A shower is a place in which a person bathes under a spray of typically warm or hot water. Indoors, there is a drain in the floor. Most showers have temperature, spray pressure and adjustable showerhead nozzle. The simplest showers have a swivelling nozzle aiming down on the user, while more complex showers have a showerhead connected to a hose that has a mounting bracket. This allows the showerer to hold the showerhead by hand to spray the water onto different parts of their body. A shower can be installed in a small shower stall or bathtub with a plastic shower curtain or door. Showering is common due to the efficiency of using it compared with a bathtub. Its use in hygiene is, therefore, common practice.

History
The original showers were neither indoor structures nor human-made but were common natural formations: waterfalls. The falling water rinsed the bathers completely clean and was more efficient than bathing in a traditional basin, which required manual transport of both fresh and waste water. Ancient people began to reproduce these natural phenomena by pouring jugs of water, often very cold, over themselves after washing. There has been evidence of early upper class Egyptian and Mesopotamians having indoor shower rooms where servants would bathe them in the privacy of their own homes. However, these were rudimentary by modern standards, having rudimentary drainage systems and water was carried, not pumped, into the room.

The ancient Greeks were the first people to have showers. Their aqueducts and sewage systems made of lead pipes allowed water to be pumped both into and out of large communal shower rooms used by elites and common citizens alike. These rooms have been discovered at the site of the city Pergamum and can also be found represented in pottery of the era. The depictions are very similar to modern locker room showers, and even included bars to hang up clothing.

The ancient Romans also followed this convention; their famous bathhouses (Thermae) can be found all around the Mediterranean and as far out as modern-day England. The Romans not only had these showers but also believed in bathing multiple times a week, if not every day. The water and sewage systems developed by the Greeks and Romans broke down and fell out of use after the fall of the Roman Empire.

Modern showers

The first mechanical shower, operated by a hand pump, was patented in England in 1767 by William Feetham, a stove maker from Ludgate Hill in London. His shower contraption used a pump to force the water into a vessel above the user's head and a chain would then be pulled to release the water from the vessel. Although the system dispensed with the servant labour of filling up and pouring out buckets of water, the showers failed to catch on with the rich as a method for piping hot water through the system was not available. The system would also recycle the same dirty water through every cycle.

This early start was greatly improved in the anonymously invented English Regency shower design of circa 1810 (there is some ambiguity among the sources). The original design was over  tall, and was made of several metal pipes painted to look like bamboo. A basin suspended above the pipes fed water into a nozzle that distributed the water over the user's shoulders. The water on the ground was drained and pumped back through the pipes into the basin, where the cycle would repeat itself. The original prototype was steadily improved upon in the following decades until it began to approximate the shower of today in its mode of operation. Hand-pumped models became fashionable at one point as well as the use of adjustable sprayers for different water flow. The reinvention of reliable indoor plumbing around 1850 allowed free-standing showers to be connected to a running water source, supplying a renewable flow of water.

Modern showers were installed in the barracks of the French army in the 1870s as an economic hygiene measure, under the guidance of François Merry Delabost, a French doctor and inventor. As surgeon-general at Bonne Nouvelle prison in Rouen, Delabost had previously replaced individual baths with mandatory communal showers for use by prisoners, arguing that they were more economical and hygienic. First six, then eight shower stalls were installed. The water was heated by a steam engine and in less than five minutes, up to eight prisoners could wash simultaneously with only twenty litres of water. The French system of communal showers was adopted by other armies, the first being that of Prussia in 1879, and by prisons in other jurisdictions. They were also adopted by boarding schools, before being installed in public bathhouses. The first shower in a public bathhouse was in 1887 in Vienna. In France, public bathhouses and showers were established by Charles Cazalet, firstly in Bordeaux in 1893 and then in Paris in 1899.

Types

Domestic
Domestic showers are most commonly stall showers or showers combined with a bathtub. A stall shower is a dedicated shower area which uses a door or curtain to contain water spray. The shower/tub format saves bathroom space and enables the area to be used for either a bath or a shower and commonly uses a sliding shower curtain or door to contain the water spray. Showers may also be in a wet room, in which there is no contained shower area, or in a dedicated shower room, which does not require containment of water spray.   Domestic showers can have a single shower head, multiple shower heads, handheld shower head(s) or other variations, all which may be adjustable as needed to varying degrees.

Public

Many modern athletic and aquatic facilities provide showers for use by patrons, commonly in gender segregated changing rooms. These can be in the form of individual stalls shielded by curtains or a door or communal shower rooms. The latter are generally large open rooms with any number of shower heads installed either directly into the walls or on posts throughout the shower area. Open showers are often provided at public swimming pools and at popular beaches. Military forces around the world set up field showers to enable the washing away of dangerous residue from modern weapons such as caustic chemicals, deadly biological agents, and radioactive materials, which can harm forces on both sides of a conflict.

Wet room
A wet room often refers to a bathroom without internal dedicated or raised areas which has an open shower. Structurally, a wet room requires the bathroom to have a gradient or slope towards a drain hole, and a foul air trap connecting the floor to the waste pipes. Depending on region, the term wet room can also encompass other rooms such as laundry rooms. In Norway, for example, any room with tap water and a drain in the floor is considered a wet room. Different jurisdictions often have special regulations concerning the design and construction of wet rooms to prevent damage from damp air or water leakage, e.g. mold.

There are also often special regulations concerning electrical installations in wet rooms. For example, since water supplies often have their own electrical grounding, there can be an added risk of injury associated with ground faults, and some jurisdictions therefore require the installation of residual-current devices in wet rooms.

Other

 Air shower, a type of bathing where high pressure air is used to blow off excess dust particles from cleanroom personnel.
 Digital shower, a shower system that works in a similar way to mixer or power showers, but provides more control over the temperature of the water with the use of a digital control panel.
 Eco shower, a shower system that comes in mixer or electric variations, but also features a regulator to regulate the flow of water with a view to saving water.
 Electric shower, a shower stall device to locally heat shower water with electrical power.
 Emergency showers, installed in laboratories and other facilities that use hazardous chemicals, are required by law in the United States; designed to deluge continuously at around  per minute  for at least 15 minutes  and should be located at most 10 seconds away from potential users.
 Mixer shower, a shower system that takes water from existing hot and cold water supplies and combines them within the unit.
 Navy shower, a method of showering that allows for significant conservation of water and energy.
 Power shower, a shower stall device that works similarly to a mixer shower by mixing existing hot and cold water feeds, but locally increases the water pressure available to the shower head by means of an electric booster pump.
 Roman shower, a shower that does not use a door or curtain.
 Steam shower, a type of bathing where a humidifying steam generator produces steam that is dispersed around a person's body.
 Vichy shower, a shower where large quantities of warm water are poured over a spa patron while the user lies within a shallow (wet) bed, similar to a massage table, but with drainage for the water.

Types of shower heads
 Fixed shower heads: Traditional fixed shower-heads are mostly common shower-faucets because they can easily connect to the plumbing fixtures without any additional hardware.
 Shower handsets: Hand-set shower-faucets are connected by a flexible hose, and can also be mounted and used like a fixed shower-head.
 Ceiling-mounted faucets: Ceiling-mounted shower-faucets are typically rain-drop shower-heads mounted in the ceiling of the shower. Water rains down, at low or medium pressure, using gravity to shower on one from directly above.
 Adjustable shower heads: Adjustable shower faucets often have numerous settings, including pulsating massage and low/medium/high-pressure flow settings.
 Shower panels: Unlike a single showerhead, these are wall-mounted with sprayers aimed horizontally at various parts of the body.

Use and ecology

Shower usage in the latter half of the 20th century skyrocketed. Personal hygiene became a primary concern, and bathing every day or multiple times a day is common among Western cultures. Showering is generally faster than bathing and can use less water. In an average home, showers are typically the third largest water use after toilets and clothes washers. The average American shower uses  and lasts for 8.2 minutes at an average flow rate of  per minute.

Showering is one of the leading ways people use water in the home, accounting for nearly 17 percent of residential indoor water use, which roughly equals to  of water annually just for showering.  The US Environmental Protection Agency recommends to retrofit home showers with a shower head that uses less than  to conserve water. However, many have hypothesized reducing flow rates of showerheads might cause users to take much longer showers. Other options to save water include using extra high pressure mist flow or design in sensors and valves to shut off or reduce water flow while people are not actively using the shower water.

Various measures can be taken to increase safety for those, especially elderly people, taking showers or baths.  Some people take more than one shower each day, normally at their normal shower time, and after exercising. People also shower to cool off in hot weather, especially those unaccustomed to the climate. Used shower water can be employed as greywater.

Cultural significance
Showering is mostly part of a daily routine primarily to promote cleanliness and prevent odor, disease and infection. Advances in science and medicine in the 19th century began to realize the benefit of regular bathing to an individual's health. As a result, most modern cultures encourage a daily personal hygiene regimen. Showering has also developed the reputation as a relaxing and generally therapeutic activity.

Structure and design
Designs for shower facilities vary by location and purpose. There are free-standing showers, but also showers which are integrated into a bathtub. Showers are separated from the surrounding area through watertight curtains (shower curtain), sliding doors, or folding doors, or shower blinds, in order to protect the space from spraying water. Showers with a level entry wet room are becoming very popular, especially due to improvements in waterproofing systems and prefabricated components.

Places such as a swimming pool, a locker room, or a military facility have multiple showers. There may be communal shower rooms without divisions, or shower stalls (typically open at the top). Many types of showers are available, including complete shower units which are all encompassing showers that include the pan, walls, and often the shower head, as well as pieced together units in which the pan, shower head, and doors are purchased separately. Each type of shower poses different installation issues.

Installation
Though the installation requirements of each of shower will differ, the installation of a shower in general requires the laying of several water transportation pipes, including a pipe for hot water and for cold water, and a drainage pipe. It is important that the wet areas of a bathroom be waterproof, and multiple layers of waterproofing can be employed.  Grout is used to fill gaps between tiles, but grout and tile setting materials are generally porous. Tiles are generally waterproof, though a shower pan must be installed beneath them as a safety to prevent water leakage. Thus small mosaic tiles offer less of a defense than large format tiles. Sub-tile waterproofing is important when tiles are being used. Best practice requires a waterproofing material to cover the walls and floor of the shower area that are then covered with tile, or in some countries with a sheet material like vinyl.

Drainage
A shower may be equipped with a second emergency drain outside of the shower in case of overflow. An emergency overflow drain is required in Australia and some European countries.

Shower head

A shower head is a perforated nozzle that distributes water over solid angle a focal point of use, generally overhead the bather. A shower uses less water than a full immersion in a bath. Some shower heads can be adjusted to spray different patterns of water, such as massage, gentle spray, strong spray, and intermittent pulse or combination modes.

Hard water may result in calcium and magnesium deposits clogging the head, reducing the flow and changing the spray pattern. For descaling, various acidic chemicals or brushes can be used or some heads have rubber-like jets that can be manually descaled. A homemade remedy is to immerse it in a solution of water and vinegar for a while, since the vinegar is able to dissolve limescale.

Some governments around the world set standards for water usage and regulate shower heads. For example, in the United States, residential and most commercial shower heads must flow no more than  per minute  per the Department of Energy ruling 10 CFR 430. Low-flow shower heads that have a water flow of equal or less than  per minute (2.0 gallons per minute), can use water more efficiently by aerating the water stream, altering nozzles through advanced flow principles or by high-speed oscillation of the spray stream. The US EPA administers a voluntary water saving program, WaterSense, which can certify low-flow shower heads.

Shower and bathtub curtains

Curtains can be used in shower or bathtub enclosures with two main purposes: to provide privacy and to prevent water from flooding or spraying outside the shower or bathtub area. Shower and bathtub curtains usually surround the bath inside the tub or shower area and are held up with railings or curtain rods high on the wall or ceiling. To accommodate the different types of bathtub shapes, railings can come in different sizes and are flexible in their design. The curtains are usually made from vinyl, cloth, or plastic.

Some people use two shower curtains: one that is inside the tub, which is mainly functional or decorative as well, and an outer shower curtain, which is purely decorative. The bottom portion of the inner curtain often comes with magnetic discs or suction cups which adhere to the bathtub itself.

Shower and bathtub doors
Shower or bathtub doors are doors (also called screens) used in bathrooms that help keep water inside a shower or bathtub and are alternatives to shower curtains. They are available in many different styles such as framed or frameless, sliding or swing. They are usually constructed of aluminium, clear glass, plexiglass or tempered glass. Shower doors can come in many different hardware finishes and glass patterns that can match other bathroom hardware such as faucets and shower heads. There are also shower doors that are in a neo angle design for use on shower pans that have the neo design as well. The design of the shower pan is extremely important as the shower door must be the type required for the pan in order to work. A shower door requires plastic lining along the edges of the door to protect against water leaking out.

Equipment
 Pressure-balanced valve, a device to provide constant shower water pressure and prevent temperature fluctuations
 Shampoo
 Shower caddy, a storage system inside the shower, typically for shampoo, conditioner, soap and other related things
 Shower cap, a cap worn while showering or bathing, to protect hair from becoming wet
 Shower radio, a radio that is waterproofed to allow it to be used in a bathroom or other wet environment
 Shower speaker, a Bluetooth speaker designed to play music while the shower is running
 Soap
 Washing mitt, a tool for applying soap to the body
 Water heat recycling units to reclaim much of the waste water's heat and recycle it to the shower head and minimize heat lost to the drain

See also
 Bathing
 Bathroom
 Bathtub
 Douche
 Navy shower
 Solar heated shower
 Steam shower
 Transfer bench
 Water recycling shower

Citations

Cited works and general references

 Books

 Web pages

Bathing
Bathrooms
Greek inventions
Hygiene
Plumbing

Shower


Mykola Danylovych Rudenko (19 de dezembro de 1920, Oblast de Dnipropetrovsk, Ucrânia - 1 de abril de 2004, Kiev, Ucrânia) foi um escritor, poeta, dramaturgo e ativista social ucraniano.

Mykola Danylovych Rudenko
Reabilitações na União Soviética

Mykola Rudenko


මොංගෝලියාව  () යනු  නැගෙනහිර සහ මධ්‍යම ආසියාවෙහි පිහිටි භූමියෙන් වටවූ රටකි.. මායිම උතුරින් රුසියාව, දකුණින්, නැගෙනහිරින් හා බටහිරින් චීනයට වේ. අගනුවර වන්නේ උලාන් බැටර් ය. එහි ජනගහනයෙන් 38% ජීවත් වේ. දේශපාලන පද්ධතිය පාර්ලිමේන්තු සමූහාණ්ඩුවක් වන මෙය විවිධ නොමැඩික් අධිරාජ්‍යයන් වන ෂියොන්නු, ෂියැංවි, රෝරන් 1206 ගල්කිෂ්කාන්  විසින් සොයාගෙන ඇත.
16 හා 17 වන ශත වර්ෂ වල  ටිබටන්  බුදුදහමේ ආභාෂය ලැබීය. 17 වන ශත වර්ෂයේ බොහෝ මොන්ගෝලියානුවන් කුයීන් රාජ්‍ය පාලනයට යටත් විය.  1911 මෙහි බිඳවැටීමත් සමඟ නිදහස ලැබු අතර 1921 තෙක් චීනයෙන් නිදහස් වීමට වෙර දැරූ අතර 1945 ජාත්‍යන්තර අවධානයට ලක්විය.1924 මොන්ගෝලියානු සමූහාණ්ඩුව විකාශය වූ අතර  දේශපාලනය ද ආරම්භ විය. 1990  බහු පක්ෂ ක්‍රමයක් හා 1992 නව ව්‍යවස්ථාවක් හා වෙළඳ ආර්ථිකයකට විවර විය.  වර්ග සැතපුම් 603,909 හා ජනගහනයෙන් මිලියන 2.9 කි. මෙහි තැනිතලා භූමිද වන අතර බෑවුම්  සහිත ප්‍රදේශ විශාලවන අතර කඳු උතුරු හා බටහිරට වන අතර, දකු‍ණට ඇත්තේ ගෝබි කාන්තාරයයි.ජනගහනයෙන් 30% ක්  නොමැඩික් සහ අර්ධ නොමැඩික්ය. මෙහි බහුතරයක් ජනයා ස්වදේශීය මොන්ගෝල් කසාන්තුවාන් සහ සුළු පිරිසක් රටේ බටහිරට වන්නට ඇත . ජනගහනයෙන් 20% දිනකට ඇ.ඩො.1.25 ට වඩා අඩුවෙන් උපයයි.

මූලාශ්‍ර
මොංගෝලියාව

මොංගෝලියාව


Algebraic K-theory is a subject area in mathematics with connections to geometry, topology, ring theory, and number theory.  Geometric, algebraic, and arithmetic objects are assigned objects called K-groups.  These are groups in the sense of abstract algebra.  They contain detailed information about the original object but are notoriously difficult to compute; for example, an important outstanding problem is to compute the K-groups of the integers.

K-theory was discovered in the late 1950s by Alexander Grothendieck in his study of intersection theory on algebraic varieties. In the modern language, Grothendieck defined only K0, the zeroth K-group, but even this single group has plenty of applications, such as the Grothendieck–Riemann–Roch theorem. Intersection theory is still a motivating force in the development of (higher) algebraic K-theory through its links with motivic cohomology and specifically Chow groups.  The subject also includes classical number-theoretic topics like quadratic reciprocity and embeddings of number fields into the real numbers and complex numbers, as well as more modern concerns like the construction of higher regulators and special values of L-functions.

The lower K-groups were discovered first, in the sense that adequate descriptions of these groups in terms of other algebraic structures were found.  For example, if F is a field, then  is isomorphic to the integers Z and is closely related to the notion of vector space dimension.  For a commutative ring R, the group  is related to the Picard group of R, and when R is the ring of integers in a number field, this generalizes the classical construction of the class group.  The group K1(R) is closely related to the group of units , and if R is a field, it is exactly the group of units.  For a number field F, the group K2(F) is related to class field theory, the Hilbert symbol, and the solvability of quadratic equations over completions.  In contrast, finding the correct definition of the higher K-groups of rings was a difficult achievement of Daniel Quillen, and many of the basic facts about the higher K-groups of algebraic varieties were not known until the work of Robert Thomason.

History

The history of K-theory was detailed by Charles Weibel.

The Grothendieck group K0

In the 19th century, Bernhard Riemann and his student Gustav Roch proved what is now known as the Riemann–Roch theorem.  If X is a Riemann surface, then the sets of meromorphic functions and meromorphic differential forms on X form vector spaces.  A line bundle on X determines subspaces of these vector spaces, and if X is projective, then these subspaces are finite dimensional.  The Riemann–Roch theorem states that the difference in dimensions between these subspaces is equal to the degree of the line bundle (a measure of twistedness) plus one minus the genus of X.  In the mid-20th century, the Riemann–Roch theorem was generalized by Friedrich Hirzebruch to all algebraic varieties.  In Hirzebruch's formulation, the Hirzebruch–Riemann–Roch theorem, the theorem became a statement about Euler characteristics: The Euler characteristic of a vector bundle on an algebraic variety (which is the alternating sum of the dimensions of its cohomology groups) equals the Euler characteristic of the trivial bundle plus a correction factor coming from characteristic classes of the vector bundle.  This is a generalization because on a projective Riemann surface, the Euler characteristic of a line bundle equals the difference in dimensions mentioned previously, the Euler characteristic of the trivial bundle is one minus the genus, and the only nontrivial characteristic class is the degree.

The subject of K-theory takes its name from a 1957 construction of Alexander Grothendieck which appeared in the Grothendieck–Riemann–Roch theorem, his generalization of Hirzebruch's theorem.  Let X be a smooth algebraic variety.  To each vector bundle on X, Grothendieck associates an invariant, its class.  The set of all classes on X was called K(X) from the German Klasse.  By definition, K(X) is a quotient of the free abelian group on isomorphism classes of vector bundles on X, and so it is an abelian group.  If the basis element corresponding to a vector bundle V is denoted [V], then for each short exact sequence of vector bundles:

Grothendieck imposed the relation .  These generators and relations define K(X), and they imply that it is the universal way to assign invariants to vector bundles in a way compatible with exact sequences.

Grothendieck took the perspective that the Riemann–Roch theorem is a statement about morphisms of varieties, not the varieties themselves.  He proved that there is a homomorphism from K(X) to the Chow groups of X coming from the Chern character and Todd class of X.  Additionally, he proved that a proper morphism  to a smooth variety Y determines a homomorphism  called the pushforward.  This gives two ways of determining an element in the Chow group of Y from a vector bundle on X: Starting from X, one can first compute the pushforward in K-theory and then apply the Chern character and Todd class of Y, or one can first apply the Chern character and Todd class of X and then compute the pushforward for Chow groups.  The Grothendieck–Riemann–Roch theorem says that these are equal.  When Y is a point, a vector bundle is a vector space, the class of a vector space is its dimension, and the Grothendieck–Riemann–Roch theorem specializes to Hirzebruch's theorem.

The group K(X) is now known as K0(X).  Upon replacing vector bundles by projective modules, K0 also became defined for non-commutative rings, where it had applications to group representations.  Atiyah and Hirzebruch quickly transported Grothendieck's construction to topology and used it to define topological K-theory.  Topological K-theory was one of the first examples of an extraordinary cohomology theory: It associates to each topological space X (satisfying some mild technical constraints) a sequence of groups Kn(X) which satisfy all the Eilenberg–Steenrod axioms except the normalization axiom.  The setting of algebraic varieties, however, is much more rigid, and the flexible constructions used in topology were not available.  While the group K0 seemed to satisfy the necessary properties to be the beginning of a cohomology theory of algebraic varieties and of non-commutative rings, there was no clear definition of the higher Kn(X).  Even as such definitions were developed, technical issues surrounding restriction and gluing usually forced Kn to be defined only for rings, not for varieties.

K0, K1, and K2

A group closely related to K1 for group rings was earlier introduced by J.H.C. Whitehead. Henri Poincaré had attempted to define the Betti numbers of a manifold in terms of a triangulation.  His methods, however, had a serious gap: Poincaré could not prove that two triangulations of a manifold always yielded the same Betti numbers.  It was clearly true that Betti numbers were unchanged by subdividing the triangulation, and therefore it was clear that any two triangulations that shared a common subdivision had the same Betti numbers.  What was not known was that any two triangulations admitted a common subdivision.  This hypothesis became a conjecture known as the Hauptvermutung (roughly "main conjecture").  The fact that triangulations were stable under subdivision led J.H.C. Whitehead to introduce the notion of simple homotopy type.  A simple homotopy equivalence is defined in terms of adding simplices or cells to a simplicial complex or cell complex in such a way that each additional simplex or cell deformation retracts into a subdivision of the old space.  Part of the motivation for this definition is that a subdivision of a triangulation is simple homotopy equivalent to the original triangulation, and therefore two triangulations that share a common subdivision must be simple homotopy equivalent.  Whitehead proved that simple homotopy equivalence is a finer invariant than homotopy equivalence by introducing an invariant called the torsion.  The torsion of a homotopy equivalence takes values in a group now called the Whitehead group and denoted Wh(π), where π is the fundamental group of the two complexes.  Whitehead found examples of non-trivial torsion and thereby proved that some homotopy equivalences were not simple.  The Whitehead group was later discovered to be a quotient of K1(Zπ), where Zπ is the integral group ring of π.  Later John Milnor used Reidemeister torsion, an invariant related to Whitehead torsion, to disprove the Hauptvermutung.

The first adequate definition of K1 of a ring was made by Hyman Bass and Stephen Schanuel.  In topological K-theory, K1 is defined using vector bundles on a suspension of the space.  All such vector bundles come from the clutching construction, where two trivial vector bundles on two halves of a space are glued along a common strip of the space.  This gluing data is expressed using the general linear group, but elements of that group coming from elementary matrices (matrices corresponding to elementary row or column operations) define equivalent gluings.  Motivated by this, the Bass–Schanuel definition of K1 of a ring R is , where GL(R) is the infinite general linear group (the union of all GLn(R)) and E(R) is the subgroup of elementary matrices.  They also provided a definition of K0 of a homomorphism of rings and proved that K0 and K1 could be fit together into an exact sequence similar to the relative homology exact sequence.

Work in K-theory from this period culminated in Bass' book Algebraic K-theory.  In addition to providing a coherent exposition of the results then known, Bass improved many of the statements of the theorems.  Of particular note is that Bass, building on his earlier work with Murthy, provided the first proof of what is now known as the fundamental theorem of algebraic K-theory.  This is a four-term exact sequence relating K0 of a ring R to K1 of R, the polynomial ring R[t], and the localization R[t, t−1].  Bass recognized that this theorem provided a description of K0 entirely in terms of K1.  By applying this description recursively, he produced negative K-groups K−n(R).  In independent work, Max Karoubi gave another definition of negative K-groups for certain categories and proved that his definitions yielded that same groups as those of Bass.

The next major development in the subject came with the definition of K2.  Steinberg studied the universal central extensions of a Chevalley group over a field and gave an explicit presentation of this group in terms of generators and relations.  In the case of the group En(k) of elementary matrices, the universal central extension is now written Stn(k) and called the Steinberg group.  In the spring of 1967, John Milnor defined K2(R) to be the kernel of the homomorphism .  The group K2 further extended some of the exact sequences known for K1 and K0, and it had striking applications to number theory.  Hideya Matsumoto's 1968 thesis showed that for a field F, K2(F) was isomorphic to:

This relation is also satisfied by the Hilbert symbol, which expresses the solvability of quadratic equations over local fields.  In particular, John Tate was able to prove that K2(Q) is essentially structured around the law of quadratic reciprocity.

Higher K-groups

In the late 1960s and early 1970s, several definitions of higher K-theory were proposed.  Swan and Gersten both produced definitions of Kn for all n, and Gersten proved that his and Swan's theories were equivalent, but the two theories were not known to satisfy all the expected properties.  Nobile and Villamayor also proposed a definition of higher K-groups.  Karoubi and Villamayor defined well-behaved K-groups for all n, but their equivalent of K1 was sometimes a proper quotient of the Bass–Schanuel K1.  Their K-groups are now called KVn and are related to homotopy-invariant modifications of K-theory.

Inspired in part by Matsumoto's theorem, Milnor made a definition of the higher K-groups of a field.  He referred to his definition as "purely ad hoc", and it neither appeared to generalize to all rings nor did it appear to be the correct definition of the higher K-theory of fields.  Much later, it was discovered by Nesterenko and Suslin and by Totaro that Milnor K-theory is actually a direct summand of the true K-theory of the field.  Specifically, K-groups have a filtration called the weight filtration, and the Milnor K-theory of a field is the highest weight-graded piece of the K-theory.  Additionally, Thomason discovered that there is no analog of Milnor K-theory for a general variety.

The first definition of higher K-theory to be widely accepted was Daniel Quillen's.  As part of Quillen's work on the Adams conjecture in topology, he had constructed maps from the classifying spaces BGL(Fq) to the homotopy fiber of , where ψq is the qth Adams operation acting on the classifying space BU.  This map is acyclic, and after modifying BGL(Fq) slightly to produce a new space BGL(Fq)+, the map became a homotopy equivalence.  This modification was called the plus construction.  The Adams operations had been known to be related to Chern classes and to K-theory since the work of Grothendieck, and so Quillen was led to define the K-theory of R as the homotopy groups of BGL(R)+.  Not only did this recover K1 and K2, the relation of K-theory to the Adams operations allowed Quillen to compute the K-groups of finite fields.

The classifying space BGL is connected, so Quillen's definition failed to give the correct value for K0.  Additionally, it did not give any negative K-groups.  Since K0 had a known and accepted definition it was possible to sidestep this difficulty, but it remained technically awkward.  Conceptually, the problem was that the definition sprung from GL, which was classically the source of K1.  Because GL knows only about gluing vector bundles, not about the vector bundles themselves, it was impossible for it to describe K0.

Inspired by conversations with Quillen, Segal soon introduced another approach to constructing algebraic K-theory under the name of Γ-objects.  Segal's approach is a homotopy analog of Grothendieck's construction of K0.  Where Grothendieck worked with isomorphism classes of bundles, Segal worked with the bundles themselves and used isomorphisms of the bundles as part of his data.  This results in a spectrum whose homotopy groups are the higher K-groups (including K0).  However, Segal's approach was only able to impose relations for split exact sequences, not general exact sequences.  In the category of projective modules over a ring, every short exact sequence splits, and so Γ-objects could be used to define the K-theory of a ring.  However, there are non-split short exact sequences in the category of vector bundles on a variety and in the category of all modules over a ring, so Segal's approach did not apply to all cases of interest.

In the spring of 1972, Quillen found another approach to the construction of higher K-theory which was to prove enormously successful.  This new definition began with an exact category, a category satisfying certain formal properties similar to, but slightly weaker than, the properties satisfied by a category of modules or vector bundles.  From this he constructed an auxiliary category using a new device called his "Q-construction."  Like Segal's Γ-objects, the Q-construction has its roots in Grothendieck's definition of K0.  Unlike Grothendieck's definition, however, the Q-construction builds a category, not an abelian group, and unlike Segal's Γ-objects, the Q-construction works directly with short exact sequences.  If C is an abelian category, then QC is a category with the same objects as C but whose morphisms are defined in terms of short exact sequences in C.  The K-groups of the exact category are the homotopy groups of ΩBQC, the loop space of the geometric realization (taking the loop space corrects the indexing).  Quillen additionally proved his " theorem" that his two definitions of K-theory agreed with each other.  This yielded the correct K0 and led to simpler proofs, but still did not yield any negative K-groups.

All abelian categories are exact categories, but not all exact categories are abelian.  Because Quillen was able to work in this more general situation, he was able to use exact categories as tools in his proofs.  This technique allowed him to prove many of the basic theorems of algebraic K-theory.  Additionally, it was possible to prove that the earlier definitions of Swan and Gersten were equivalent to Quillen's under certain conditions.

K-theory now appeared to be a homology theory for rings and a cohomology theory for varieties.  However, many of its basic theorems carried the hypothesis that the ring or variety in question was regular.  One of the basic expected relations was a long exact sequence (called the "localization sequence") relating the K-theory of a variety X and an open subset U.  Quillen was unable to prove the existence of the localization sequence in full generality.  He was, however, able to prove its existence for a related theory called G-theory (or sometimes K′-theory).  G-theory had been defined early in the development of the subject by Grothendieck.  Grothendieck defined G0(X) for a variety X to be the free abelian group on isomorphism classes of coherent sheaves on X, modulo relations coming from exact sequences of coherent sheaves.  In the categorical framework adopted by later authors, the K-theory of a variety is the K-theory of its category of vector bundles, while its G-theory is the K-theory of its category of coherent sheaves.  Not only could Quillen prove the existence of a localization exact sequence for G-theory, he could prove that for a regular ring or variety, K-theory equaled G-theory, and therefore K-theory of regular varieties had a localization exact sequence.  Since this sequence was fundamental to many of the facts in the subject, regularity hypotheses pervaded early work on higher K-theory.

Applications of algebraic K-theory in topology

The earliest application of algebraic K-theory to topology was Whitehead's construction of Whitehead torsion.  A closely related construction was found by C. T. C. Wall in 1963.  Wall found that a space X dominated by a finite complex has a generalized Euler characteristic taking values in a quotient of K0(Zπ), where π is the fundamental group of the space.  This invariant is called Wall's finiteness obstruction because X is homotopy equivalent to a finite complex if and only if the invariant vanishes.  Laurent Siebenmann in his thesis found an invariant similar to Wall's that gives an obstruction to an open manifold being the interior of a compact manifold with boundary.  If two manifolds with boundary M and N have isomorphic interiors (in TOP, PL, or DIFF as appropriate), then the isomorphism between them defines an h-cobordism between M and N.

Whitehead torsion was eventually reinterpreted in a more directly K-theoretic way.  This reinterpretation happened through the study of h-cobordisms.  Two n-dimensional manifolds M and N are h-cobordant if there exists an -dimensional manifold with boundary W whose boundary is the disjoint union of M and N and for which the inclusions of M and N into W are homotopy equivalences (in the categories TOP, PL, or DIFF).  Stephen Smale's h-cobordism theorem asserted that if , W is compact, and M, N, and W are simply connected, then W is isomorphic to the cylinder  (in TOP, PL, or DIFF as appropriate).  This theorem proved the Poincaré conjecture for .

If M and N are not assumed to be simply connected, then an h-cobordism need not be a cylinder.  The s-cobordism theorem, due independently to Mazur, Stallings, and Barden, explains the general situation: An h-cobordism is a cylinder if and only if the Whitehead torsion of the inclusion  vanishes.  This generalizes the h-cobordism theorem because the simple connectedness hypotheses imply that the relevant Whitehead group is trivial.  In fact the s-cobordism theorem implies that there is a bijective correspondence between isomorphism classes of h-cobordisms and elements of the Whitehead group.

An obvious question associated with the existence of h-cobordisms is their uniqueness.  The natural notion of equivalence is isotopy.  Jean Cerf proved that for simply connected smooth manifolds M of dimension at least 5, isotopy of h-cobordisms is the same as a weaker notion called pseudo-isotopy.  Hatcher and Wagoner studied the components of the space of pseudo-isotopies and related it to a quotient of K2(Zπ).

The proper context for the s-cobordism theorem is the classifying space of h-cobordisms.  If M is a CAT manifold, then HCAT(M) is a space that classifies bundles of h-cobordisms on M.  The s-cobordism theorem can be reinterpreted as the statement that the set of connected components of this space is the Whitehead group of π1(M).  This space contains strictly more information than the Whitehead group; for example, the connected component of the trivial cobordism describes the possible cylinders on M and in particular is the obstruction to the uniqueness of a homotopy between a manifold and .  Consideration of these questions led Waldhausen to introduce his algebraic K-theory of spaces.  The algebraic K-theory of M is a space A(M) which is defined so that it plays essentially the same role for higher K-groups as K1(Zπ1(M)) does for M.  In particular, Waldhausen showed that there is a map from A(M) to a space Wh(M) which generalizes the map  and whose homotopy fiber is a homology theory.

In order to fully develop A-theory, Waldhausen made significant technical advances in the foundations of K-theory.  Waldhausen introduced Waldhausen categories, and for a Waldhausen category C he introduced a simplicial category S⋅C (the S is for Segal) defined in terms of chains of cofibrations in C.  This freed the foundations of K-theory from the need to invoke analogs of exact sequences.

Algebraic topology and algebraic geometry in algebraic K-theory

Quillen suggested to his student Kenneth Brown that it might be possible to create a theory of sheaves of spectra of which K-theory would provide an example.  The sheaf of K-theory spectra would, to each open subset of a variety, associate the K-theory of that open subset.  Brown developed such a theory for his thesis.  Simultaneously, Gersten had the same idea.  At a Seattle conference in autumn of 1972, they together discovered a spectral sequence converging from the sheaf cohomology of , the sheaf of Kn-groups on X, to the K-group of the total space.  This is now called the Brown–Gersten spectral sequence.

Spencer Bloch, influenced by Gersten's work on sheaves of K-groups, proved that on a regular surface, the cohomology group  is isomorphic to the Chow group CH2(X) of codimension 2 cycles on X.  Inspired by this, Gersten conjectured that for a regular local ring R with fraction field F, Kn(R) injects into Kn(F) for all n.  Soon Quillen proved that this is true when R contains a field, and using this he proved that

for all p.  This is known as  Bloch's formula.  While progress has been made on Gersten's conjecture since then, the general case remains open.

Lichtenbaum conjectured that special values of the zeta function of a number field could be expressed in terms of the K-groups of the ring of integers of the field.  These special values were known to be related to the étale cohomology of the ring of integers.  Quillen therefore generalized Lichtenbaum's conjecture, predicting the existence of a spectral sequence like the Atiyah–Hirzebruch spectral sequence in topological K-theory.  Quillen's proposed spectral sequence would start from the étale cohomology of a ring R and, in high enough degrees and after completing at a prime  invertible in R, abut to the -adic completion of the K-theory of R.  In the case studied by Lichtenbaum, the spectral sequence would degenerate, yielding Lichtenbaum's conjecture.

The necessity of localizing at a prime  suggested to Browder that there should be a variant of K-theory with finite coefficients.  He introduced K-theory groups Kn(R; Z/Z) which were Z/Z-vector spaces, and he found an analog of the Bott element in topological K-theory.  Soule used this theory to construct "étale Chern classes", an analog of topological Chern classes which took elements of algebraic K-theory to classes in étale cohomology.  Unlike algebraic K-theory, étale cohomology is highly computable, so étale Chern classes provided an effective tool for detecting the existence of elements in K-theory.  William G. Dwyer and Eric Friedlander then invented an analog of K-theory for the étale topology called étale K-theory.  For varieties defined over the complex numbers, étale K-theory is isomorphic to topological K-theory.  Moreover, étale K-theory admitted a spectral sequence similar to the one conjectured by Quillen.  Thomason proved around 1980 that after inverting the Bott element, algebraic K-theory with finite coefficients became isomorphic to étale K-theory.

Throughout the 1970s and early 1980s, K-theory on singular varieties still lacked adequate foundations.  While it was believed that Quillen's K-theory gave the correct groups, it was not known that these groups had all of the envisaged properties.  For this, algebraic K-theory had to be reformulated.  This was done by Thomason in a lengthy monograph which he co-credited to his dead friend Thomas Trobaugh, who he said gave him a key idea in a dream.  Thomason combined Waldhausen's construction of K-theory with the foundations of intersection theory described in volume six of Grothendieck's Séminaire de Géométrie Algébrique du Bois Marie.  There, K0 was described in terms of complexes of sheaves on algebraic varieties.  Thomason discovered that if one worked with in derived category of sheaves, there was a simple description of when a complex of sheaves could be extended from an open subset of a variety to the whole variety.  By applying Waldhausen's construction of K-theory to derived categories, Thomason was able to prove that algebraic K-theory had all the expected properties of a cohomology theory.

In 1976, Keith Dennis discovered an entirely novel technique for computing K-theory based on Hochschild homology.  This was based around the existence of the Dennis trace map, a homomorphism from K-theory to Hochschild homology.  While the Dennis trace map seemed to be successful for calculations of K-theory with finite coefficients, it was less successful for rational calculations.  Goodwillie, motivated by his "calculus of functors", conjectured the existence of a theory intermediate to K-theory and Hochschild homology.  He called this theory topological Hochschild homology because its ground ring should be the sphere spectrum (considered as a ring whose operations are defined only up to homotopy).  In the mid-1980s, Bokstedt gave a definition of topological Hochschild homology that satisfied nearly all of Goodwillie's conjectural properties, and this made possible further computations of K-groups.  Bokstedt's version of the Dennis trace map was a transformation of spectra .  This transformation factored through the fixed points of a circle action on THH, which suggested a relationship with cyclic homology.  In the course of proving an algebraic K-theory analog of the Novikov conjecture, Bokstedt, Hsiang, and Madsen introduced topological cyclic homology, which bore the same relationship to topological Hochschild homology as cyclic homology did to Hochschild homology.
The Dennis trace map to topological Hochschild homology factors through topological cyclic homology, providing an even more detailed tool for calculations.  In 1996, Dundas, Goodwillie, and McCarthy proved that topological cyclic homology has in a precise sense the same local structure as algebraic K-theory, so that if a calculation in K-theory or topological cyclic homology is possible, then many other "nearby" calculations follow.

Lower K-groups
The lower K-groups were discovered first, and given various ad hoc descriptions, which remain useful. Throughout, let A be a ring.

K0
The functor K0 takes a ring A to the Grothendieck group of the set of isomorphism classes of its finitely generated projective modules, regarded as a monoid under direct sum. Any ring homomorphism A → B gives a map K0(A) → K0(B) by mapping (the class of) a projective A-module M to M ⊗A B, making K0 a covariant functor.

If the ring A is commutative, we can define a subgroup of  K0(A) as the set

where :

is the map sending every (class of a) finitely generated projective A-module M to the rank of the free -module  (this module is indeed free, as any finitely generated projective module over a local ring is free). This subgroup  is known as the reduced zeroth K-theory of A.

If B is a ring without an identity element, we can extend the definition of K0 as follows.  Let A = B⊕Z be the extension of B to a ring with unity obtaining by adjoining an identity element (0,1).  There is a short exact sequence B → A → Z and we define K0(B) to be the kernel of the corresponding map K0(A) → K0(Z) = Z.

Examples

 (Projective) modules over a field k are vector spaces and K0(k) is isomorphic to Z, by dimension.
 Finitely generated projective modules over a local ring A are free and so in this case once again K0(A) is isomorphic to Z, by rank.
 For A a Dedekind domain, K0(A) = Pic(A) ⊕ Z, where Pic(A) is the Picard group of A,

An algebro-geometric variant of this construction is applied to the category of algebraic varieties; it associates with a given algebraic variety X the Grothendieck's K-group of the category of locally free sheaves (or coherent sheaves) on X. Given a compact topological space X, the topological K-theory Ktop(X) of (real) vector bundles over X coincides with K0 of the ring of continuous real-valued functions on X.

Relative K0
Let I be an ideal of A and define the "double" to be a subring of the Cartesian product A×A:

The relative K-group is defined in terms of the "double"

where the map is induced by projection along the first factor.

The relative K0(A,I) is isomorphic to K0(I), regarding I as a ring without identity.  The independence from A is an analogue of the Excision theorem in homology.

K0 as a ring
If A is a commutative ring, then the tensor product of projective modules is again projective, and so tensor product induces a multiplication turning K0 into a commutative ring with the class [A] as identity.  The exterior product similarly induces a λ-ring structure.
The Picard group embeds as a subgroup of the group of units K0(A)∗.

K1
Hyman Bass provided this definition, which generalizes the group of units of a ring: K1(A) is the abelianization of the infinite general linear group:

Here

is the direct limit of the GL(n), which embeds in GL(n + 1) as the upper left block matrix, and  is its commutator subgroup. Define an elementary matrix to be one which is the sum of an identity matrix and a single off-diagonal element (this is a subset of the elementary matrices used in linear algebra). Then Whitehead's lemma states that the group E(A) generated by elementary matrices equals the commutator subgroup [GL(A), GL(A)]. Indeed, the group GL(A)/E(A) was first defined and studied by Whitehead, and is called the Whitehead group of the ring A.

Relative K1
The relative K-group is defined in terms of the "double"

There is a natural exact sequence

Commutative rings and fields
For A a commutative ring, one can define a determinant det: GL(A) → A* to the group of units of A, which vanishes on E(A) and thus descends to a map det : K1(A) → A*.  As E(A) ◅ SL(A), one can also define the special Whitehead group SK1(A) := SL(A)/E(A). This map splits via the map A* → GL(1, A) → K1(A) (unit in the upper left corner), and hence is onto, and has the special Whitehead group as kernel, yielding the split short exact sequence:

which is a quotient of the usual split short exact sequence defining the special linear group, namely

The determinant is split by including the group of units  A* = GL1(A) into the general linear group GL(A), so  K1(A) splits as the direct sum of the group of units and the special Whitehead group: K1(A) ≅ A* ⊕ SK1 (A).

When A is a Euclidean domain (e.g. a field, or the integers) SK1(A) vanishes, and the determinant map is an isomorphism from K1(A) to A∗. This is false in general for PIDs, thus providing one of the rare mathematical features of Euclidean domains that do not generalize to all PIDs.  An explicit PID such that SK1 is nonzero was given by Ischebeck in 1980 and by Grayson in 1981. If A is a Dedekind domain whose quotient field is an algebraic number field (a finite extension of the rationals) then  shows that SK1(A) vanishes.

The vanishing of SK1 can be interpreted as saying that K1 is generated by the image of GL1 in GL.  When this fails, one can ask whether K1 is generated by the image of GL2.  For a Dedekind domain, this is the case: indeed, K1 is generated by the images of GL1 and SL2 in GL.  The subgroup of SK1 generated by SL2 may be studied by Mennicke symbols.  For Dedekind domains with all quotients by maximal  ideals finite, SK1 is a torsion group.

For a non-commutative ring, the determinant cannot in general be defined, but the map GL(A) → K1(A) is a generalisation of the determinant.

Central simple algebras
In the case of a central simple algebra A over a field F, the reduced norm provides a generalisation of the determinant giving a map K1(A) → F∗ and SK1(A) may be defined as the kernel.  Wang's theorem states that if A has prime degree then SK1(A) is trivial, and this may be extended to square-free degree.  Wang also showed that SK1(A) is trivial for any central simple algebra over a number field, but Platonov has given examples of algebras of degree prime squared for which SK1(A) is non-trivial.

K2

John Milnor found the right definition of K2: it is the center of the Steinberg group St(A) of A.

It can also be defined as the kernel of the map

or as the Schur multiplier of the group of elementary matrices.

For a field, K2 is determined by Steinberg symbols: this leads to Matsumoto's theorem.

One can compute that K2 is zero for any finite field.  The computation of K2(Q) is complicated: Tate proved

and remarked that the proof followed Gauss's first proof of the Law of Quadratic Reciprocity.

For non-Archimedean local fields, the group K2(F) is the direct sum of a finite cyclic group of order m, say, and a divisible group K2(F)m.

We have K2(Z) = Z/2, and in general K2 is finite for the ring of integers of a number field.

We further have K2(Z/n) = Z/2 if n is divisible by 4, and otherwise zero.

Matsumoto's theorem
Matsumoto's theorem states that for a field k, the second K-group is given by

Matsumoto's original theorem is even more general: For any root system, it gives a presentation for the unstable K-theory. This presentation is different from the one given here only for symplectic root systems. For non-symplectic root systems, the unstable second K-group with respect to the root system is exactly the stable K-group for GL(A). Unstable second K-groups (in this context) are defined by taking the kernel of the universal central extension of the Chevalley group of universal type for a given root system. This construction yields the kernel of the Steinberg extension for the root systems An (n > 1) and, in the limit, stable second K-groups.

Long exact sequences
If A is a Dedekind domain with field of fractions F then there is a long exact sequence

where p runs over all prime ideals of A.

There is also an extension of the exact sequence for relative K1 and K0:

Pairing
There is a pairing on K1 with values in K2.  Given commuting matrices X and Y over A, take elements x and y in the Steinberg group  with X,Y as images.  The commutator  is an element of K2.  The map is not always surjective.

Milnor K-theory

The above expression for K2 of a field k led Milnor to the following definition of "higher" K-groups by

thus as graded parts of a quotient of the tensor algebra of the multiplicative group k× by the two-sided ideal, generated by the

For n = 0,1,2 these coincide with those below, but for n ≧ 3 they differ in general.  For example, we have K(Fq) = 0 for n ≧ 2
but KnFq is nonzero for odd n (see below).

The tensor product on the tensor algebra induces a product  making  a graded ring which is graded-commutative.

The images of elements  in  are termed symbols, denoted .  For integer m invertible in k there is a map

where  denotes the group of m-th roots of unity in some separable extension of k.  This extends to

satisfying the defining relations of the Milnor K-group.  Hence   may be regarded as a map on , called the Galois symbol map.

The relation between étale (or Galois) cohomology of the field and Milnor K-theory modulo 2 is the Milnor conjecture, proven by Vladimir Voevodsky. The analogous statement for odd primes is the Bloch-Kato conjecture, proved by Voevodsky, Rost, and others.

Higher K-theory
The accepted definitions of higher K-groups were given by , after a few years during which several incompatible definitions were suggested.  The object of the program was to find definitions of K(R) and K(R,I) in terms of classifying spaces so that
R ⇒ K(R) and (R,I) ⇒ K(R,I) are functors into a homotopy category of spaces and the long exact sequence for relative K-groups arises as the long exact homotopy sequence of a fibration K(R,I) → K(R) → K(R/I).

Quillen gave two constructions, the "plus-construction" and the "Q-construction", the latter subsequently modified in different ways.  The two constructions yield the same K-groups.

The +-construction
One possible definition of higher algebraic K-theory of rings was given by Quillen

Here πn is a homotopy group, GL(R) is the direct limit of the general linear groups over R for the size of the matrix tending to infinity, B is the classifying space construction of homotopy theory, and the + is Quillen's plus construction. He originally found this idea while studying the group cohomology of  and noted some of his calculations were related to .

This definition only holds for n > 0 so one often defines the higher algebraic K-theory via

Since BGL(R)+ is path connected and K0(R) discrete, this definition doesn't differ in higher degrees and also holds for n = 0.

The Q-construction

The Q-construction gives the same results as the +-construction, but it applies in more general situations. Moreover, the definition is more direct in the sense that the K-groups, defined via the Q-construction are functorial by definition. This fact is not automatic in the plus-construction.

Suppose  is an exact category; associated to  a new category  is defined, objects of which are those of  and morphisms from M′ to M″ are isomorphism classes of diagrams

where the first arrow is an admissible epimorphism and the second arrow is an admissible monomorphism. Note the morphisms in  are analogous to the definitions of morphisms in the category of motives, where morphisms are given as correspondences  such thatis a diagram where the arrow on the left is a covering map (hence surjective) and the arrow on the right is injective. This category can then be turned into a topological space using the classifying space construction  , which is defined to be the geometric realisation of the nerve of . Then, the i-th K-group of the exact category  is then defined as

with a fixed zero-object . Note the classifying space of a groupoid  moves the homotopy groups up one degree, hence the shift in degrees for  being  of a space.

This definition coincides with the above definition of K0(P). If P is the category of finitely generated projective R-modules, this definition agrees with the above BGL+
definition of Kn(R) for all n.

More generally, for a scheme X, the higher K-groups of X are defined to be the K-groups of (the exact category of) locally free coherent sheaves on X.

The following variant of this is also used: instead of finitely generated projective (= locally free) modules, take finitely generated modules. The resulting K-groups are usually written Gn(R). When R is a noetherian regular ring, then G- and K-theory coincide. Indeed, the global dimension of regular rings is finite, i.e. any finitely generated module has a finite projective resolution P* → M, and a simple argument shows that the canonical map K0(R) → G0(R) is an isomorphism, with [M]=Σ ± [Pn].  This isomorphism extends to the higher K-groups, too.

The S-construction

A third construction of K-theory groups is the S-construction, due to Waldhausen. It applies to categories with cofibrations (also called Waldhausen categories). This is a more general concept than exact categories.

Examples
While the Quillen algebraic K-theory has provided deep insight into various aspects of algebraic geometry and topology, the K-groups have proved particularly difficult to compute except in a few isolated but interesting cases. (See also: K-groups of a field.)

Algebraic K-groups of finite fields
The first and one of the most important calculations of the higher algebraic K-groups of a ring were made by Quillen himself for the case of finite fields:

If  Fq is the finite field with q elements, then:

 K0(Fq) = Z,
 K2i(Fq) = 0 for i ≥1,
 K2i–1(Fq) = Z/(q i − 1)Z for i ≥ 1.
 reproved Quillen's computation using different methods.

Algebraic K-groups of rings of integers
Quillen proved that if A is the ring of algebraic integers in an algebraic number field F (a finite extension of the rationals), then the algebraic K-groups of A are finitely generated.  Armand Borel used this to calculate Ki(A) and Ki(F) modulo torsion. For example, for the integers Z, Borel proved that (modulo torsion)

 Ki (Z)/tors.=0 for positive i unless i=4k+1 with k positive
 K4k+1 (Z)/tors.= Z for positive k.

The torsion subgroups of K2i+1(Z), and the orders of the finite groups K4k+2(Z) have recently been determined, but whether the latter groups are cyclic, and whether the groups K4k(Z) vanish depends upon Vandiver's conjecture about the class groups of cyclotomic integers. See Quillen–Lichtenbaum conjecture for more details.

Applications and open questions
Algebraic K-groups are used in conjectures on special values of L-functions and the formulation of a non-commutative main conjecture of Iwasawa theory and in construction of higher regulators.

Parshin's conjecture concerns the higher algebraic K-groups for smooth varieties over finite fields, and states that in this case the groups vanish up to torsion.

Another fundamental conjecture due to Hyman Bass (Bass' conjecture) says that all  of the groups Gn(A) are finitely generated when A is a finitely generated Z-algebra. (The groups
Gn(A) are the K-groups of the category of finitely generated A-modules)

See also
Additive K-theory
Bloch's formula
Fundamental theorem of algebraic K-theory
Basic theorems in algebraic K-theory
K-theory
K-theory of a category
K-group of a field
K-theory spectrum
Redshift conjecture
Topological K-theory
Rigidity (K-theory)

Notes

References

  (lower K-groups)

  (Quillen's Q-construction)
  (relation of Q-construction to plus-construction)
.  Errata

  (survey article)

Further reading

Pedagogical references
 Higher Algebraic K-Theory: an overview
.  Errata

Historical references

 Bokstedt, M., Topological Hochschild homology. Preprint, Bielefeld, 1986.
 Bokstedt, M., Hsiang, W. C., Madsen, I., The cyclotomic trace and algebraic K-theory of spaces. Invent. Math., 111(3) (1993), 465–539.

 Brown, K., Gersten, S., Algebraic K-theory as generalized sheaf cohomology, Algebraic K-theory I, Lecture Notes in Math., vol. 341, Springer-Verlag, 1973, pp. 266–292.

 Dennis, R. K., Higher algebraic K-theory and Hochschild homology, unpublished preprint (1976).

 Grothendieck, Alexander, Classes de fasiceaux et theoreme de Riemann–Roch, mimeographed notes, Princeton 1957.

 Milnor, J., Introduction to Algebraic K-theory, Princeton Univ. Press, 1971.
 Nobile, A., Villamayor, O., Sur la K-theorie algebrique,  Annales Scientifiques de l'École Normale Supérieure, 4e serie, 1, no. 3, 1968, 581–616.
 Quillen, Daniel, Cohomology of groups, Proc. ICM Nice 1970, vol. 2, Gauthier-Villars, Paris, 1971, 47–52.
 Quillen, Daniel, Higher algebraic K-theory I, Algebraic K-theory I, Lecture Notes in Math., vol. 341, Springer Verlag, 1973, 85–147.
 Quillen, Daniel, Higher algebraic K-theory, Proc. Intern. Congress Math., Vancouver, 1974, vol. I, Canad. Math. Soc., 1975, pp. 171–176.

 Siebenmann, Larry, The Obstruction to Finding a Boundary for an Open Manifold of Dimension Greater than Five, Thesis, Princeton University (1965).

 Steinberg, R., Generateurs, ,relations et revetements de groupes algebriques, ́Colloq. Theorie des Groupes Algebriques, Gauthier-Villars, Paris, 1962, pp. 113–127. (French)
 Swan, Richard, Nonabelian homological algebra and K-theory, Proc. Sympos. Pure Math., vol. XVII, 1970, pp. 88–123.
 Thomason, R. W., Algebraic K-theory and étale cohomology, Ann. Scient. Ec. Norm. Sup. 18, 4e serie (1985), 437–552; erratum 22 (1989), 675–677.
 Thomason, R. W., Le principe de sciendage et l'inexistence d'une K-theorie de Milnor globale, Topology  31, no. 3, 1992, 571–588.

 Waldhausen, F., Algebraic K-theory of topological spaces. I, in Algebraic and geometric topology (Proc. Sympos. Pure Math., Stanford Univ., Stanford, Calif., 1976), Part 1, pp. 35–60, Proc. Sympos. Pure Math., XXXII, Amer. Math. Soc., Providence, R.I., 1978.
 Waldhausen, F., Algebraic K-theory of spaces, in Algebraic and geometric topology (New Brunswick, N.J., 1983), Lecture Notes in Mathematics, vol. 1126 (1985), 318–419.

External links
 K theory preprint archive

Algebraic geometry

Algebraic K-theory


Pasul Cavnic (găsit și sub numele de Pasul Rotunda) este o trecătoare în Carpații Orientali situată în Munții Lăpușului, la altitudinea de 1080 m, care face legătura între valea Cavnicului și Depresiunea Lăpușului, respectiv între orașele Cavnic și Târgu Lăpuș.

Date geografice
Se află localizată între vârfurile Roții (1216 mm) – aflat la nord-est și Bolchiș (1266 m) – aflat la sud-vest. Ascensiunea se face dinspre nord-vest pe valea râului Cavnic și dinspre sud-est pe valea râului Strâmbu-Băiuț.

Este traversat de DJ109F.

În apropiere se află trecătorile Neteda (Rotunda) la nord, Gutâi la nord-vest și Botiza la est.

Oportunități turistice de vecinătate
Ruinele de la Logolda – ruine ale unei topitorii pentru aur, deschise în 1862 de către o societate engleză
Cascada Izvor (Piciorul Calului) –  situată în zona minei și topitoriei de aur Logolda, pe cursul unui pârâu care izvorăște dintr-o fostă galerie minieră
 Tăul Negru

Referințe

Bibliografie
  Istvan, Dumitru & Popescu, Stelian & Pop, Ioan; Gutâi; Colecția Munții Noștri, Nr. 45; Editura Sport-Turism; București; 1990; ISBN 973-41-0066-1

Carpații Orientali
Cavnic

Pasul Cavnic


Ndong Mboula, né le , est un musicien gabonais originaire du Woleu-Ntem. Il est considéré comme l'un des doyens de la musique gabonaise notamment grâce à son titre Etouk Nzom, sorti en 2006 ou plus récemment en 2020 avec Elone 2.0.

Biographie
Ndong Mboula est né en 1955 au nord du Gabon.

Son morceau Etouk Nzom fut un véritable tube dans les années 2006-2008, ce qui le positionnera comme un des meilleurs artistes du pays.

En 2007, il sort le titre Moïse.

Après plusieurs années d'absence dans la musique mais plusieurs concerts dans sa ville natale, il revient en collaboration avec Ndoman pour le morceau Elone 2.0 qui fut un hit avec notamment plus d'1 million de vues sur YouTube.

Il a a son actif 4 albums et 2 Balafon Music Awards.

Il à notamment fortement influencé le musicien Sima Mboula.

Discographie

 Etouk Nzom (2007)
 Monsieur le Roi (2020)

Prix et récompenses

Vie privée
En 2022, il est  d'empoisonnement mais s'en sort avec quelques séquelles.

Notes et références

Naissance en janvier 1955
Musicien gabonais

Ndong Mboula


Lacticaseibacillus paracasei (comumente abreviado como Lc. paracasei) é uma espécie gram-positiva e homofermentativa de bactérias do ácido lático que são comumente usadas na fermentação de produtos lácteos e como culturas probióticas. Lc. paracasei é uma bactéria que opera por comensalismo. É comumente encontrado em muitos habitats humanos, como tratos intestinais e bocas humanas, bem como esgotos, silagens e produtos lácteos mencionados anteriormente. O nome inclui morfologia, uma bactéria em forma de bastonete (forma de bacilo) com largura de 2,0 a 4,0μm e comprimento de 0,8 a 1,0μm.

Cepas de L. paracasei foram isoladas de uma variedade de ambientes, incluindo produtos lácteos, plantas ou fermentações de plantas, e do trato gastrointestinal humano e animal. Um período prolongado de refrigeração antes do trânsito gastrointestinal in vitro (TGI) não afetou ou influenciou muito fracamente a resistência celular.

Lacticaseibacillus paracasei é genotipicamente e fenotipicamente intimamente relacionado com outros membros do grupo Lacticaseibacillus casei que também inclui Lacticaseibacillus casei, Lacticaseibacillus zeae e Lacticaseibacillus rhamnosus. No entanto, essas espécies são facilmente diferenciadas umas das outras por tipagem de sequência multilocus, filogenia do genoma central ou identidade nucleotídica média. Suas propriedades fermentativas permitem que ele seja usado como processadores biológicos de alimentos e suplementos para dietas e distúrbios médicos, especialmente no trato gastrointestinal.

Embora os probióticos sejam considerados seguros, eles podem causar interações bactérias-hospedeiro e consequências adversas para a saúde. Em certos casos existe o risco de bacteremia quando se utilizam probióticos. Atualmente, a cepa probiótica, frequência, dose e duração das terapias probióticas não estão estabelecidas.

Fisiologia
Lacticaseibacillus paracasei é um microrganismo gram-positivo, homofermentativo, não formador de esporos. Como Lc. paracasei é homofermentativo, o ácido lático é produzido como o principal produto do metabolismo da hexose enquanto o lactato e o acetato são produzidos a partir das pentoses. As células de Lc. paracasei são tipicamente em forma de bastão, com uma faixa de tamanho de 2,0 μm a 4,0 μm de largura e 0,8 a 1,0 μm de comprimento. O organismo é imóvel. Lc. as células paracasei geralmente têm extremidades quadradas e podem existir em forma única ou em cadeias.

Lacticaseibacillus paracasei cresce otimamente em uma faixa de temperatura entre 10 e 37 °C. Nenhum crescimento ocorre acima de 40 °C. O organismo é capaz de sobreviver por aproximadamente 40 segundos em uma temperatura máxima de 72 °C. A capacidade de sobrevivência de Lc. paracasei foi notavelmente maior quando armazenado sob refrigeração (4 °C). Em contraste, a menor sobrevivência foi observada durante o armazenamento não refrigerado(22 °C)  O congelamento a -20 graus C e -70 graus C teve muito menos efeito adverso na viabilidade do que o armazenamento a 7 graus C.

Lacticaseibacillus paracasei existe temporariamente como um habitante comum do trato gastrointestinal humano como parte da microbiota normal. Legumes fermentados naturalmente, leite e carne também podem conter cepas de L. paracasei.

Filogenia
Lacticaseibacillus paracasei pertence ao reino Bacteria. Lc. paracasei faz parte do filo Bacillota, da classe Bacilli, da ordem Lactobacillales e da família Lactobacillaceae, respectivamente. O argumento sobre a nomenclatura de L. paracasei versus L. casei foi intensamente debatido, pois muitas cepas de L. casei ou L. paracasei para as quais os dados de sequência estão disponíveis nos bancos de dados são rotuladas incorretamente. Em 1989, foi proposto que L. paracasei fosse designado uma subespécie ( paracasei ) para contabilizar as espécies com as quais compartilha homologia de DNA. Foi demonstrado que seus nomes têm sido usados de forma intercambiável na literatura científica. A homologia da sequência de RNA 16S confirmou a relação entre essas espécies mas a filogenia do genoma central confirmou que as espécies intimamente relacionadas Lc. casei, Lc. paracasei, Lc. rhamnosus e Lc. zeae são espécies separadas.

Historicamente, a diferença entre Lacticaseibacillus paracasei e outros lactobacilos foi baseada em características bioquímicas. Existe uma identidade de sequência de aproximadamente 90% entre casei, paracasei e rhamnosus. No entanto, existem alguns critérios diferenciais que são comumente usados para diferenciá-los. Esses critérios diferenciais incluem exigências nutricionais e ambiente de crescimento. L. paracasei foi encontrado para mostrar diferenças específicas com outros lactobacilos em que é um pouco resistente ao calor, cresce bem no queijo maturado e tem alta atividade proteolítica.

Genômica
O genoma de Lacticaseibacillus paracasei contém DNA circular e varia ligeiramente entre as diferentes cepas isoladas. Em média, os genomas são de 2,9 a 3,0 milhões de pares de bases (comumente abreviado Mb). Tem um conteúdo de GC entre 46,2 e 46,6% e prevê-se que codifique cerca de 2800 a 3100 proteínas. A diferença nos genomas dessas cepas está nos envelopes celulares variantes, proteínas secretoras e polissacarídeos. Muitas das proteínas comumente codificadas são hidrolases de parede celular associadas à superfície celular que protegem a célula contra a apoptose. Essas enzimas demonstraram fornecer proteção celular às células epiteliais humanas.

A diversidade genética para os diferentes genomas de L. paracasei foi avaliada usando tipagem de sequência multilocus (MLST) e polimorfismo de comprimento de fragmento amplificado (AFLP). MLST é uma técnica utilizada para classificar micróbios pelo uso de fragmentos de DNA de genes essenciais do organismo. AFLP é uma ferramenta de Reação em Cadeia da Polimerase (PCR) usada no perfil de DNA para amplificar um fragmento de DNA desejado com o uso de enzimas de restrição e ligantes.

Aplicações clínicas e de pesquisa
Lacticaseibacillus paracasei foi identificado como uma bactéria que possui propriedades probióticas. L. paracasei IMPC2.1 pode ser um quimioprofilático em células gastrointestinais. As células gastrointestinais são suscetíveis à apoptose e ao crescimento celular de ambas as cepas IMPC2.1 mortas pelo calor e viáveis. Lc. paracasei 8700:2 foi isolado de mucosa gastrointestinal humana saudável e fezes humanas. A cepa 8700:2 também inibe Salmonella enterica e Helicobacter pylori, dois patógenos comumente encontrados no trato gastrointestinal. A cepa 8700:2 decompõe a oligofrutose e a inulina, enquanto também cresce rapidamente em ambas e produz ácido lático como produto final.

Uma formulação de bactérias vivas incluindo Lc. paracasei pode ser usado em combinação com terapias convencionais para tratar a colite ulcerativa. Uma revisão sistemática forneceu evidências significativas de efeitos clínicos e imunológicos benéficos de Lc. paracasei LP-33 no tratamento da rinite alérgica.

O ácido lipoteicóico da parede celular de um Lacticaseibacillus paracasei D3-5 morto pelo calor melhora o intestino permeável relacionado ao envelhecimento, a inflamação e melhora as funções físicas e cognitivas em camundongos.

A ingestão de leite fermentado fortificado com LP-33 (Lactobacillus paracasei 33) por 30 dias pode melhorar de forma eficaz e segura a qualidade de vida de pacientes com rinite alérgica, podendo servir como tratamento alternativo para a rinite alérgica.

Lacticaseibacillus paracasei BRAP01 são as cepas dominantes que induzem a produção de IFN-γ/IL-10 em indivíduos taiwaneses.

Lacticaseibacillus paracasei HB89 atenua as alergias do trato respiratório estimuladas pelo PM .

Preocupações com a saúde
A manipulação da microbiota intestinal é complexa e pode causar interações bactérias-hospedeiro. Embora os probióticos sejam considerados seguros, quando utilizados por via oral existe o risco de passagem de bactérias viáveis do trato gastrointestinal para os órgãos internos (translocação bacteriana) e posterior bacteremia, o que pode causar consequências adversas à saúde. Algumas pessoas, como aquelas com comprometimento imunológico, síndrome do intestino curto, cateteres venosos centrais, doença valvar cardíaca e bebês prematuros, podem ter maior risco de eventos adversos.

Atualmente, a cepa probiótica, a frequência, a dose e a duração da terapia probiótica não estão estabelecidas. Bactérias vivas podem não ser essenciais porque os efeitos benéficos dos probióticos parecem ser mediados por seu DNA e por fatores solúveis secretados, e seus efeitos terapêuticos podem ser obtidos por administração sistêmica ao invés de administração oral.

História
LAB (Lactic Acid Bacteria) foram classificadas e agrupadas no início de 1900 depois de ganhar a atenção dos cientistas após observar as interações das bactérias em diferentes alimentos, especialmente produtos lácteos. Em 1991, Martinus Beijerinck, um microbiologista holandês, separou Lactobacillus como bactérias gram-positivas do grupo LAB anteriormente conhecido. L. paracasei foi recentemente classificado como parte do grupo de probióticos Lacticaseibacillus casei. O nome Lc. paracasei foi proposto para rejeição em 1996 por Dicks, Duplessis, Dellaglio e Lauer, mas trabalhos posteriores confirmaram a validade da espécie.

Referências

Ligações externas
 Tipo de cepa de Lactobacillus paracasei em Bac Dive - o Metadatabase de Diversidade Bacteriana

Probióticos
Bactérias descritas em 1989
Lactobacillaceae

Lacticaseibacillus paracasei


කොටි ත්‍රස්තවාදී කතකි. පොන්නයියා සෙල්වනායගම්ට රැකවරණය දුන් තැනැත්තිය වේ.

ආශ්‍රිත

එල්.ටී.ටී.ඊ. ත්‍රස්තවාදියෝ

බාලේන්ද්‍රන් ජෙයකුමාරි


An Nervilia petaloidea in uska species han Liliopsida nga ginhulagway ni Cedric Errol Carr. An Nervilia petaloidea in nahilalakip ha genus nga Nervilia, ngan familia nga Orchidaceae. Waray hini subspecies nga nakalista.

Mga kasarigan

Nervilia

Nervilia petaloidea


Broadway kan henvise til:

 Broadway - en gade på Manhattan i New York City
 Broadway (teaterdistrikt) - opkaldt efter gaden, omtales Broadway for New Yorks revy- og teaterliv generelt
 Broadway (Brooklyn) - en gade i Brooklyn i New York City
 Broadway (Queens) - en gade i Queens i New York City

Bygninger
 Broadway Mansions - et hotel i Shanghai
 Broadway Theatre - teater i New York City
 Broadway Tower - et tårn i England

Film og skuespil
 Broadway: Brooklyn -en dansk dokumentarfilm fra 2003
 Broadway Danny Rose - en amerikansk film instrueret af Woody Allen fra 1984
 Broadway (film fra 1942), en amerikansk film
 Broadway (skuespil), et amerikansk skuespil skrevet af George Abbott

Broadway (flertydig)


Earl of Inchcape is a title in the Peerage of the United Kingdom. It was created in 1929 for the Scottish shipping magnate and public servant James Mackay, 1st Viscount Inchcape. He had been created Baron Inchcape, of Strathnaver in the County of Sutherland, in 1911, and Viscount Inchcape, of Strathnaver in the County of Sutherland, in 1924. He was made Viscount Glenapp, of Strathnaver in the County of Sutherland, at the same time as he was given the earldom. These titles are also in the Peerage of the United Kingdom.

The family seat is Carlock House near Ballantrae, Ayrshire. The former family seat was Glenapp Castle, which is now a luxury hotel. It was sold by the Mackay family in 1982, having been acquired by the 1st Earl of Inchcape in 1917. The present Earl of Inchcape, however, still owns the Glenapp Estate.

Earls of Inchcape (1929)
James Lyle Mackay, 1st Earl of Inchcape (1852–1932)
Kenneth Mackay, 2nd Earl of Inchcape (1887–1939)
Kenneth James William Mackay, 3rd Earl of Inchcape (1917–1994)
Kenneth Peter Lyle Mackay, 4th Earl of Inchcape (b. 1943)
The heir apparent is the present holder's only son Fergus James Kenneth Mackay, Viscount Glenapp (b. 1979).
The heir apparent's heir apparent is his son, the Hon. Alexander David James Mackay (b. 2017).

Line of succession

  James Lyle Mackay, 1st Earl of Inchcape (1852–1932)
  Kenneth Mackay, 2nd Earl of Inchcape (1887–1939)
  Kenneth James William Mackay, 3rd Earl of Inchcape (1917–1994)
  Kenneth Peter Lyle Mackay, 4th Earl of Inchcape (born 1943)
 (1) Fergus James Kenneth Mackay, Viscount Glenapp (b. 1979)
 (2) Hon. Alexander David James Mackay (b. 2017)
 (3) Hon. James Jonathan Thorn Mackay (b. 1947)
 (4) Aidan James Turner Mackay (b. 1978)
 (5) Hon. Shane Mackay (b. 1973)
 (6) Hon. Ivan Cholmeley Mackay (b. 1976)
 (7)  Simon Brooke Mackay, Baron Tanlaw (born 1934)
 (8) Hon. James Brooke Mackay (b. 1961)
 (9) Hon. Brooke Brooke Mackay (b. 1982)

See also
Inchcape
Inchcape plc

References

Work cited

Earldoms in the Peerage of the United Kingdom
Noble titles created in 1929

Earl of Inchcape


Pentziinae, podtribus glavočika, dio tribusa Anthemideae. Postoji šest rodova s juga Afrike, a tipični je Pentzia .

Podtribus je opisan 2007.

Rodovi
 Cymbopappus B. Nord.
 Foveolina Källersjö
 Marasmodes DC.
 Myxopappus Källersjö
 Oncosiphon Kallersjo
 Pentzia Thunb.

Izvori

Anthemideae

Pentziinae


The Valdivian temperate forests (NT0404) is an ecoregion on the west coast of southern South America, in Chile and Argentina. It is part of the Neotropical realm. The forests are named after the city of Valdivia. The Valdivian temperate rainforests are characterized by their dense understories of bamboos, ferns, and for being mostly dominated by evergreen angiosperm trees with some deciduous specimens, though conifer trees are also common.

Setting
Temperate rain forests comprise a relatively narrow Chilean coastal strip, between the Pacific Ocean to the west and the southern Andes Mountains to the east, from roughly 37° to 48° south latitude. North of 42°, the Chilean Coast Range stretches on, with just the north–south running Chilean Central Valley between it and the Andes. South of 42°, the coast range continues as a chain of offshore islands (including Chiloé Island and the Chonos Archipelago), while the "Central Valley" is submerged and continues as the Gulf of Corcovado. Much of the ecoregion was once covered by the Patagonian Ice Sheet and other glaciers during the peak of the last ice age, with ice descending from the Andes mountains; numerous bodies of water within the Chilean Lake District (in the central part of the ecoregion) are the remnants of ancient glacial valleys. The southern part of the region features many glacier-carved fjords.

To the north, the Valdivian forests give way to the Mediterranean forests, woodlands, and scrub of the Chilean Matorral ecoregion. A few coastal enclaves of Valdivian forest grow in north-central Chile (such as Bosque de Fray Jorge National Park) as remains of the last glacial maximum. To the south lies the Magellanic subpolar forests ecoregion. The temperate Valdivian, matorral, and Magellanic ecoregions are isolated from the subtropical/tropical forests that dominate northern South America by such landscapes as the Atacama desert (north of the matorral), the Andes Mountains, and the dry, rain-shadow Patagonian steppe east of the Andes. As a result, the temperate forest regions have evolved in relative isolation, with a high degree of endemic species.

Due to a similar location geographically and geologically (along the Pacific Plate), the coast-hugging temperate rainforests of the Pacific Northwest (from Northern California to Southern Alaska, roughly 40°-60° north latitude) exist in similar settings, with the Rocky Mountains to the east and the Pacific Ocean to the west. East of the Rocky Mountains, the North American prairie grassland stretches from south-central Canada to Texas, not unlike the Argentine grasslands to the east of the Andes. Similar to the Atacama region of Chile, the Baja California and Sonoran deserts in the US/Mexican states of California and Baja California act as climatic borders for the northwest's rainforests.

Climate
Since the forest is located at around 40 degrees south, it is strongly influenced by the westerlies. The water vapour held by the westerlies condenses as they encounter the windward slope of the Chilean Coast Range and the Andes, creating orographic rainfall. Average annual precipitation varies from 1,000 mm at the northern edge of the ecoregion to more than 6,000 mm per year in the south. The northern portion of the ecoregion has a Submediterranean climate, with rainfall concentrated in the winter months. This seasonality decreases towards the south.

Average annual temperatures are fairly uniform within the area, especially at coastal locations where annual temperature differences between localities never exceed 7 °C. The northward-flowing oceanic Humboldt Current creates humid and foggy conditions near the coast. In the summer the temperature can climb to 16.5 °C (62 °F), while during winter the temperature can drop below 7 °C (45 °F). Winter temperatures are lower at higher elevations. The tree line is at about 2,400 m in the northern part of the ecoregion (35° S), and descends to 1,000 m in the south of the Valdivian region.

Flora

The Valdivian temperate rain forests are temperate broadleaf and mixed forests. The Valdivian and Magellanic temperate rainforests are the only temperate rain forests in South America and one of a small number of temperate rain forests in the world. Together they are the second largest in the world, after the Pacific temperate rain forests of North America (which stretches from Alaska to northern California). The Valdivian forests are a refuge for the Antarctic flora, and share many plant families with the temperate rainforests of New Zealand, Tasmania, and Australia. Fully half the species of woody plants are endemic to this ecoregion.

Chusquea quila is a bamboo that grows in humid areas below 500 m, where Chusquea culeou becomes more dominant above. Chusquea quila can form pure stands called quilantales. Very few plants can grow under this species. Other notable species are the nalca or Chilean rhubarb (Gunnera tinctoria) and the ferns Lophosoria quadripinnata and Parablechnum cordatum. Chile's national flower, the copihue (Lapageria rosea) is a pioneer species that grows in disturbed areas of the Valdivian rain forest.

The maximum plant species richness is found at latitudes 40 to 43° S.

Forest ecosystems

There are four main types of forest ecosystems in the Valdivian ecoregion.

Deciduous forests. At the northern end of the ecoregion are deciduous forests, dominated by deciduous species of southern beech, including rauli (Nothofagus alpina) and roble (N. obliqua). This is a transitional zone to the Mediterranean-climate region to the north. It grows from 35 to 36º S latitude along Chilean Coast Range, where it is known as Maulino forest. Nothofagus glauca and N. Alessandrii are predominant trees in Maulino forest. Deciduous Nothofagus forests also grow along the Andes of central Chile as far north as 33º S latitude, from approximately 1,200 meters elevation up to the tree line at 2,400 meters elevation. A krummholz of Nothofagus antarctica and N. pumilio grows near the tree line. The southern limit of these forests is 38º S latitude. Tall coniferous pehuén monkey-puzzle trees (Araucaria araucana) grow at the southern edge of the deciduous forests, from the coast at Nahuelbuta National Park to the Andes.

Valdivian laurel-leaved forests. Valdivian laurel-leaved forests, characterized by a variety of broadleaf evergreen trees, including Laureliopsis philippiana, Aextoxicon punctatum, Eucryphia cordifolia, Caldcluvia paniculata, and Weinmannia trichosperma, with an understory of Myrceugenia planipes, the arrayán (Luma apiculata) and other plants. Old-growth Valdivian evergreen forest (siempreverde) tend to form stratified canopy made up of two or three layers.

Patagonian Andean forests. The third forest type is the Patagonian Andean forests, which are distributed at higher elevations along the Andes mountain front, and are dominated by evergreen conifers, including pehuén (Araucaria araucana) and alerce (Fitzroya cupressoides). The alerce looks like a giant sequoia, and is a rival in longevity to the bristlecone pine, some with growth rings recording 3,625 years of local weather cycles. Closer to the treeline, the conifers give way to Andean scrublands of deciduous Nothofagus antarctica.

Northern Patagonian forests. The fourth and last type is the Northern Patagonian forests, which dominate the southern half of the ecoregion, with evergreen species such as the broadleaf Nothofagus dombeyi, Nothofagus betuloides and Drimys winteri and the coniferous podocarps, including Podocarpus nubigenus.

Distribution of plants follow Rapoport's rule with plant species distribution increasing at higher latitudes and decreasing at those closer to the equator, incidents of endemism becoming more frequent as one moves equatorward.

Lowland soils near the coast contain more available nutrients than more inland soils.

Origin and evolution
The flora of the forests has inherited and developed its characteristics due to a variety of causes. Its Neotropical affinities reflect its current geographic connection to the remaining South America. Its "Gondwanan" species are a legacy of the ancient supercontinent of Gondwana, a landmass formerly composed of South America, Africa, India, Antarctica and Australia. The high degree of endemic species and monospecific genera is thought to be linked to the geographic isolation came into being as result of the uplift of the Andes. There are a few "boreal" components in the Valdivian temperate rain forest which arrived by long-distance transport. Yet another component are the species from nearby South American ecosystems that adapted to the temperate rainforest.

During the Llanquihue glaciation much of the area west of Llanquihue Lake remained ice-free during the Last Glacial Maximum and had sparsely distributed vegetation dominated by Nothofagus species. Valdivian temperate rain forest proper was reduced to scattered remnants in to the west of the Andes. More specifically the refugia of the Valdivian temperate rain forest between latitudes 41 and 37° S were; the coastal region, the lower slopes of the Chilean Coast Range and the westernmost Chilean Central Valley all of which remained free of disturbance by the glacial, glacifluvial and periglacial disturbance through the glaciation.

Fauna

Some of the threatened mammals of the Valdivian forests include the monito del monte (Dromiciops gliroides), an arboreal marsupial, the southern pudú (Pudu puda) the world's smallest deer, and the kodkod (Leopardus guigna), South America's smallest cat. Since the beginning of the 20th century, there have also been nonnative wild boars living in the Valdivian forests. The Chilean climbing mouse (Irenomys tarsalis) and Chilean shrew opossum (Rhyncholestes raphanurus) are endemic to the ecoregion.

Most mammal genera in Valdivian forests are also found in semi-arid parts of Patagonia. Relative to similar forest in North America there is a low diversity of mammals in Chilean temperate forests.

The slender-billed parakeet (Enicognathus leptorhynchus) is endemic. Near-endemic and limited-range birds include the Chilean pigeon (Patagioenas araucana), Black-throated huet-huet (Pteroptochos tarnii), Chucao tapaculo (Scelorchilus rubecula), and Ochre-flanked tapaculo (Eugralla paradoxa). Hummingbirds are common in the Valdivian forests because of the presence of plants like the maqui (Aristotelia chilensis) and the copihue.

Conservation

The Valdivian forests include stands of huge trees, especially Nothofagus and Fitzroya, which can live to a great age. These magnificent rainforests are endangered by extensive logging and their replacement by fast-growing pines and eucalyptus, which are more sought-after by the pulp and paper industry. The native trees that are cleared to make way for these monocultures are often exported as woodchips to Japan.  A start at conservation was made in November 2003 when a consortium of conservation groups, both local and international, bought at auction of a bankrupt logging firm 147,500 acres (600 km2) of biologically rich rainforest in the Valdivian Coastal Range. Gianni Lopez, Executive Director of CONAMA, Chile's national environmental agency remarked, "Ten years ago the existence of protected areas not owned by the government was unthinkable." Among the efforts supporting conservation, has been a growing ecotourism industry.

Protected areas
24.16% of the Valdivian temperate forests ecoregion is in protected areas.

List of protected areas

Argentina
Los Alerces National Park
Los Arrayanes National Park
Lago Puelo National Park
Lanín National Park
Nahuel Huapí National Park

Chile
Public:
Alerce Andino National Park
Alerce Costero National Park
Altos de Lircay National Reserve
Altos de Pemehue National Reserve
Bosque de Fray Jorge National Park
Carlos Anwandter Sanctuary
Chiloé National Park
China Muerta National Reserve
Conguillío National Park
Corcovado National Park
Coyhaique National Reserve
Futaleufú National Reserve
Hornopirén National Park
Huerquehue National Park
Isla Guamblin National Park
La Campana National Park
Lago Rosselot National Reserve
Laguna del Laja National Park
Laguna San Rafael National Park
Lahuen Ñadi Natural Monument
Llanquihue National Reserve
Los Bellotos del Melado National Reserve
Los Huemules de Niblinto National Reserve
Los Queules National Reserve
Los Ruiles National Reserve
Melimoyu National Park
Mocho-Choshuenco National Reserve
Nahuelbuta National Park
Nonguén National Park
Península de Hualpén Nature Sanctuary
Pumalín Douglas Tompkins National Park
Puyehue National Park
Queulat National Park
Radal Siete Tazas National Park
Ralco National Reserve
Río Los Cipreses National Reserve
Tolhuaca National Park
Vicente Perez Rosales National Park
Villarrica National Park
Private:
Huilo-Huilo Biological Reserve (owned by Víctor Peterman)
Llancahue (administered by the Austral University of Chile)
Los Vertientes Private Nature Reserve
Meullín-Puye Nature Sanctuary (administered by the Kreen Foundation)
Oncol Park (owned by Celulosa Arauco y Constitución)
Área Costera Protegida Punta Curiñanco (owned by CODEFF)
Tantauco Park (owned by Sebastián Piñera)
Valdivian Coastal Reserve (owned by The Nature Conservancy)

See also
Chilean Native Forest Law
Chilean matorral
Magellanic subpolar forests
Maulino forest

Notes and references

External links
Explore the Valdivian Coastal Reserve

Valdivian temperate rain forests/Juan Fernandez Islands (World Wildlife Fund)
Research at Chiloé Rain Forest
Flora species from Valdivian rain forests

 01
Temperate broadleaf and mixed forests
Ecoregions of Argentina
Ecoregions of Chile
Andean forests
Ecology of Patagonia
Natural history of Argentina
Natural history of Chile

Chilean Coast Range
Geography of Los Lagos Region
Geography of Los Ríos Region
Neotropical ecoregions

Valdivian temperate forests


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