text stringlengths 7 145k | page_title stringlengths 1 105 | source stringclasses 1
value | token_count int64 3 30.2k | id stringlengths 2 8 | url stringlengths 31 168 | topic stringclasses 198
values | section stringlengths 4 49 ⌀ | sublist stringlengths 0 628 ⌀ | method stringclasses 2
values |
|---|---|---|---|---|---|---|---|---|---|
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 mod... | Binomial distribution | Wikipedia | 393 | 3876 | https://en.wikipedia.org/wiki/Binomial%20distribution | Mathematics | Statistics and probability | null | vital_articles |
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 ... | Binomial distribution | Wikipedia | 469 | 3876 | https://en.wikipedia.org/wiki/Binomial%20distribution | Mathematics | Statistics and probability | null | vital_articles |
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 ... | Binomial distribution | Wikipedia | 508 | 3876 | https://en.wikipedia.org/wiki/Binomial%20distribution | Mathematics | Statistics and probability | null | vital_articles |
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... | Binomial distribution | Wikipedia | 332 | 3876 | https://en.wikipedia.org/wiki/Binomial%20distribution | Mathematics | Statistics and probability | null | vital_articles |
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 ... | Binomial distribution | Wikipedia | 454 | 3876 | https://en.wikipedia.org/wiki/Binomial%20distribution | Mathematics | Statistics and probability | null | vital_articles |
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... | Binomial distribution | Wikipedia | 482 | 3876 | https://en.wikipedia.org/wiki/Binomial%20distribution | Mathematics | Statistics and probability | null | vital_articles |
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 num... | Binomial distribution | Wikipedia | 227 | 3876 | https://en.wikipedia.org/wiki/Binomial%20distribution | Mathematics | Statistics and probability | null | vital_articles |
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... | Binary relation | Wikipedia | 450 | 3931 | https://en.wikipedia.org/wiki/Binary%20relation | Mathematics | Set theory | null | vital_articles |
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 binar... | Binary relation | Wikipedia | 500 | 3931 | https://en.wikipedia.org/wiki/Binary%20relation | Mathematics | Set theory | null | vital_articles |
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 intersect... | Binary relation | Wikipedia | 508 | 3931 | https://en.wikipedia.org/wiki/Binary%20relation | Mathematics | Set theory | null | vital_articles |
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 grandm... | Binary relation | Wikipedia | 435 | 3931 | https://en.wikipedia.org/wiki/Binary%20relation | Mathematics | Set theory | null | vital_articles |
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... | Binary relation | Wikipedia | 361 | 3931 | https://en.wikipedia.org/wiki/Binary%20relation | Mathematics | Set theory | null | vital_articles |
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 t... | Binary relation | Wikipedia | 314 | 3931 | https://en.wikipedia.org/wiki/Binary%20relation | Mathematics | Set theory | null | vital_articles |
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... | Binary relation | Wikipedia | 416 | 3931 | https://en.wikipedia.org/wiki/Binary%20relation | Mathematics | Set theory | null | vital_articles |
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 ... | Binary relation | Wikipedia | 434 | 3931 | https://en.wikipedia.org/wiki/Binary%20relation | Mathematics | Set theory | null | vital_articles |
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... | Binary relation | Wikipedia | 488 | 3931 | https://en.wikipedia.org/wiki/Binary%20relation | Mathematics | Set theory | null | vital_articles |
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 morphi... | Binary relation | Wikipedia | 469 | 3931 | https://en.wikipedia.org/wiki/Binary%20relation | Mathematics | Set theory | null | vital_articles |
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... | Binary relation | Wikipedia | 484 | 3931 | https://en.wikipedia.org/wiki/Binary%20relation | Mathematics | Set theory | null | vital_articles |
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 relat... | Binary relation | Wikipedia | 249 | 3931 | https://en.wikipedia.org/wiki/Binary%20relation | Mathematics | Set theory | null | vital_articles |
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 s... | Bijection | Wikipedia | 460 | 3942 | https://en.wikipedia.org/wiki/Bijection | Mathematics | Functions: General | null | vital_articles |
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... | Bijection | Wikipedia | 438 | 3942 | https://en.wikipedia.org/wiki/Bijection | Mathematics | Functions: General | null | vital_articles |
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 i... | Bijection | Wikipedia | 164 | 3942 | https://en.wikipedia.org/wiki/Bijection | Mathematics | Functions: General | null | vital_articles |
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 num... | Bijection | Wikipedia | 505 | 3942 | https://en.wikipedia.org/wiki/Bijection | Mathematics | Functions: General | null | vital_articles |
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 b... | Bijection | Wikipedia | 439 | 3942 | https://en.wikipedia.org/wiki/Bijection | Mathematics | Functions: General | null | vital_articles |
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(... | Bijection | Wikipedia | 470 | 3942 | https://en.wikipedia.org/wiki/Bijection | Mathematics | Functions: General | null | vital_articles |
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 some... | Bijection | Wikipedia | 122 | 3942 | https://en.wikipedia.org/wiki/Bijection | Mathematics | Functions: General | null | vital_articles |
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 b... | Biochemistry | Wikipedia | 394 | 3954 | https://en.wikipedia.org/wiki/Biochemistry | Biology and health sciences | Chemistry | null | vital_articles |
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... | Biochemistry | Wikipedia | 418 | 3954 | https://en.wikipedia.org/wiki/Biochemistry | Biology and health sciences | Chemistry | null | vital_articles |
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 hi... | Biochemistry | Wikipedia | 508 | 3954 | https://en.wikipedia.org/wiki/Biochemistry | Biology and health sciences | Chemistry | null | vital_articles |
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 amount... | Biochemistry | Wikipedia | 499 | 3954 | https://en.wikipedia.org/wiki/Biochemistry | Biology and health sciences | Chemistry | null | vital_articles |
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... | Biochemistry | Wikipedia | 466 | 3954 | https://en.wikipedia.org/wiki/Biochemistry | Biology and health sciences | Chemistry | null | vital_articles |
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 f... | Biochemistry | Wikipedia | 422 | 3954 | https://en.wikipedia.org/wiki/Biochemistry | Biology and health sciences | Chemistry | null | vital_articles |
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... | Biochemistry | Wikipedia | 301 | 3954 | https://en.wikipedia.org/wiki/Biochemistry | Biology and health sciences | Chemistry | null | vital_articles |
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 comm... | Biochemistry | Wikipedia | 409 | 3954 | https://en.wikipedia.org/wiki/Biochemistry | Biology and health sciences | Chemistry | null | vital_articles |
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... | Biochemistry | Wikipedia | 447 | 3954 | https://en.wikipedia.org/wiki/Biochemistry | Biology and health sciences | Chemistry | null | vital_articles |
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 bacteri... | Biochemistry | Wikipedia | 499 | 3954 | https://en.wikipedia.org/wiki/Biochemistry | Biology and health sciences | Chemistry | null | vital_articles |
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 releva... | Biochemistry | Wikipedia | 495 | 3954 | https://en.wikipedia.org/wiki/Biochemistry | Biology and health sciences | Chemistry | null | vital_articles |
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 (anaerob... | Biochemistry | Wikipedia | 281 | 3954 | https://en.wikipedia.org/wiki/Biochemistry | Biology and health sciences | Chemistry | null | vital_articles |
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 glu... | Biochemistry | Wikipedia | 366 | 3954 | https://en.wikipedia.org/wiki/Biochemistry | Biology and health sciences | Chemistry | null | vital_articles |
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. Th... | Biochemistry | Wikipedia | 322 | 3954 | https://en.wikipedia.org/wiki/Biochemistry | Biology and health sciences | Chemistry | null | vital_articles |
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 a... | Biochemistry | Wikipedia | 399 | 3954 | https://en.wikipedia.org/wiki/Biochemistry | Biology and health sciences | Chemistry | null | vital_articles |
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 b... | Boolean algebra (structure) | Wikipedia | 477 | 3959 | https://en.wikipedia.org/wiki/Boolean%20algebra%20%28structure%29 | Mathematics | Order theory | null | vital_articles |
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 fo... | Boolean algebra (structure) | Wikipedia | 460 | 3959 | https://en.wikipedia.org/wiki/Boolean%20algebra%20%28structure%29 | Mathematics | Order theory | null | vital_articles |
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 equi... | Boolean algebra (structure) | Wikipedia | 333 | 3959 | https://en.wikipedia.org/wiki/Boolean%20algebra%20%28structure%29 | Mathematics | Order theory | null | vital_articles |
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... | Boolean algebra (structure) | Wikipedia | 505 | 3959 | https://en.wikipedia.org/wiki/Boolean%20algebra%20%28structure%29 | Mathematics | Order theory | null | vital_articles |
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 t... | Boolean algebra (structure) | Wikipedia | 388 | 3959 | https://en.wikipedia.org/wiki/Boolean%20algebra%20%28structure%29 | Mathematics | Order theory | null | vital_articles |
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 ,... | Boolean algebra (structure) | Wikipedia | 450 | 3959 | https://en.wikipedia.org/wiki/Boolean%20algebra%20%28structure%29 | Mathematics | Order theory | null | vital_articles |
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 axiomatiza... | Boolean algebra (structure) | Wikipedia | 465 | 3959 | https://en.wikipedia.org/wiki/Boolean%20algebra%20%28structure%29 | Mathematics | Order theory | null | vital_articles |
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 | vital_articles |
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 th... | Bicycle | Wikipedia | 448 | 3973 | https://en.wikipedia.org/wiki/Bicycle | Technology | Transportation | null | vital_articles |
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 ... | Bicycle | Wikipedia | 316 | 3973 | https://en.wikipedia.org/wiki/Bicycle | Technology | Transportation | null | vital_articles |
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 Lallem... | Bicycle | Wikipedia | 495 | 3973 | https://en.wikipedia.org/wiki/Bicycle | Technology | Transportation | null | vital_articles |
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 ra... | Bicycle | Wikipedia | 510 | 3973 | https://en.wikipedia.org/wiki/Bicycle | Technology | Transportation | null | vital_articles |
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 ... | Bicycle | Wikipedia | 452 | 3973 | https://en.wikipedia.org/wiki/Bicycle | Technology | Transportation | null | vital_articles |
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 tra... | Bicycle | Wikipedia | 456 | 3973 | https://en.wikipedia.org/wiki/Bicycle | Technology | Transportation | null | vital_articles |
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 bendin... | Bicycle | Wikipedia | 374 | 3973 | https://en.wikipedia.org/wiki/Bicycle | Technology | Transportation | null | vital_articles |
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 tub... | Bicycle | Wikipedia | 489 | 3973 | https://en.wikipedia.org/wiki/Bicycle | Technology | Transportation | null | vital_articles |
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 gear... | Bicycle | Wikipedia | 512 | 3973 | https://en.wikipedia.org/wiki/Bicycle | Technology | Transportation | null | vital_articles |
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 ... | Bicycle | Wikipedia | 413 | 3973 | https://en.wikipedia.org/wiki/Bicycle | Technology | Transportation | null | vital_articles |
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 ... | Bicycle | Wikipedia | 429 | 3973 | https://en.wikipedia.org/wiki/Bicycle | Technology | Transportation | null | vital_articles |
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... | Bicycle | Wikipedia | 491 | 3973 | https://en.wikipedia.org/wiki/Bicycle | Technology | Transportation | null | vital_articles |
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 f... | Bicycle | Wikipedia | 449 | 3973 | https://en.wikipedia.org/wiki/Bicycle | Technology | Transportation | null | vital_articles |
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, ... | Bicycle | Wikipedia | 499 | 3973 | https://en.wikipedia.org/wiki/Bicycle | Technology | Transportation | null | vital_articles |
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 al... | Bicycle | Wikipedia | 506 | 3973 | https://en.wikipedia.org/wiki/Bicycle | Technology | Transportation | null | vital_articles |
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 headquart... | Bicycle | Wikipedia | 344 | 3973 | https://en.wikipedia.org/wiki/Bicycle | Technology | Transportation | null | vital_articles |
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... | Bicycle | Wikipedia | 466 | 3973 | https://en.wikipedia.org/wiki/Bicycle | Technology | Transportation | null | vital_articles |
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 busine... | Bicycle | Wikipedia | 485 | 3973 | https://en.wikipedia.org/wiki/Bicycle | Technology | Transportation | null | vital_articles |
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, whi... | Bicycle | Wikipedia | 435 | 3973 | https://en.wikipedia.org/wiki/Bicycle | Technology | Transportation | null | vital_articles |
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 bicyc... | Bicycle | Wikipedia | 83 | 3973 | https://en.wikipedia.org/wiki/Bicycle | Technology | Transportation | null | vital_articles |
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... | Bicarbonate | Wikipedia | 407 | 3982 | https://en.wikipedia.org/wiki/Bicarbonate | Physical sciences | Carbonic oxyanions | Chemistry | vital_articles |
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 especiall... | Bicarbonate | Wikipedia | 475 | 3982 | https://en.wikipedia.org/wiki/Bicarbonate | Physical sciences | Carbonic oxyanions | Chemistry | vital_articles |
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 800... | Bicarbonate | Wikipedia | 140 | 3982 | https://en.wikipedia.org/wiki/Bicarbonate | Physical sciences | Carbonic oxyanions | Chemistry | vital_articles |
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 c... | Banach space | Wikipedia | 467 | 3989 | https://en.wikipedia.org/wiki/Banach%20space | Mathematics | Linear algebra | null | vital_articles |
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 me... | Banach space | Wikipedia | 461 | 3989 | https://en.wikipedia.org/wiki/Banach%20space | Mathematics | Linear algebra | null | vital_articles |
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 t... | Banach space | Wikipedia | 407 | 3989 | https://en.wikipedia.org/wiki/Banach%20space | Mathematics | Linear algebra | null | vital_articles |
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;... | Banach space | Wikipedia | 474 | 3989 | https://en.wikipedia.org/wiki/Banach%20space | Mathematics | Linear algebra | null | vital_articles |
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 Su... | Banach space | Wikipedia | 372 | 3989 | https://en.wikipedia.org/wiki/Banach%20space | Mathematics | Linear algebra | null | vital_articles |
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 uniform... | Banach space | Wikipedia | 483 | 3989 | https://en.wikipedia.org/wiki/Banach%20space | Mathematics | Linear algebra | null | vital_articles |
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 ... | Banach space | Wikipedia | 383 | 3989 | https://en.wikipedia.org/wiki/Banach%20space | Mathematics | Linear algebra | null | vital_articles |
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... | Banach space | Wikipedia | 505 | 3989 | https://en.wikipedia.org/wiki/Banach%20space | Mathematics | Linear algebra | null | vital_articles |
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 ... | Banach space | Wikipedia | 394 | 3989 | https://en.wikipedia.org/wiki/Banach%20space | Mathematics | Linear algebra | null | vital_articles |
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 ... | Banach space | Wikipedia | 452 | 3989 | https://en.wikipedia.org/wiki/Banach%20space | Mathematics | Linear algebra | null | vital_articles |
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 subs... | Banach space | Wikipedia | 511 | 3989 | https://en.wikipedia.org/wiki/Banach%20space | Mathematics | Linear algebra | null | vital_articles |
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
... | Banach space | Wikipedia | 498 | 3989 | https://en.wikipedia.org/wiki/Banach%20space | Mathematics | Linear algebra | null | vital_articles |
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 b... | Banach space | Wikipedia | 511 | 3989 | https://en.wikipedia.org/wiki/Banach%20space | Mathematics | Linear algebra | null | vital_articles |
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... | Banach space | Wikipedia | 491 | 3989 | https://en.wikipedia.org/wiki/Banach%20space | Mathematics | Linear algebra | null | vital_articles |
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 ... | Banach space | Wikipedia | 467 | 3989 | https://en.wikipedia.org/wiki/Banach%20space | Mathematics | Linear algebra | null | vital_articles |
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 ... | Banach space | Wikipedia | 493 | 3989 | https://en.wikipedia.org/wiki/Banach%20space | Mathematics | Linear algebra | null | vital_articles |
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 ... | Banach space | Wikipedia | 511 | 3989 | https://en.wikipedia.org/wiki/Banach%20space | Mathematics | Linear algebra | null | vital_articles |
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 onl... | Banach space | Wikipedia | 475 | 3989 | https://en.wikipedia.org/wiki/Banach%20space | Mathematics | Linear algebra | null | vital_articles |
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... | Banach space | Wikipedia | 511 | 3989 | https://en.wikipedia.org/wiki/Banach%20space | Mathematics | Linear algebra | null | vital_articles |
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... | Banach space | Wikipedia | 383 | 3989 | https://en.wikipedia.org/wiki/Banach%20space | Mathematics | Linear algebra | null | vital_articles |
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... | Boat | Wikipedia | 512 | 3996 | https://en.wikipedia.org/wiki/Boat | Technology | Maritime transport | null | vital_articles |
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 Museu... | Boat | Wikipedia | 495 | 3996 | https://en.wikipedia.org/wiki/Boat | Technology | Maritime transport | null | vital_articles |
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 deck... | Boat | Wikipedia | 468 | 3996 | https://en.wikipedia.org/wiki/Boat | Technology | Maritime transport | null | vital_articles |
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 s... | Boat | Wikipedia | 486 | 3996 | https://en.wikipedia.org/wiki/Boat | Technology | Maritime transport | null | vital_articles |
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 t... | Boat | Wikipedia | 245 | 3996 | https://en.wikipedia.org/wiki/Boat | Technology | Maritime transport | null | vital_articles |
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 | Wikipedia | 512 | 3997 | https://en.wikipedia.org/wiki/Blood | Biology and health sciences | Biology | null | vital_articles |
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 o... | Blood | Wikipedia | 427 | 3997 | https://en.wikipedia.org/wiki/Blood | Biology and health sciences | Biology | null | vital_articles |
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... | Blood | Wikipedia | 405 | 3997 | https://en.wikipedia.org/wiki/Blood | Biology and health sciences | Biology | null | vital_articles |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.