with the w*-topology.
In this identification, the maximal ideal space can be viewed as a w*-compact subset of the unit ball in the dual
Not every unital commutative Banach algebra is of the form for some compact Hausdorff space However, this statement holds if one places in the smaller category of commutative C*-algebras.
Gelfand's representation theorem for commutative C*-algebras states that every commutative unital C*-algebra is isometrically isomorphic to a space.
The Hausdorff compact space here is again the maximal ideal space, also called the spectrum of in the C*-algebra context.
Bidual
If is a normed space, the (continuous) dual of the dual is called , or of
For every normed space there is a natural map, | Banach space | Wikipedia | 498 | 3989 | https://en.wikipedia.org/wiki/Banach%20space | Mathematics | Linear algebra | null |
This defines as a continuous linear functional on that is, an element of The map is a linear map from to
As a consequence of the existence of a norming functional for every this map is isometric, thus injective.
For example, the dual of is identified with and the dual of is identified with the space of bounded scalar sequences.
Under these identifications, is the inclusion map from to It is indeed isometric, but not onto.
If is surjective, then the normed space is called reflexive (see below).
Being the dual of a normed space, the bidual is complete, therefore, every reflexive normed space is a Banach space.
Using the isometric embedding it is customary to consider a normed space as a subset of its bidual.
When is a Banach space, it is viewed as a closed linear subspace of If is not reflexive, the unit ball of is a proper subset of the unit ball of
The Goldstine theorem states that the unit ball of a normed space is weakly*-dense in the unit ball of the bidual.
In other words, for every in the bidual, there exists a net in so that
The net may be replaced by a weakly*-convergent sequence when the dual is separable.
On the other hand, elements of the bidual of that are not in cannot be weak*-limit of in since is weakly sequentially complete.
Banach's theorems
Here are the main general results about Banach spaces that go back to the time of Banach's book () and are related to the Baire category theorem.
According to this theorem, a complete metric space (such as a Banach space, a Fréchet space or an F-space) cannot be equal to a union of countably many closed subsets with empty interiors.
Therefore, a Banach space cannot be the union of countably many closed subspaces, unless it is already equal to one of them; a Banach space with a countable Hamel basis is finite-dimensional.
The Banach–Steinhaus theorem is not limited to Banach spaces.
It can be extended for example to the case where is a Fréchet space, provided the conclusion is modified as follows: under the same hypothesis, there exists a neighborhood of in such that all in are uniformly bounded on
This result is a direct consequence of the preceding Banach isomorphism theorem and of the canonical factorization of bounded linear maps. | Banach space | Wikipedia | 511 | 3989 | https://en.wikipedia.org/wiki/Banach%20space | Mathematics | Linear algebra | null |
This is another consequence of Banach's isomorphism theorem, applied to the continuous bijection from onto sending to the sum
Reflexivity
The normed space is called reflexive when the natural map
is surjective. Reflexive normed spaces are Banach spaces.
This is a consequence of the Hahn–Banach theorem.
Further, by the open mapping theorem, if there is a bounded linear operator from the Banach space onto the Banach space then is reflexive.
Indeed, if the dual of a Banach space is separable, then is separable.
If is reflexive and separable, then the dual of is separable, so is separable.
Hilbert spaces are reflexive. The spaces are reflexive when More generally, uniformly convex spaces are reflexive, by the Milman–Pettis theorem.
The spaces are not reflexive.
In these examples of non-reflexive spaces the bidual is "much larger" than
Namely, under the natural isometric embedding of into given by the Hahn–Banach theorem, the quotient is infinite-dimensional, and even nonseparable.
However, Robert C. James has constructed an example of a non-reflexive space, usually called "the James space" and denoted by such that the quotient is one-dimensional.
Furthermore, this space is isometrically isomorphic to its bidual.
When is reflexive, it follows that all closed and bounded convex subsets of are weakly compact.
In a Hilbert space the weak compactness of the unit ball is very often used in the following way: every bounded sequence in has weakly convergent subsequences.
Weak compactness of the unit ball provides a tool for finding solutions in reflexive spaces to certain optimization problems.
For example, every convex continuous function on the unit ball of a reflexive space attains its minimum at some point in
As a special case of the preceding result, when is a reflexive space over every continuous linear functional in attains its maximum on the unit ball of
The following theorem of Robert C. James provides a converse statement.
The theorem can be extended to give a characterization of weakly compact convex sets.
On every non-reflexive Banach space there exist continuous linear functionals that are not norm-attaining.
However, the Bishop–Phelps theorem states that norm-attaining functionals are norm dense in the dual of
Weak convergences of sequences | Banach space | Wikipedia | 491 | 3989 | https://en.wikipedia.org/wiki/Banach%20space | Mathematics | Linear algebra | null |
A sequence in a Banach space is weakly convergent to a vector if converges to for every continuous linear functional in the dual The sequence is a weakly Cauchy sequence if converges to a scalar limit for every in
A sequence in the dual is weakly* convergent to a functional if converges to for every in
Weakly Cauchy sequences, weakly convergent and weakly* convergent sequences are norm bounded, as a consequence of the Banach–Steinhaus theorem.
When the sequence in is a weakly Cauchy sequence, the limit above defines a bounded linear functional on the dual that is, an element of the bidual of and is the limit of in the weak*-topology of the bidual.
The Banach space is weakly sequentially complete if every weakly Cauchy sequence is weakly convergent in
It follows from the preceding discussion that reflexive spaces are weakly sequentially complete.
An orthonormal sequence in a Hilbert space is a simple example of a weakly convergent sequence, with limit equal to the vector.
The unit vector basis of for or of is another example of a weakly null sequence, that is, a sequence that converges weakly to
For every weakly null sequence in a Banach space, there exists a sequence of convex combinations of vectors from the given sequence that is norm-converging to
The unit vector basis of is not weakly Cauchy.
Weakly Cauchy sequences in are weakly convergent, since -spaces are weakly sequentially complete.
Actually, weakly convergent sequences in are norm convergent. This means that satisfies Schur's property.
Results involving the basis
Weakly Cauchy sequences and the basis are the opposite cases of the dichotomy established in the following deep result of H. P. Rosenthal.
A complement to this result is due to Odell and Rosenthal (1975).
By the Goldstine theorem, every element of the unit ball of is weak*-limit of a net in the unit ball of When does not contain every element of is weak*-limit of a in the unit ball of
When the Banach space is separable, the unit ball of the dual equipped with the weak*-topology, is a metrizable compact space and every element in the bidual defines a bounded function on : | Banach space | Wikipedia | 467 | 3989 | https://en.wikipedia.org/wiki/Banach%20space | Mathematics | Linear algebra | null |
This function is continuous for the compact topology of if and only if is actually in considered as subset of
Assume in addition for the rest of the paragraph that does not contain
By the preceding result of Odell and Rosenthal, the function is the pointwise limit on of a sequence of continuous functions on it is therefore a first Baire class function on
The unit ball of the bidual is a pointwise compact subset of the first Baire class on
Sequences, weak and weak* compactness
When is separable, the unit ball of the dual is weak*-compact by the Banach–Alaoglu theorem and metrizable for the weak* topology, hence every bounded sequence in the dual has weakly* convergent subsequences.
This applies to separable reflexive spaces, but more is true in this case, as stated below.
The weak topology of a Banach space is metrizable if and only if is finite-dimensional. If the dual is separable, the weak topology of the unit ball of is metrizable.
This applies in particular to separable reflexive Banach spaces.
Although the weak topology of the unit ball is not metrizable in general, one can characterize weak compactness using sequences.
A Banach space is reflexive if and only if each bounded sequence in has a weakly convergent subsequence.
A weakly compact subset in is norm-compact. Indeed, every sequence in has weakly convergent subsequences by Eberlein–Šmulian, that are norm convergent by the Schur property of
Type and cotype
A way to classify Banach spaces is through the probabilistic notion of type and cotype, these two measure how far a Banach space is from a Hilbert space.
Schauder bases
A Schauder basis in a Banach space is a sequence of vectors in with the property that for every vector there exist defined scalars depending on such that
Banach spaces with a Schauder basis are necessarily separable, because the countable set of finite linear combinations with rational coefficients (say) is dense.
It follows from the Banach–Steinhaus theorem that the linear mappings are uniformly bounded by some constant
Let denote the coordinate functionals which assign to every in the coordinate of in the above expansion.
They are called biorthogonal functionals. When the basis vectors have norm the coordinate functionals have norm in the dual of | Banach space | Wikipedia | 493 | 3989 | https://en.wikipedia.org/wiki/Banach%20space | Mathematics | Linear algebra | null |
Most classical separable spaces have explicit bases.
The Haar system is a basis for
The trigonometric system is a basis in when
The Schauder system is a basis in the space
The question of whether the disk algebra has a basis remained open for more than forty years, until Bočkarev showed in 1974 that admits a basis constructed from the Franklin system.
Since every vector in a Banach space with a basis is the limit of with of finite rank and uniformly bounded, the space satisfies the bounded approximation property.
The first example by Enflo of a space failing the approximation property was at the same time the first example of a separable Banach space without a Schauder basis.
Robert C. James characterized reflexivity in Banach spaces with a basis: the space with a Schauder basis is reflexive if and only if the basis is both shrinking and boundedly complete.
In this case, the biorthogonal functionals form a basis of the dual of
Tensor product
Let and be two -vector spaces. The tensor product of and is a -vector space with a bilinear mapping which has the following universal property:
If is any bilinear mapping into a -vector space then there exists a unique linear mapping such that
The image under of a couple in is denoted by and called a simple tensor.
Every element in is a finite sum of such simple tensors.
There are various norms that can be placed on the tensor product of the underlying vector spaces, amongst others the projective cross norm and injective cross norm introduced by A. Grothendieck in 1955.
In general, the tensor product of complete spaces is not complete again. When working with Banach spaces, it is customary to say that the projective tensor product of two Banach spaces and is the of the algebraic tensor product equipped with the projective tensor norm, and similarly for the injective tensor product
Grothendieck proved in particular that
where is a compact Hausdorff space, the Banach space of continuous functions from to and the space of Bochner-measurable and integrable functions from to and where the isomorphisms are isometric.
The two isomorphisms above are the respective extensions of the map sending the tensor to the vector-valued function
Tensor products and the approximation property
Let be a Banach space. The tensor product is identified isometrically with the closure in of the set of finite rank operators.
When has the approximation property, this closure coincides with the space of compact operators on | Banach space | Wikipedia | 511 | 3989 | https://en.wikipedia.org/wiki/Banach%20space | Mathematics | Linear algebra | null |
For every Banach space there is a natural norm linear map
obtained by extending the identity map of the algebraic tensor product. Grothendieck related the approximation problem to the question of whether this map is one-to-one when is the dual of
Precisely, for every Banach space the map
is one-to-one if and only if has the approximation property.
Grothendieck conjectured that and must be different whenever and are infinite-dimensional Banach spaces.
This was disproved by Gilles Pisier in 1983.
Pisier constructed an infinite-dimensional Banach space such that and are equal. Furthermore, just as Enflo's example, this space is a "hand-made" space that fails to have the approximation property. On the other hand, Szankowski proved that the classical space does not have the approximation property.
Some classification results
Characterizations of Hilbert space among Banach spaces
A necessary and sufficient condition for the norm of a Banach space to be associated to an inner product is the parallelogram identity:
It follows, for example, that the Lebesgue space is a Hilbert space only when
If this identity is satisfied, the associated inner product is given by the polarization identity. In the case of real scalars, this gives:
For complex scalars, defining the inner product so as to be -linear in antilinear in the polarization identity gives:
To see that the parallelogram law is sufficient, one observes in the real case that is symmetric, and in the complex case, that it satisfies the Hermitian symmetry property and The parallelogram law implies that is additive in
It follows that it is linear over the rationals, thus linear by continuity.
Several characterizations of spaces isomorphic (rather than isometric) to Hilbert spaces are available.
The parallelogram law can be extended to more than two vectors, and weakened by the introduction of a two-sided inequality with a constant : Kwapień proved that if
for every integer and all families of vectors then the Banach space is isomorphic to a Hilbert space.
Here, denotes the average over the possible choices of signs
In the same article, Kwapień proved that the validity of a Banach-valued Parseval's theorem for the Fourier transform characterizes Banach spaces isomorphic to Hilbert spaces. | Banach space | Wikipedia | 475 | 3989 | https://en.wikipedia.org/wiki/Banach%20space | Mathematics | Linear algebra | null |
Lindenstrauss and Tzafriri proved that a Banach space in which every closed linear subspace is complemented (that is, is the range of a bounded linear projection) is isomorphic to a Hilbert space. The proof rests upon Dvoretzky's theorem about Euclidean sections of high-dimensional centrally symmetric convex bodies. In other words, Dvoretzky's theorem states that for every integer any finite-dimensional normed space, with dimension sufficiently large compared to contains subspaces nearly isometric to the -dimensional Euclidean space.
The next result gives the solution of the so-called . An infinite-dimensional Banach space is said to be homogeneous if it is isomorphic to all its infinite-dimensional closed subspaces. A Banach space isomorphic to is homogeneous, and Banach asked for the converse.
An infinite-dimensional Banach space is hereditarily indecomposable when no subspace of it can be isomorphic to the direct sum of two infinite-dimensional Banach spaces.
The Gowers dichotomy theorem asserts that every infinite-dimensional Banach space contains, either a subspace with unconditional basis, or a hereditarily indecomposable subspace and in particular, is not isomorphic to its closed hyperplanes.
If is homogeneous, it must therefore have an unconditional basis. It follows then from the partial solution obtained by Komorowski and Tomczak–Jaegermann, for spaces with an unconditional basis, that is isomorphic to
Metric classification
If is an isometry from the Banach space onto the Banach space (where both and are vector spaces over ), then the Mazur–Ulam theorem states that must be an affine transformation.
In particular, if this is maps the zero of to the zero of then must be linear. This result implies that the metric in Banach spaces, and more generally in normed spaces, completely captures their linear structure.
Topological classification
Finite dimensional Banach spaces are homeomorphic as topological spaces, if and only if they have the same dimension as real vector spaces.
Anderson–Kadec theorem (1965–66) proves that any two infinite-dimensional separable Banach spaces are homeomorphic as topological spaces. Kadec's theorem was extended by Torunczyk, who proved that any two Banach spaces are homeomorphic if and only if they have the same density character, the minimum cardinality of a dense subset.
Spaces of continuous functions | Banach space | Wikipedia | 511 | 3989 | https://en.wikipedia.org/wiki/Banach%20space | Mathematics | Linear algebra | null |
When two compact Hausdorff spaces and are homeomorphic, the Banach spaces and are isometric. Conversely, when is not homeomorphic to the (multiplicative) Banach–Mazur distance between and must be greater than or equal to see above the results by Amir and Cambern.
Although uncountable compact metric spaces can have different homeomorphy types, one has the following result due to Milutin:
The situation is different for countably infinite compact Hausdorff spaces.
Every countably infinite compact is homeomorphic to some closed interval of ordinal numbers
equipped with the order topology, where is a countably infinite ordinal.
The Banach space is then isometric to . When are two countably infinite ordinals, and assuming the spaces and are isomorphic if and only if .
For example, the Banach spaces
are mutually non-isomorphic.
Examples
Derivatives
Several concepts of a derivative may be defined on a Banach space. See the articles on the Fréchet derivative and the Gateaux derivative for details.
The Fréchet derivative allows for an extension of the concept of a total derivative to Banach spaces. The Gateaux derivative allows for an extension of a directional derivative to locally convex topological vector spaces.
Fréchet differentiability is a stronger condition than Gateaux differentiability.
The quasi-derivative is another generalization of directional derivative that implies a stronger condition than Gateaux differentiability, but a weaker condition than Fréchet differentiability.
Generalizations
Several important spaces in functional analysis, for instance the space of all infinitely often differentiable functions or the space of all distributions on are complete but are not normed vector spaces and hence not Banach spaces.
In Fréchet spaces one still has a complete metric, while LF-spaces are complete uniform vector spaces arising as limits of Fréchet spaces. | Banach space | Wikipedia | 383 | 3989 | https://en.wikipedia.org/wiki/Banach%20space | Mathematics | Linear algebra | null |
A boat is a watercraft of a large range of types and sizes, but generally smaller than a ship, which is distinguished by its larger size or capacity, its shape, or its ability to carry boats.
Small boats are typically used on inland waterways such as rivers and lakes, or in protected coastal areas. However, some boats (such as whaleboats) were intended for offshore use. In modern naval terms, a boat is a vessel small enough to be carried aboard a ship.
Boats vary in proportion and construction methods with their intended purpose, available materials, or local traditions. Canoes have been used since prehistoric times and remain in use throughout the world for transportation, fishing, and sport. Fishing boats vary widely in style partly to match local conditions. Pleasure craft used in recreational boating include ski boats, pontoon boats, and sailboats. House boats may be used for vacationing or long-term residence. Lighters are used to move cargo to and from large ships unable to get close to shore. Lifeboats have rescue and safety functions.
Boats can be propelled by manpower (e.g. rowboats and paddle boats), wind (e.g. sailboats), and inboard/outboard motors (including gasoline, diesel, and electric).
History
Differentiation from other prehistoric watercraft
The earliest watercraft are considered to have been rafts. These would have been used for voyages such as the settlement of Australia sometime between 50,000 and 60,000 years ago.
A boat differs from a raft by obtaining its buoyancy by having most of its structure exclude water with a waterproof layer, e.g. the planks of a wooden hull, the hide covering (or tarred canvas) of a currach. In contrast, a raft is buoyant because it joins components that are themselves buoyant, for example, logs, bamboo poles, bundles of reeds, floats (such as inflated hides, sealed pottery containers or, in a modern context, empty oil drums). The key difference between a raft and a boat is that the former is a "flow through" structure, with waves able to pass up through it. Consequently, except for short river crossings, a raft is not a practical means of transport in colder regions of the world as the users would be at risk of hypothermia. Today that climatic limitation restricts rafts to between 40° north and 40° south, with, in the past, similar boundaries that have moved as the world's climate has varied. | Boat | Wikipedia | 512 | 3996 | https://en.wikipedia.org/wiki/Boat | Technology | Maritime transport | null |
Types
The earliest boats may have been either dugouts or hide boats. The oldest recovered boat in the world, the Pesse canoe, found in the Netherlands, is a dugout made from the hollowed tree trunk of a Pinus sylvestris that was constructed somewhere between 8200 and 7600 BC. This canoe is exhibited in the Drents Museum in Assen, Netherlands. Other very old dugout boats have also been recovered. Hide boats, made from covering a framework with animal skins, could be equally as old as logboats, but such a structure is much less likely to survive in an archaeological context.
Plank-built boats are considered, in most cases, to have developed from the logboat. There are examples of logboats that have been expanded: by deforming the hull under the influence of heat, by raising up the sides with added planks, or by splitting down the middle and adding a central plank to make it wider. (Some of these methods have been in quite recent usethere is no simple developmental sequence). The earliest known plank-built boats are from the Nile, dating to the third millennium BC. Outside Egypt, the next earliest are from England. The Ferriby boats are dated to the early part of the second millennium BC and the end of the third millennium. Plank-built boats require a level of woodworking technology that was first available in the neolithic with more complex versions only becoming achievable in the Bronze Age.
Types
Boats can be categorized by their means of propulsion. These divide into:
Unpowered. This involves drifting with the tide or a river current.
Powered by the crew-members on board, using oars, paddles or a punting pole or quant.
Powered by sail.
Towedeither by humans or animals from a river or canal bank (or in very shallow water, by walking on the sea or river bed) or by another vessel.
Powered by machinery, such as internal combustion engines, steam engines or by batteries and an electric motor.Any one vessel may use more than one of these methods at different times or in combination.
A number of large vessels are usually referred to as boats. Submarines are a prime example. Other types of large vessels which are traditionally called boats include Great Lakes freighters, riverboats, and ferryboats. Though large enough to carry their own boats and heavy cargo, these vessels are designed for operation on inland or protected coastal waters.
Terminology | Boat | Wikipedia | 495 | 3996 | https://en.wikipedia.org/wiki/Boat | Technology | Maritime transport | null |
The hull is the main, and in some cases only, structural component of a boat. It provides both capacity and buoyancy. The keel is a boat's "backbone", a lengthwise structural member to which the perpendicular frames are fixed. On some boats, a deck covers the hull, in part or whole. While a ship often has several decks, a boat is unlikely to have more than one. Above the deck are often lifelines connected to stanchions, bulwarks perhaps topped by gunnels, or some combination of the two. A cabin may protrude above the deck forward, aft, along the centerline, or cover much of the length of the boat. Vertical structures dividing the internal spaces are known as bulkheads.
The forward end of a boat is called the bow, the aft end the stern. Facing forward the right side is referred to as starboard and the left side as port.
Building materials
Until the mid-19th century, most boats were made of natural materials, primarily wood, although bark and animal skins were also used. Early boats include the birch bark canoe, the animal hide-covered kayak and coracle and the dugout canoe made from a single log.
By the mid-19th century, some boats had been built with iron or steel frames but still planked in wood. In 1855 ferro-cement boat construction was patented by the French, who coined the name "ferciment". This is a system by which a steel or iron wire framework is built in the shape of a boat's hull and covered over with cement. Reinforced with bulkheads and other internal structures it is strong but heavy, easily repaired, and, if sealed properly, will not leak or corrode.
As the forests of Britain and Europe continued to be over-harvested to supply the keels of larger wooden boats, and the Bessemer process (patented in 1855) cheapened the cost of steel, steel ships and boats began to be more common. By the 1930s boats built entirely of steel from frames to plating were seen replacing wooden boats in many industrial uses and fishing fleets. Private recreational boats of steel remain uncommon. In 1895 WH Mullins produced steel boats of galvanized iron and by 1930 became the world's largest producer of pleasure boats. | Boat | Wikipedia | 468 | 3996 | https://en.wikipedia.org/wiki/Boat | Technology | Maritime transport | null |
Mullins also offered boats in aluminum from 1895 through 1899 and once again in the 1920s, but it was not until the mid-20th century that aluminium gained widespread popularity. Though much more expensive than steel, aluminum alloys exist that do not corrode in salt water, allowing a similar load carrying capacity to steel at much less weight.
Around the mid-1960s, boats made of fiberglass (aka "glass fiber") became popular, especially for recreational boats. Fiberglass is also known as "GRP" (glass-reinforced plastic) in the UK, and "FRP" (for fiber-reinforced plastic) in the US. Fiberglass boats are strong and do not rust, corrode, or rot. Instead, they are susceptible to structural degradation from sunlight and extremes in temperature over their lifespan. Fiberglass structures can be made stiffer with sandwich panels, where the fiberglass encloses a lightweight core such as balsa or foam.
Cold molding is a modern construction method, using wood as the structural component. In one cold molding process, very thin strips of wood are layered over a form. Each layer is coated with resin, followed by another directionally alternating layer laid on top. Subsequent layers may be stapled or otherwise mechanically fastened to the previous, or weighted or vacuum bagged to provide compression and stabilization until the resin sets. An alternative process uses thin sheets of plywood shaped over a disposable male mold, and coated with epoxy.
Propulsion
The most common means of boat propulsion are as follows:
Engine
Inboard motor
Stern drive (Inboard/outboard)
Outboard motor
Paddle wheel
Water jet (jetboat, personal water craft)
Fan (hovercraft, air boat)
Man (rowing, paddling, setting pole etc.)
Wind (sailing)
Buoyancy
A boat displaces its weight in water, regardless whether it is made of wood, steel, fiberglass, or even concrete. If weight is added to the boat, the volume of the hull drawn below the waterline will increase to keep the balance above and below the surface equal. Boats have a natural or designed level of buoyancy. Exceeding it will cause the boat first to ride lower in the water, second to take on water more readily than when properly loaded, and ultimately, if overloaded by any combination of structure, cargo, and water, sink. | Boat | Wikipedia | 486 | 3996 | https://en.wikipedia.org/wiki/Boat | Technology | Maritime transport | null |
As commercial vessels must be correctly loaded to be safe, and as the sea becomes less buoyant in brackish areas such as the Baltic, the Plimsoll line was introduced to prevent overloading.
European Union classification
Since 1998 all new leisure boats and barges built in Europe between 2.5m and 24m must comply with the EU's Recreational Craft Directive (RCD). The Directive establishes four categories that permit the allowable wind and wave conditions for vessels in each class:
Class A - the boat may safely navigate any waters.
Class B - the boat is limited to offshore navigation. (Winds up to Force 8 & waves up to 4 metres)
Class C - the boat is limited to inshore (coastal) navigation. (Winds up to Force 6 & waves up to 2 metres)
Class D - the boat is limited to rivers, canals and small lakes. (Winds up to Force 4 & waves up to 0.5 metres)
Europe is the main producer of recreational boats (the second production in the world is located in Poland). European brands are known all over the world - in fact, these are the brands that created RCD and set the standard for shipyards around the world. | Boat | Wikipedia | 245 | 3996 | https://en.wikipedia.org/wiki/Boat | Technology | Maritime transport | null |
Blood is a body fluid in the circulatory system of humans and other vertebrates that delivers necessary substances such as nutrients and oxygen to the cells, and transports metabolic waste products away from those same cells.
Blood is composed of blood cells suspended in blood plasma. Plasma, which constitutes 55% of blood fluid, is mostly water (92% by volume), and contains proteins, glucose, mineral ions, and hormones. The blood cells are mainly red blood cells (erythrocytes), white blood cells (leukocytes), and (in mammals) platelets (thrombocytes). The most abundant cells are red blood cells. These contain hemoglobin, which facilitates oxygen transport by reversibly binding to it, increasing its solubility. Jawed vertebrates have an adaptive immune system, based largely on white blood cells. White blood cells help to resist infections and parasites. Platelets are important in the clotting of blood.
Blood is circulated around the body through blood vessels by the pumping action of the heart. In animals with lungs, arterial blood carries oxygen from inhaled air to the tissues of the body, and venous blood carries carbon dioxide, a waste product of metabolism produced by cells, from the tissues to the lungs to be exhaled. Blood is bright red when its hemoglobin is oxygenated and dark red when it is deoxygenated.
Medical terms related to blood often begin with hemo-, hemato-, haemo- or haemato- from the Greek word () for "blood". In terms of anatomy and histology, blood is considered a specialized form of connective tissue, given its origin in the bones and the presence of potential molecular fibers in the form of fibrinogen.
Functions
Blood performs many important functions within the body, including:
Supply of oxygen to tissues (bound to hemoglobin, which is carried in red cells)
Supply of nutrients such as glucose, amino acids, and fatty acids (dissolved in the blood or bound to plasma proteins (e.g., blood lipids))
Removal of waste such as carbon dioxide, urea, and lactic acid
Immunological functions, including circulation of white blood cells, and detection of foreign material by antibodies
Coagulation, the response to a broken blood vessel, the conversion of blood from a liquid to a semisolid gel to stop bleeding
Messenger functions, including the transport of hormones and the signaling of tissue damage
Regulation of core body temperature
Hydraulic functions | Blood | Wikipedia | 512 | 3997 | https://en.wikipedia.org/wiki/Blood | Biology and health sciences | Biology | null |
Constituents
In mammals
Blood accounts for 7% of the human body weight, with an average density around 1060 kg/m3, very close to pure water's density of 1000 kg/m3. The average adult has a blood volume of roughly or 1.3 gallons, which is composed of plasma and formed elements. The formed elements are the two types of blood cell or corpuscle – the red blood cells, (erythrocytes) and white blood cells (leukocytes), and the cell fragments called platelets that are involved in clotting. By volume, the red blood cells constitute about 45% of whole blood, the plasma about 54.3%, and white cells about 0.7%.
Whole blood (plasma and cells) exhibits non-Newtonian fluid dynamics.
Cells
One microliter of blood contains:
4.7 to 6.1 million (male), 4.2 to 5.4 million (female) erythrocytes: Red blood cells contain the blood's hemoglobin and distribute oxygen. Mature red blood cells lack a nucleus and organelles in mammals. The red blood cells (together with endothelial vessel cells and other cells) are also marked by glycoproteins that define the different blood types. The proportion of blood occupied by red blood cells is referred to as the hematocrit, and is normally about 45%. The combined surface area of all red blood cells of the human body would be roughly 2,000 times as great as the body's exterior surface.
4,000–11,000 leukocytes: White blood cells are part of the body's immune system; they destroy and remove old or aberrant cells and cellular debris, as well as attack infectious agents (pathogens) and foreign substances. The cancer of leukocytes is called leukemia.
200,000–500,000 thrombocytes: Also called platelets, they take part in blood clotting (coagulation). Fibrin from the coagulation cascade creates a mesh over the platelet plug.
Plasma | Blood | Wikipedia | 427 | 3997 | https://en.wikipedia.org/wiki/Blood | Biology and health sciences | Biology | null |
About 55% of blood is blood plasma, a fluid that is the blood's liquid medium, which by itself is straw-yellow in color. The blood plasma volume totals of 2.7–3.0 liters (2.8–3.2 quarts) in an average human. It is essentially an aqueous solution containing 92% water, 8% blood plasma proteins, and trace amounts of other materials. Plasma circulates dissolved nutrients, such as glucose, amino acids, and fatty acids (dissolved in the blood or bound to plasma proteins), and removes waste products, such as carbon dioxide, urea, and lactic acid.
Other important components include:
Serum albumin
Blood-clotting factors (to facilitate coagulation)
Immunoglobulins (antibodies)
lipoprotein particles
Various other proteins
Various electrolytes (mainly sodium and chloride)
The term serum refers to plasma from which the clotting proteins have been removed. Most of the proteins remaining are albumin and immunoglobulins.
Acidity
Blood pH is regulated to stay within the narrow range of 7.35 to 7.45, making it slightly basic (compensation). Extra-cellular fluid in blood that has a pH below 7.35 is too acidic, whereas blood pH above 7.45 is too basic. A pH below 6.9 or above 7.8 is usually lethal. Blood pH, partial pressure of oxygen (pO2), partial pressure of carbon dioxide (pCO2), and bicarbonate (HCO3−) are carefully regulated by a number of homeostatic mechanisms, which exert their influence principally through the respiratory system and the urinary system to control the acid–base balance and respiration, which is called compensation. An arterial blood gas test measures these. Plasma also circulates hormones transmitting their messages to various tissues. The list of normal reference ranges for various blood electrolytes is extensive.
In non-mammals | Blood | Wikipedia | 405 | 3997 | https://en.wikipedia.org/wiki/Blood | Biology and health sciences | Biology | null |
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