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A grid compass [ 1 ] known as well as grid steering compass , is a navigating instrument. It is a design of magnetic compass that facilitates steering a steady course without the risk of parallax error.
The grid compass is the simplest steering compass from the pilot's or helmsman's point of view, because he doesn't need to watch the number (or the division mark) of the wanted course. He has only to steer the craft so that the N/S compass needle lies parallel between the lines of the overlay disc. The principle is similar to the compass-controlled autopilot. [ 2 ] Although sophisticated electronics have taken over for commercial navigation, light aircraft, gliders and yachtsmen still use the grid compass because of its simplicity and ease of use.
The compass card is in the form of a bold parallel sided arrow which indicates magnetic north. Some models have an east/west cross bar as well. Overlaying this but in the same gimbal or suspension is a transparent plate which can be rotated around the same axis as the compass card but has sufficient friction (or a mechanical clamp) to stay fixed relative to the gimbal system once set to a course. Across this disk are engraved a series of parallel lines. The outer edge of this disk is marked in clockwise in degrees, the radial line meeting 0º being parallel to the engraved lines, so that a course can be laid for any bearing from 0º to 359º. By keeping the arrow on the card and the lines on the overlay parallel, the pilot or helmsman can keep the course set. The frequency of the degree markings depend upon the size of the compass.
To set a course the rotating ring is (unlocked and) turned so that the heading in degrees on the ring alines with the centre line of the craft. The craft comes on to the required course when the arrow on the compass card is parallel with the lines on the ring. [ 3 ]
The grid steering compasses (Type P8 to Type P11) were fitted in World War II Spitfire aeroplanes , replacing the old P4 series of instruments. They were used for course setting and reading, and as a check compass on aircraft fitted with a remote indicating compass. [ 2 ] | https://en.wikipedia.org/wiki/Grid_compass |
Latticial metal complex or grid complex is a supramolecular complex of several metal atoms and coordinating ligands which form a grid-like structural motif. The structure formation usually occurs while on thermodynamic molecular self-assembly . They have properties that make them interesting for information technology as the future storage materials. [ 1 ] Chelate ligands are used as ligands in tetrahedral or octahedral structures, which mostly use nitrogen atoms in pyridine like ring systems other than donor centers. Suitable metal ions are in accordance with octahedral coordinating transition metal ions such as Mn or rare tetrahedral Coordinating such as Ag used. [ 1 ]
The nomenclature is based on [n × m] G, n corresponds to the number of ligands above the metal ion level, m the number below ones. In case of using only one ligand type, the homoleptic grid is formed in a square [nxn] structure. When using different ligands arise heteroleptic complexes, however, compete with the homoleptic . The number of metal ions is always n + m.
The grid complexes exhibit pH -dependent changes in the optical absorption , electronic spin states and reversible redox states. The latticial metal complexes may thus be used theoretically for information storage and processing in the future. [ 2 ] [ 3 ] [ 4 ]
An interwoven grid complex has been used to template the synthesis of a doubly-twisted [2]catenane (otherwise known as a Solomon Link ). [ 5 ] The unique arrangement of interwoven ligands around the planar array of iron, zinc, or cobalt ions generated the crossing points required to covalently trap the interlocked structure using ring-closing metathesis. Building on this discovery, 2 × 2 interwoven grids were used to template the synthesis of more topologically complex molecules: a six-crossing doubly-interlocked [2]catenane and a granny knot. [ 6 ] In 2021, the first report of a 3 × 3 interwoven grid was published. It was used to template the synthesis of a molecular Endless Knot . [ 7 ] | https://en.wikipedia.org/wiki/Grid_complex |
Grid dip oscillator ( GDO ), also called grid dip meter , gate dip meter , dip meter , or just dipper , is a type of electronic instrument that measures the resonant frequency of nearby unconnected radio frequency tuned circuits . It is a variable-frequency oscillator that circulates a small-amplitude signal through an exposed coil, whose electromagnetic field can interact with adjacent circuitry. The oscillator loses power when its coil is near a circuit that resonates at the same frequency. A meter on the GDO registers the amplitude drop, or "dip", hence the name.
Dip oscillators have been widely used by amateur radio operators for measuring the properties of resonant circuits, filters, and antennas . They can also be used for transmission line testing, as signal generators, and for measuring inductance and capacitance of components. Measurement with a GDO is called "dipping" a circuit. [ 1 ]
Central to the dip meter is a high-frequency variable-frequency oscillator with a calibrated tuning capacitor and matching interchangeable coils, as shown in the circuit diagram. Resonance is indicated by a dip in amplitude of the signal within the GDO, by a meter on the device.
When the oscillator's exposed coil is in the vicinity of another resonant circuit, the coupled pair behaves as a low-Q transformer whose coupling is most effective when their respective resonant frequencies match. The degree of coupling affects the frequency and amplitude of oscillation in the dip meter, which is sensed in any of several ways, the simplest and most usual of which is a built-in microammeter . The distance between the coil and the tested circuit needs to be adjusted carefully so that the GDO amplitude is significantly affected by the coupled circuit, but its frequency is not. [ 1 ] [ 2 ] : 1–8
Grid dip oscillators were first developed in the 1920s and were built with vacuum tubes . The devices displayed the amplitude of the tube's grid current, hence GDO.
Modern dip meters are solid-state devices, and are sometimes called gate dip oscillators or emitter dip oscillators in reference to the analogous part of the transistor whose current is measured instead of a vacuum tube grid. [ 1 ] Solid-state versions of the grid dip oscillator are more versatile, since they can operate at higher Q and lower amplitude, and are not tethered by a power cord.
Dip meters of all types are being supplanted by antenna analyzers , which are more complex, but perform many of the same functions more conveniently, and require less dexterity. [ 3 ]
The dip meter can be used either to measure the relative power lost to a nearby circuit (in which case the amplitude shown on the meter "dips") or to measure the relative power absorbed from a nearby powered circuit (in which case, the meter amplitude peaks). In either mode of operation, some experimentation is needed to find a distance between the pickup coil and the circuit under test, to ensure that the two circuits are close enough to transfer power, but not coupled so closely that the circuit supplied with power overwhelms the responding circuit, and forces it to oscillate regardless of the frequency. [ 4 ]
The heart of the instrument is a tunable LC circuit whose exposed external coil will loosely couple to the measured resonant circuit when held moderately close by. The measuring distance must be skillfully adjusted to be close enough to provide sufficient coupling for the dip to show clearly, but far enough that the meter and the tested circuit oscillate independently, so neither device's frequency is significantly distorted by the other, and so that power lost to the external circuit does not swamp the dip meter's oscillator.
The coil and the test circuit can either be inductively or capacitively coupled: Coupling is inductive if the coil wires are held parallel to the nearest wires of the circuit being tested, capacitive if the coil wires and the circuit wires are held perpendicular. Depending on the context desired for the measurement, the circuit under test can be temporarily disconnected from its surroundings to avoid distortion by the parts it is normally attached to, or left wired in place to measure the response of the combined system.
In ordinary use, only the oscillator in the dip meter is powered, and the circuit under test's only power is what it drains from the signal in the GDO coil. When both circuits are resonant at the same frequency, power transfer from the coil to the adjacent tested circuit reaches a maximum, consequently the dip meter's oscillator amplitude reaches a minimum due to the power lost to the circuit under test. [ 1 ] : 25–10
The operator adjusts the frequency of the GDO until its meter shows its lowest reading (the "dip"). The frequency is read from the dial on the GDO, or the frequency can be measured by finding the dip meter's signal on a well-calibrated radio receiver. Some modern GDOs have a built-in frequency meter, which makes over-coupling somewhat less troublesome. [ 1 ]
Some dip meters can be used in reverse, as ultra-short-range, tuned field strength meters . The operator finds the frequency at which the highest rise in meter power occurs in the unpowered GDO, when its coil is held near the wires of an active resonant circuit. Since the power in the circuit under test must be high enough to register on the meter, this uncommon method is risky both for the operator and the equipment. | https://en.wikipedia.org/wiki/Grid_dip_oscillator |
The Wireless Grid Fabric in communication is a MIMOS Berhad innovation for WiMAX multi-hop relay networks (IEEE802.16j) [ 1 ] for rural area communication.
The idea of the Wireless Grid Fabric involves using multihop base stations (MR-BS) to forward messages to and from the network. Each relay station (RS) covers approximately two square kilometers of area with omnidirectional antennas. Each such square is called a cell. In this scheme, the network's scalability depends not on the number of nodes but the number of cells, each of which contains several nodes.
In any rural area community supported by a Wireless Grid Fabric, it is assumed that the main traffic (content) is self-created by the population (peer–to-peer), [ 2 ] such as video streaming, VOIP , IPTV , and others which are all multicast-based. The Wireless Grid Fabric network has many advantages over other mesh technologies (i.e. WiFi-Mesh and Fixed WiMAX-Mesh), as it achieves hundreds of Mbit/s with mobility for hundreds of mobiles per service deployment. | https://en.wikipedia.org/wiki/Grid_fabric |
The grid method (also known as the box method or matrix method ) of multiplication is an introductory approach to multi-digit multiplication calculations that involve numbers larger than ten. Because it is often taught in mathematics education at the level of primary school or elementary school , this algorithm is sometimes called the grammar school method. [ 1 ]
Compared to traditional long multiplication , the grid method differs in clearly breaking the multiplication and addition into two steps, and in being less dependent on place value.
Whilst less efficient than the traditional method, grid multiplication is considered to be more reliable , in that children are less likely to make mistakes. Most pupils will go on to learn the traditional method, once they are comfortable with the grid method; but knowledge of the grid method remains a useful "fall back", in the event of confusion. It is also argued that since anyone doing a lot of multiplication would nowadays use a pocket calculator, efficiency for its own sake is less important; equally, since this means that most children will use the multiplication algorithm less often, it is useful for them to become familiar with a more explicit (and hence more memorable) method.
Use of the grid method has been standard in mathematics education in primary schools in England and Wales since the introduction of a National Numeracy Strategy with its "numeracy hour" in the 1990s. It can also be found included in various curricula elsewhere. Essentially the same calculation approach, but not with the explicit grid arrangement, is also known as the partial products algorithm or partial products method .
The grid method can be introduced by thinking about how to add up the number of points in a regular array, for example the number of squares of chocolate in a chocolate bar. As the size of the calculation becomes larger, it becomes easier to start counting in tens; and to represent the calculation as a box which can be sub-divided, rather than drawing a multitude of dots. [ 2 ] [ 3 ]
At the simplest level, pupils might be asked to apply the method to a calculation like 3 × 17. Breaking up ("partitioning") the 17 as (10 + 7), this unfamiliar multiplication can be worked out as the sum of two simple multiplications:
so 3 × 17 = 30 + 21 = 51.
This is the "grid" or "boxes" structure which gives the multiplication method its name.
Faced with a slightly larger multiplication, such as 34 × 13, pupils may initially be encouraged to also break this into tens.
So, expanding 34 as 10 + 10 + 10 + 4 and 13 as 10 + 3, the product 34 × 13 might be represented:
Totalling the contents of each row, it is apparent that the final result of the calculation is (100 + 100 + 100 + 40) + (30 + 30 + 30 + 12) = 340 + 102 = 442.
Once pupils have become comfortable with the idea of splitting the whole product into contributions from separate boxes, it is a natural step to group the tens together, so that the calculation 34 × 13 becomes
giving the addition
so 34 × 13 = 442.
This is the most usual form for a grid calculation. In countries such as the UK where teaching of the grid method is usual, pupils may spend a considerable period of time regularly setting out calculations like the above, until the method is entirely comfortable and familiar.
The grid method extends straightforwardly to calculations involving larger numbers.
For example, to calculate 345 × 28, the student could construct the grid with six easy multiplications
to find the answer 6900 + 2760 = 9660.
However, by this stage (at least in standard current UK teaching practice) pupils may be starting to be encouraged to set out such a calculation using the traditional long multiplication form without having to draw up a grid.
Traditional long multiplication can be related to a grid multiplication in which only one of the numbers is broken into tens and units parts to be multiplied separately:
The traditional method is ultimately faster and much more compact; but it requires two significantly more difficult multiplications which pupils may at first struggle with [ citation needed ] . Compared to the grid method, traditional long multiplication may also be more abstract [ citation needed ] and less manifestly clear [ citation needed ] , so some pupils find it harder to remember what is to be done at each stage and why [ citation needed ] . Pupils may therefore be encouraged for quite a period to use the simpler grid method alongside the more efficient traditional long multiplication method, as a check and a fall-back.
While not normally taught as a standard method for multiplying fractions , the grid method can readily be applied to simple cases where it is easier to find a product by breaking it down.
For example, the calculation 2 1 / 2 × 1 1 / 2 can be set out using the grid method
to find that the resulting product is 2 + 1 / 2 + 1 + 1 / 4 = 3 3 / 4
The grid method can also be used to illustrate the multiplying out of a product of binomials , such as ( a + 3)( b + 2), a standard topic in elementary algebra (although one not usually met until secondary school ):
Thus ( a + 3)( b + 2) = ab + 3 b + 2 a + 6.
32-bit CPUs usually lack an instruction to multiply two 64-bit integers. However, most CPUs support a "multiply with overflow" instruction, which takes two 32-bit operands, multiplies them, and puts the 32-bit result in one register and the overflow in another, resulting in a carry. For example, these include the umull instruction added in the ARMv4t instruction set or the pmuludq instruction added in SSE2 which operates on the lower 32 bits of an SIMD register containing two 64-bit lanes.
On platforms that support these instructions, a slightly modified version of the grid method is used.
The differences are:
This would be the routine in C:
This would be the routine in ARM assembly:
Mathematically, the ability to break up a multiplication in this way is known as the distributive law , which can be expressed in algebra as the property that a ( b + c ) = ab + ac . The grid method uses the distributive property twice to expand the product, once for the horizontal factor, and once for the vertical factor.
Historically the grid calculation (tweaked slightly) was the basis of a method called lattice multiplication , which was the standard method of multiple-digit multiplication developed in medieval Arabic and Hindu mathematics. Lattice multiplication was introduced into Europe by Fibonacci at the start of the thirteenth century along with Arabic numerals themselves; although, like the numerals also, the ways he suggested to calculate with them were initially slow to catch on. Napier's bones were a calculating help introduced by the Scot John Napier in 1617 to assist lattice-method calculations. | https://en.wikipedia.org/wiki/Grid_method_multiplication |
A projected coordinate system – also called a projected coordinate reference system , planar coordinate system , or grid reference system – is a type of spatial reference system that represents locations on Earth using Cartesian coordinates ( x , y ) on a planar surface created by a particular map projection . [ 1 ] Each projected coordinate system, such as " Universal Transverse Mercator WGS 84 Zone 26N," is defined by a choice of map projection (with specific parameters), a choice of geodetic datum to bind the coordinate system to real locations on the earth, an origin point, and a choice of unit of measure. [ 2 ] Hundreds of projected coordinate systems have been specified for various purposes in various regions.
When the first standardized coordinate systems were created during the 20th century, such as the Universal Transverse Mercator , State Plane Coordinate System , and British National Grid , they were commonly called grid systems ; the term is still common in some domains such as the military that encode coordinates as alphanumeric grid references . However, the term projected coordinate system has recently become predominant to clearly differentiate it from other types of spatial reference system . The term is used in international standards such as the EPSG and ISO 19111 (also published by the Open Geospatial Consortium as Abstract Specification 2), and in most geographic information system software. [ 3 ] [ 2 ]
The map projection and the geographic coordinate system (GCS, latitude and longitude) date to the Hellenistic period , proliferating during the Enlightenment Era of the 18th century. However, their use as the basis for specifying precise locations, rather than latitude and longitude, is a 20th century innovation.
Among the earliest was the State Plane Coordinate System (SPCS), which was developed in the United States during the 1930s for surveying and engineering, because calculations such as distance are much simpler in a Cartesian coordinate system than the three-dimensional trigonometry of GCS. In the United Kingdom , the first version of the British National Grid was released in 1938, based on earlier experiments during World War I by the Army and the Ordnance Survey . [ 4 ]
During World War II , modern warfare practices required soldiers to quickly and accurately measure and report their location, leading to the printing of grids on maps by the U.S. Army Map Service (AMS) and other combatants. [ 5 ] Initially, each theater of war was mapped in a custom projection with its own grid and coding system, but this resulted in confusion. This led to the development of the Universal Transverse Mercator coordinate system , possibly adopted from a system originally developed by the German Wehrmacht . [ 6 ] To facilitate unambiguous reporting, the alphanumeric Military Grid Reference System (MGRS) was then created as an encoding scheme for UTM coordinates to make them easier to communicate. [ 5 ]
After the War, UTM gradually gained users, especially in the scientific community. Because UTM zones do not align with political boundaries, several countries followed the United Kingdom in creating their own national or regional grid systems based on custom projections. The use and invention of such systems especially proliferated during the 1980s with the emergence of geographic information systems . GIS requires locations to be specified as precise coordinates and performs numerous calculations on them, making Cartesian geometry preferable to spherical trigonometry when computing power was at a premium. In recent years, the rise of global GIS datasets and satellite navigation , along with an abundance of processing speed in personal computers, have led to a resurgence in the use of GCS. That said, projected coordinate systems are still very common in the GIS data stored in the spatial data infrastructures (SDI) of local areas, such as cities, counties, states and provinces, and small countries.
Because the purpose of any coordinate system is to accurately and unambiguously measure, communicate, and perform calculations on locations, it must be defined precisely. The EPSG Geodetic Parameter Dataset is the most common mechanism for publishing such definitions in a machine-readable form, and forms the basis for many GIS and other location-aware software programs. [ 3 ] A projected SRS specification consists of three parts:
To establish the position of a geographic location on a map , a map projection is used to convert geodetic coordinates to plane coordinates on a map; it projects the datum ellipsoidal coordinates and height onto a flat surface of a map. The datum, along with a map projection applied to a grid of reference locations, establishes a grid system for plotting locations. Conformal projections are generally preferred. Common map projections include the transverse Mercator (used in Universal Transverse Mercator , the British National Grid , the State Plane Coordinate System for some states), Lambert conformal conic (some states in the SPCS ), and Mercator ( Swiss coordinate system ).
Map projection formulas depend on the geometry of the projection as well as parameters dependent on the particular location at which the map is projected. The set of parameters can vary based on the type of project and the conventions chosen for the projection. For the transverse Mercator projection used in UTM, the parameters associated are the latitude and longitude of the natural origin, the false northing and false easting, and an overall scale factor. [ 7 ] Given the parameters associated with particular location or grin, the projection formulas for the transverse Mercator are a complex mix of algebraic and trigonometric functions. [ 7 ] : 45–54
Every map projection has a natural origin , e.g., at which the ellipsoid and flat map surfaces coincide, at which point the projection formulas generate a coordinate of (0,0). [ 7 ] To ensure that the northing and easting coordinates on a map are not negative (thus making measurement, communication, and computation easier), map projections may set up a false origin , specified in terms of false northing and false easting values, that offset the true origin. For example, in UTM, the origin of each northern zone is a point on the equator 500 km west of the central meridian of the zone (the edge of the zone itself is just under 400 km to the west). This has the desirable effect of making all coordinates within the zone positive values, being east and north of the origin. Because of this, they are often referred to as the easting and northing .
Grid north ( GN ) is a navigational term referring to the direction northwards along the grid lines of a map projection . It is contrasted with true north (the direction of the North Pole ) and magnetic north (the direction in which a compass needle points). Many topographic maps , including those of the United States Geological Survey and Great Britain's Ordnance Survey , indicate the difference between grid north, true north, and magnetic north. [ 8 ]
The grid lines on Ordnance Survey maps divide the UK into one-kilometre squares, east of an imaginary zero point in the Atlantic Ocean, west of Cornwall. The grid lines point to a Grid North, varying slightly from True North. This variation is zero on the central meridian (north-south line) of the map, which is at two degrees west of the Prime Meridian , and greatest at the map edges. The difference between grid north and true north is very small and can be ignored for most navigation purposes. The difference exists because the correspondence between a flat map and the round Earth is necessarily imperfect.
At the South Pole , grid north conventionally points northwards along the Prime Meridian . [ 9 ] Since the meridians converge at the poles, true east and west directions change rapidly in a condition similar to gimbal lock . Grid north solves this problem.
Locations in a projected coordinate system, like any cartesian coordinate system, are measured and reported as easting/northing or ( x , y ) pairs. The pair is usually represented conventionally with easting first, northing second. For example, the peak of Mount Assiniboine (at 50°52′10″N 115°39′03″W / 50.86944°N 115.65083°W / 50.86944; -115.65083 on the British Columbia / Alberta border in Canada ) in UTM Zone 11 is at (0594934mE, 5636174mN) , meaning that is almost 600km east of the false origin for Zone 11 (95km east of the true central meridian at 117°W) and 5.6 million meters north of the equator .
While such precise numbers are easy to store and calculate in GIS and other computer databases, they can be difficult for humans to remember and communicate. Thus, since the mid 20th century, there have been alternative encodings that shorten the numbers or convert the numbers into some form of alphanumeric string.
For example, a truncated grid reference may be used where the general location is already known to participants and may be assumed. [ 10 ] Because the (leading) most significant digits specify the part of the world and the (trailing) least significant digits provide a precision that is not needed in most circumstances, they may be unnecessary for some uses. This permits users to shorten the example coordinates to 949-361 by concealing 05nnn34 56nnn74 , assuming the significant digits (3,4, and 5 in this case) are known to both parties. [ 11 ]
Alphanumeric encodings typically use codes to replace the most significant digits by partitioning the world up into large grid squares. For example, in the Military Grid Reference System , the above coordinate is in grid 11U (representing UTM Zone 11 5xxxxxx mN), and grid cell NS within that (representing the second digit 5xxxxxmE x6xxxxxm N), and as many remaining digits as are needed are reported, yielding an MGRS grid reference of 11U NS 949 361 (or 11U NS 9493 3617 or 11U NS 94934 36174).
The Ordnance Survey National Grid (United Kingdom) and other national grid systems use similar approaches. In Ordnance Survey maps, each Easting and Northing grid line is given a two-digit code, based on the British national grid reference system with an origin point just off the southwest coast of the United Kingdom . The area is divided into 100 km squares, each of which is denoted by a two-letter code. Within each 100 km square, a numerical grid reference is used. Since the Eastings and Northings are one kilometre apart, a combination of a Northing and an Easting will give a four-digit grid reference describing a one-kilometre square on the ground. The convention is the grid reference numbers call out the lower-left corner of the desired square. In the example map above, the town Little Plumpton lies in the square 6901, even though the writing which labels the town is in 6802 and 6902, most of the buildings (the orange boxed symbols) are in square 6901.
The more digits added to a grid reference, the more precise the reference becomes. To locate a specific building in Little Plumpton, a further two digits are added to the four-digit reference to create a six-digit reference. The extra two digits describe a position within the 1-kilometre square. Imagine (or draw or superimpose a Romer ) a further 10x10 grid within the current grid square. Any of the 100 squares in the superimposed 10×10 grid can be accurately described using a digit from 0 to 9 (with 0 0 being the bottom left square and 9 9 being the top right square).
For the church in Little Plumpton, this gives the digits 6 and 7 (6 on the left to right axis (Eastings) and 7 on the bottom to top axis (Northings). These are added to the four-figure grid reference after the two digits describing the same coordinate axis , and thus our six-figure grid reference for the church becomes 696017. This reference describes a 100-metre by 100-metre square, and not a single point, but this precision is usually sufficient for navigation purposes. The symbols on the map are not precise in any case, for example the church in the example above would be approximately 100x200 metres if the symbol was to scale, so in fact, the middle of the black square represents the map position of the real church, independently of the actual size of the church.
Grid references comprising larger numbers for greater precision could be determined using large-scale maps and an accurate Romer . This might be used in surveying but is not generally used for land navigating for walkers or cyclists, etc. The growing availability and decreasing cost of handheld GPS receivers enables determination of accurate grid references without needing a map, but it is important to know how many digits the GPS displays to avoid reading off just the first six digits. A GPS unit commonly gives a ten-digit grid reference, based on two groups of five numbers for the Easting and Northing values. Each successive increase in precision (from 6 digit to 8 digit to 10 digit) pinpoints the location more precisely by a factor of 10. Since, in the UK at least, a 6-figure grid reference identifies a square of 100-metre sides, an 8-figure reference would identify a 10-metre square, and a 10-digit reference a 1-metre square. In order to give a standard 6-figure grid reference from a 10-figure GPS readout, the 4th, 5th, 9th and 10th digits must be omitted, so it is important not to read just the first 6 digits. | https://en.wikipedia.org/wiki/Grid_north |
This is an extended usage of word gridlock specifically in economics to describe the common situation that occurs in the competition within an industry or a company. The similar usage of gridlock can be seen in politics as well. See also gridlock (Politics) . However, the term gridlock was first used in engineering as, "A state of severe road congestion arising when continuous queues of vehicles block an entire network of intersecting streets, bringing traffic in all directions to a complete standstill; a traffic jam of this kind." [ 1 ]
In microeconomics, gridlock refers to the difficult situation of firms to invest in long-term and sustainable business, especially in a fiercely competitive industry where most firms focus on the short-term returns. It is very hard for any player to make move towards sustainable and health direction, since if one party initiates the long-term investment, which is not very profitable in the short-term, the other competitors can easily take advantage of the situation to outcompete that firm, and even drive that firm out of market. As a result, no party will take the risk to make a move. This phenomenon is also called first-mover disadvantage. In the end, the unhealthy situation of the industry can be hardly improved by the market leaders themselves. In the business as in traffic, when gridlock happens, an industry is unable to function at a healthy level, which can be highly problematic and costly.
Not only in an industry, gridlock can also happen at company level. If there are several groups with conflicting interests exist within one company competing to gain the control and power, this can create a situation in which business transactions can not be completed until the problem is solved. [ 2 ]
This kind of dilemma is also very often cited in game theory . If all the parties can cooperate, for example, companies can collaboratively invest in a sustainable way, the gridlock can be easily solved and the industry will function in a healthy way. But this is not happening because of the desire to maximize one's own benefit given the uncertainty about the others’ commitment to cooperation.
Naturally to solve this problem, it requires institutions within the industry or company to regulate rules and prevent the inefficient situation to happen. Thus institution plays a role of traffic lights or traffic police in the conjunction of business world.
In the book Institutional Economics an introduction, professor John Groenewegen, Antoon Spithoven and Annette van den Berg explained approaches of analyzing this kind of situation and suggest the solutions to similar problems from institutional prospective. See also Institutional economics . | https://en.wikipedia.org/wiki/Gridlock_(economics) |
A gridshell is a structure which derives its strength from its double curvature (in a similar way that a fabric structure derives strength from double curvature), but is constructed of a grid or lattice.
The grid can be made of any material, but is most often wood (similar to garden trellis) or steel.
Gridshells were pioneered in the 1896 by Russian engineer Vladimir Shukhov in constructions of exhibition pavilions of the All-Russia industrial and art exhibition 1896 in Nizhny Novgorod . [ 1 ]
Large span timber gridshells are commonly constructed by initially laying out the main lath members flat in a regular square or rectangular lattice, and subsequently deforming this into the desired doubly curved form. This can be achieved by pushing the members up from the ground, as in the Mannheim Multihalle. [ 2 ] More recent projects such as the Savill Garden gridshell were constructed by laying the laths on top of a sizeable temporary scaffolding structure which is removed in phases to let the laths settle into the desired curvature. | https://en.wikipedia.org/wiki/Gridshell |
The Grieco elimination is an organic reaction describing the elimination reaction of an aliphatic primary alcohol through a selenide to a terminal alkene . [ 1 ] [ 2 ] It is named for Paul Grieco .
The alcohol first reacts with o -nitrophenylselenocyanate and tributylphosphine to form a selenide via a nucleophilic substitution on the electron-deficient selenium . In the second step, the selenide is oxidized with hydrogen peroxide to give a selenoxide . This structure decomposes to form an alkene by an E i elimination mechanism with expulsion of a selenol in a fashion similar to that of the Cope elimination . This reaction takes part in the synthesis of ring C of the Danishefsky Taxol synthesis .
The elimination step is common with the Clive-Reich-Sharpless olefination that uses PhSeX as the selenium source. [ 3 ] | https://en.wikipedia.org/wiki/Grieco_elimination |
The Grieco three-component condensation is an organic chemistry reaction that produces nitrogen-containing six-member heterocycles via a multi-component reaction of an aldehyde , a nitrogen component, such as aniline , and an electron-rich alkene . The reaction is catalyzed by trifluoroacetic acid or Lewis acids such as ytterbium trifluoromethanesulfonate (Yb(OTf) 3 ). The reaction is named for Paul Grieco , who first reported it in 1985. [ 1 ] [ 2 ] In the original paper the nitrogen component were benzylamine, methyl amine or ammonium chloride, the reaction now also include anilines, similar to the earlier Povarov reaction .
The reaction process involves the formation of an aryl immonium ion intermediate followed by an aza Diels-Alder reaction with an alkene. Imines are electron-poor, and thus usually function as the dienophile. Here, however, the alkene is electron-rich, so it reacts well with the immonium diene in an Inverse electron-demand Diels–Alder reaction .
Researchers have extended the Grieco three-component reaction to reactants or catalysts immobilized on solid support, which greatly expands the application of this reaction to various combinatorial chemistry settings. Kielyov and Armstrong [ 3 ] were the first to report a solid-supported version of this reaction, they found that this reaction works well for each reactants immobilized on solid support. Kobayashi and co-workers [ 4 ] show that a polymer-supported scandium catalyst catalyze the Grieco reaction with high efficiency.
Given the effectiveness of the reaction and the commercial availability of various Grieco partners, the Grieco three-component coupling is very useful for preparing quinoline libraries for drug discovery . | https://en.wikipedia.org/wiki/Grieco_three-component_condensation |
The Griesbaum coozonolysis is a name reaction in organic chemistry that allows for the preparation of tetrasubstituted ozonides (1,2,4-trioxolanes) by the reaction of O -methyl oximes with a carbonyl compound in the presence of ozone. Contrary to their usual roles as intermediates in ozonolysis and other oxidative alkene cleavage reactions, 1,2,4-trioxolanes are relatively stable compounds and are isolable. [ 1 ] [ 2 ] [ 3 ] [ 4 ] [ 5 ]
The oxime first reacts with the ozone to form the corresponding carbonyl oxide, undergoes 1,3-dipolar cycloaddition with the carbonyl reactant to form the cyclic ozonide, as usual for the Criegee intermediate in the ozonolysis of alkenes.
If no carbonyl compound is used, the carbonyl oxide may dimerize and form 1,2,4,5-tetraoxanes.
This chemical reaction article is a stub . You can help Wikipedia by expanding it . | https://en.wikipedia.org/wiki/Griesbaum_coozonolysis |
In the mathematics of coding theory , the Griesmer bound , named after James Hugo Griesmer, is a bound on the length of linear binary codes of dimension k and minimum distance d .
There is also a very similar version for non-binary codes.
For a binary linear code, the Griesmer bound is:
Let N ( k , d ) {\displaystyle N(k,d)} denote the minimum length of a binary code of dimension k and distance d . Let C be such a code. We want to show that
Let G be a generator matrix of C . We can always suppose that the first row of G is of the form r = (1, ..., 1, 0, ..., 0) with weight d .
The matrix G ′ {\displaystyle G'} generates a code C ′ {\displaystyle C'} , which is called the residual code of C . {\displaystyle C.} C ′ {\displaystyle C'} obviously has dimension k ′ = k − 1 {\displaystyle k'=k-1} and length n ′ = N ( k , d ) − d . {\displaystyle n'=N(k,d)-d.} C ′ {\displaystyle C'} has a distance d ′ , {\displaystyle d',} but we don't know it. Let u ∈ C ′ {\displaystyle u\in C'} be such that w ( u ) = d ′ {\displaystyle w(u)=d'} . There exists a vector v ∈ F 2 d {\displaystyle v\in \mathbb {F} _{2}^{d}} such that the concatenation ( v | u ) ∈ C . {\displaystyle (v|u)\in C.} Then w ( v ) + w ( u ) = w ( v | u ) ⩾ d . {\displaystyle w(v)+w(u)=w(v|u)\geqslant d.} On the other hand, also ( v | u ) + r ∈ C , {\displaystyle (v|u)+r\in C,} since r ∈ C {\displaystyle r\in C} and C {\displaystyle C} is linear: w ( ( v | u ) + r ) ⩾ d . {\displaystyle w((v|u)+r)\geqslant d.} But
so this becomes d − w ( v ) + w ( u ) ⩾ d {\displaystyle d-w(v)+w(u)\geqslant d} . By summing this with w ( v ) + w ( u ) ⩾ d , {\displaystyle w(v)+w(u)\geqslant d,} we obtain d + 2 w ( u ) ⩾ 2 d {\displaystyle d+2w(u)\geqslant 2d} . But w ( u ) = d ′ , {\displaystyle w(u)=d',} so we get 2 d ′ ⩾ d . {\displaystyle 2d'\geqslant d.} As d ′ {\displaystyle d'} is integral, we get d ′ ⩾ ⌈ d 2 ⌉ . {\displaystyle d'\geqslant \left\lceil {\tfrac {d}{2}}\right\rceil .} This implies
so that
By induction over k we will eventually get
Note that at any step the dimension decreases by 1 and the distance is halved, and we use the identity
for any integer a and positive integer k .
For a linear code over F q {\displaystyle \mathbb {F} _{q}} , the Griesmer bound becomes:
The proof is similar to the binary case and so it is omitted. | https://en.wikipedia.org/wiki/Griesmer_bound |
In mathematics , the Griess algebra is a commutative non-associative algebra on a real vector space of dimension 196884 that has the Monster group M as its automorphism group . It is named after mathematician R. L. Griess , who constructed it in 1980 and subsequently used it in 1982 to construct M . The Monster fixes (vectorwise) a 1-space in this algebra and acts absolutely irreducibly on the 196883-dimensional orthogonal complement of this 1-space.
(The Monster preserves the standard inner product on the 196884-space.)
Griess's construction was later simplified by Jacques Tits and John H. Conway .
The Griess algebra is the same as the degree 2 piece of the monster vertex algebra , and the Griess product is one of the vertex algebra products.
This algebra -related article is a stub . You can help Wikipedia by expanding it . | https://en.wikipedia.org/wiki/Griess_algebra |
The Griess test is an analytical chemistry test which detects the presence of nitrite ion in solution . One of its most important uses is the determination of nitrite in drinking water . The Griess diazotization reaction, on which the Griess reagent relies, was first described in 1858 by Peter Griess . [ 1 ] [ 2 ] The test has also been widely used for the detection of nitrates (N- oxidation state = +5), which are a common component of explosives , as they can be reduced to nitrites (N- oxidation state = +3) and detected with the Griess test. [ 3 ]
Nitrite is detected and analyzed by the formation of a red pink colour upon treatment of a nitrite-containing sample with the Griess reagent, which consists of two components in an acidic solution: an aniline derivative and a coupling agent. The most common arrangements use sulfanilamide and N-(1-naphthyl)ethylenediamine : [ 3 ] a typical commercial Griess reagent contains 0.2% N-(1-naphthyl)ethylenediamine dihydrochloride, and 2% sulfanilamide in 5% phosphoric acid . [ 4 ] This diamine is used in place of the simpler and cheaper 1-naphthylamine because the latter is a potent carcinogen and moreover the diamine forms a more polar and hence a much more soluble dye in acidic aqueous medium. [ 5 ] Other aniline derivatives that have been used include sulfanilic acid , nitroaniline , and p -aminoacetophenone. [ 3 ]
The Griess test involves two subsequent reactions. When sulfanilamide is added, the nitrite ion reacts with it in the Griess diazotization reaction to form a diazonium salt , which then reacts with N-(1-naphthyl)ethylenediamine in an azo coupling reaction, forming a pink-red azo dye .
Using a spectrophotometer , it is possible to quantitatively determine the nitrite concentration. The detection limit of the Griess test generally ranges between 0.02 and 2 μM, depending on the exact details of the specific components used in the Griess reagent. [ 3 ]
The test was used in forensics for many years to test for the traces of nitroglycerine . Caustic soda is used to break down sample containing nitroglycerine to produce nitrite ions.
The test involves the taking of a sample with ether and its division into two bowls. Caustic soda is added to the first bowl followed by the Griess reagent; if the solution turns pink within ten seconds, this indicates the presence of nitrites. The test itself is positive if, after adding only Griess reagent to the second bowl, the solution there remains clear.
The convictions of Judith Ward and the Birmingham Six were assisted by Frank Skuse 's flawed interpretation of Griess test results. [ 6 ] | https://en.wikipedia.org/wiki/Griess_test |
Griffith's experiment , [ 1 ] performed by Frederick Griffith and reported in 1928, [ 2 ] was the first experiment suggesting that bacteria are capable of transferring genetic information through a process known as transformation . [ 3 ] [ 4 ] Griffith's findings were followed by research in the late 1930s and early 40s that isolated DNA as the material that communicated this genetic information.
Pneumonia was a serious cause of death in the wake of the post-WWI Spanish influenza pandemic , and Griffith was studying the possibility of creating a vaccine . Griffith used two strains of pneumococcus ( Diplococcus pneumoniae ) bacteria which infect mice – a type III-S (smooth) which was virulent , and a type II-R (rough) strain which was nonvirulent. The III-S strain synthesized a polysaccharide capsule that protected itself from the host's immune system , resulting in the death of the host, while the II-R strain did not have that protective capsule and was defeated by the host's immune system. A German bacteriologist, Fred Neufeld , had discovered the three pneumococcal types (Types I, II, and III) and discovered the quellung reaction to identify them in vitro . [ 5 ] Until Griffith's experiment, bacteriologists believed that the types were fixed and unchangeable, from one generation to another.
In this experiment, bacteria from the III-S strain were killed by heat, and their remains were added to II-R strain bacteria. While neither alone harmed the mice, the combination was able to kill its host. Griffith was also able to isolate both live II-R and live III-S strains of pneumococcus from the blood of these dead mice. Griffith concluded that the type II-R had been "transformed" into the lethal III-S strain by a "transforming principle" that was somehow part of the dead III-S strain bacteria.
Scientific advances since then have revealed that the "transforming principle" Griffith observed was the DNA of the III-s strain bacteria. While the bacteria had been killed, the DNA had survived the heating process and was taken up by the II-R strain bacteria. The III-S strain DNA contains the genes that form the smooth protective polysaccharide capsule. Equipped with this gene, the former II-R strain bacteria were now protected from the host's immune system and could kill the host. The exact nature of the transforming principle (DNA) was verified in the experiments done by Avery, McLeod and McCarty and by Hershey and Chase . | https://en.wikipedia.org/wiki/Griffith's_experiment |
In statistical mechanics , the Griffiths inequality , sometimes also called Griffiths–Kelly–Sherman inequality or GKS inequality , named after Robert B. Griffiths , is a correlation inequality for ferromagnetic spin systems. Informally, it says that in ferromagnetic spin systems, if the 'a-priori distribution' of the spin is invariant under spin flipping, the correlation of any monomial of the spins is non-negative; and the two point correlation of two monomial of the spins is non-negative.
The inequality was proved by Griffiths for Ising ferromagnets with two-body interactions, [ 1 ] then generalised by Kelly and Sherman to interactions involving an arbitrary number of spins, [ 2 ] and then by Griffiths to systems with arbitrary spins. [ 3 ] A more general formulation was given by Ginibre , [ 4 ] and is now called the Ginibre inequality .
Let σ = { σ j } j ∈ Λ {\displaystyle \textstyle \sigma =\{\sigma _{j}\}_{j\in \Lambda }} be a configuration of (continuous or discrete) spins on a lattice Λ . If A ⊂ Λ is a list of lattice sites, possibly with duplicates, let σ A = ∏ j ∈ A σ j {\displaystyle \textstyle \sigma _{A}=\prod _{j\in A}\sigma _{j}} be the product of the spins in A .
Assign an a-priori measure dμ(σ) on the spins;
let H be an energy functional of the form
where the sum is over lists of sites A , and let
be the partition function . As usual,
stands for the ensemble average .
The system is called ferromagnetic if, for any list of sites A , J A ≥ 0 . The system is called invariant under spin flipping if, for any j in Λ , the measure μ is preserved under the sign flipping map σ → τ , where
In a ferromagnetic spin system which is invariant under spin flipping,
for any list of spins A .
In a ferromagnetic spin system which is invariant under spin flipping,
for any lists of spins A and B .
The first inequality is a special case of the second one, corresponding to B = ∅.
Observe that the partition function is non-negative by definition.
Proof of first inequality : Expand
then
where n A (j) stands for the number of times that j appears in A . Now, by invariance under spin flipping,
if at least one n(j) is odd, and the same expression is obviously non-negative for even values of n . Therefore, Z < σ A >≥0, hence also < σ A >≥0.
Proof of second inequality . For the second Griffiths inequality, double the random variable, i.e. consider a second copy of the spin, σ ′ {\displaystyle \sigma '} , with the same distribution of σ {\displaystyle \sigma } . Then
Introduce the new variables
The doubled system ⟨ ⟨ ⋅ ⟩ ⟩ {\displaystyle \langle \langle \;\cdot \;\rangle \rangle } is ferromagnetic in τ , τ ′ {\displaystyle \tau ,\tau '} because − H ( σ ) − H ( σ ′ ) {\displaystyle -H(\sigma )-H(\sigma ')} is a polynomial in τ , τ ′ {\displaystyle \tau ,\tau '} with positive coefficients
Besides the measure on τ , τ ′ {\displaystyle \tau ,\tau '} is invariant under spin flipping because d μ ( σ ) d μ ( σ ′ ) {\displaystyle d\mu (\sigma )d\mu (\sigma ')} is.
Finally the monomials σ A {\displaystyle \sigma _{A}} , σ B − σ B ′ {\displaystyle \sigma _{B}-\sigma '_{B}} are polynomials in τ , τ ′ {\displaystyle \tau ,\tau '} with positive coefficients
The first Griffiths inequality applied to ⟨ ⟨ σ A ( σ B − σ B ′ ) ⟩ ⟩ {\displaystyle \langle \langle \sigma _{A}(\sigma _{B}-\sigma '_{B})\rangle \rangle } gives the result.
More details are in [ 5 ] and. [ 6 ]
The Ginibre inequality is an extension, found by Jean Ginibre, [ 4 ] of the Griffiths inequality.
Let (Γ, μ ) be a probability space . For functions f , h on Γ, denote
Let A be a set of real functions on Γ such that. for every f 1 , f 2 ,..., f n in A , and for any choice of signs ±,
Then, for any f , g ,− h in the convex cone generated by A ,
Let
Then
Now the inequality follows from the assumption and from the identity | https://en.wikipedia.org/wiki/Griffiths_inequality |
The Grignard reaction ( French: [ɡʁiɲaʁ] ) is an organometallic chemical reaction in which, according to the classical definition, carbon alkyl , allyl , vinyl , or aryl magnesium halides ( Grignard reagent ) are added to the carbonyl groups of either an aldehyde or ketone under anhydrous conditions. [ 1 ] [ 2 ] [ 3 ] This reaction is important for the formation of carbon–carbon bonds . [ 4 ] [ 5 ]
Grignard reactions and reagents were discovered by and are named after the French chemist François Auguste Victor Grignard ( University of Nancy , France), who described them in 1900. [ 6 ] He was awarded the 1912 Nobel Prize in Chemistry for this work. [ 7 ]
The reaction of an organic halide with magnesium is not a Grignard reaction, but provides a Grignard reagent. [ 8 ]
Classically, the Grignard reaction refers to the reaction between a ketone or aldehyde group with a Grignard reagent to form a primary or tertiary alcohol. [ 1 ] However, some chemists understand the definition to mean all reactions of any electrophiles with Grignard reagents. [ 9 ] Therefore, there is some dispute about the modern definition of the Grignard reaction. In the Merck Index , published online by the Royal Society of Chemistry , the classical definition is acknowledged, followed by "A more modern interpretation extends the scope of the reaction to include the addition of Grignard reagents to a wide variety of electrophilic substrates." [ 9 ] This variety of definitions illustrates that there is some dispute within the chemistry community about the definition of a Grignard reaction.
Shown below are some reactions involving Grignard reagents , but they themselves are not classically understood as Grignard reactions.
Because carbon is more electronegative than magnesium, the carbon attached to magnesium acts as a nucleophile and attacks the electrophilic carbon atom in the polar bond of a carbonyl group. The addition of the Grignard reagent to the carbonyl group typically proceeds through a six-membered ring transition state , as shown below. [ 10 ]
Based on the detection of radical coupling side products, an alternative single electron transfer (SET) mechanism that involves the initial formation of a ketyl radical intermediate has also been proposed. [ 11 ] A recent computational study suggests that the operative mechanism (polar vs. radical) is substrate-dependent, with the reduction potential of the carbonyl compound serving as a key parameter. [ 12 ]
The Grignard reaction is conducted under anhydrous conditions . [ 3 ] Otherwise, the reaction will fail because the Grignard reagent will act as a base rather than a nucleophile and pick up a labile proton rather than attacking the electrophilic site. This will result in no formation of the desired product as the R-group of the Grignard reagent will become protonated while the MgX portion will stabilize the deprotonated species.
To prevent this, Grignard reactions are completed in an inert atmosphere to remove all water from the reaction flask and ensure that the desired product is formed. [ 13 ] Additionally, if there are acidic protons in the starting material, as shown in the figure on the right, one can overcome this by protecting the acidic site of the reactant by turning it into an ether or a silyl ether to eliminate the labile proton from the solution prior to the Grignard reaction.
Other variations of the Grignard reagent have been discovered to improve the chemoselectivity of the Grignard reaction, which include but are not limited to: Turbo-Grignards, organocerium reagents, and organocuprate (Gilman) reagents.
Turbo-Grignards are Grignard reagents modified with lithium chloride . Compared to conventional Grignard reagents, Turbo-Grignards are more chemoselective ; esters , amides , and nitriles do not react with the Turbo-Grignard reagent. [ 14 ]
The behavior of Grignard reagents can be usefully modified in the present of other metals. Copper(I) salts give organocuprates that preferentially effect 1,4 addition . [ 15 ] Cerium trichloride allows selective 1,2-additions to the same substrates. Nickel and palladium halides catalyze cross coupling reactions . | https://en.wikipedia.org/wiki/Grignard_reaction |
Grignard reagents or Grignard compounds are chemical compounds with the general formula R−Mg−X , where X is a halogen and R is an organic group , normally an alkyl or aryl . Two typical examples are methylmagnesium chloride Cl−Mg−CH 3 and phenylmagnesium bromide (C 6 H 5 )−Mg−Br . They are a subclass of the organomagnesium compounds .
Grignard compounds are popular reagents in organic synthesis for creating new carbon–carbon bonds . For example, when reacted with another halogenated compound R'−X' in the presence of a suitable catalyst , they typically yield R−R' and the magnesium halide MgXX' as a byproduct; and the latter is insoluble in the solvents normally used.
Grignard reagents are rarely isolated as solids. Instead, they are normally handled as solutions in solvents such as diethyl ether or tetrahydrofuran using air-free techniques . Grignard reagents are complex with the magnesium atom bonded to two ether ligands as well as the halide and organyl ligands.
The discovery of the Grignard reaction in 1900 was recognized with the Nobel Prize awarded to Victor Grignard in 1912.
Traditionally Grignard reagents are prepared by treating an organic halide (normally organobromine) with magnesium metal. Ethers are required to stabilize the organomagnesium compound . Water and air, which rapidly destroy the reagent by protonolysis or oxidation, are excluded. [ 1 ] Although the reagents still need to be dry, ultrasound can allow Grignard reagents to form in wet solvents by activating the magnesium such that it consumes the water. [ 2 ]
As is common for reactions involving solids and solution, the formation of Grignard reagents is often subject to an induction period . During this stage, the passivating oxide on the magnesium is removed. After this induction period, the reactions can be highly exothermic . This exothermicity must be considered when a reaction is scaled-up from laboratory to production plant. [ 3 ] Most organohalides will work, but carbon-fluorine bonds are generally unreactive, except with specially activated magnesium (through Rieke metals ).
Typically the reaction to form Grignard reagents involves the use of magnesium ribbon. All magnesium is coated with a passivating layer of magnesium oxide , which inhibits reactions with the organic halide. Many methods have been developed to weaken this passivating layer, thereby exposing highly reactive magnesium to the organic halide. Mechanical methods include crushing of the Mg pieces in situ, rapid stirring, and sonication . [ 4 ] Iodine , methyl iodide , and 1,2-dibromoethane are common activating agents. The use of 1,2-dibromoethane is advantageous as its action can be monitored by the observation of bubbles of ethylene . Furthermore, the side-products are innocuous:
The amount of Mg consumed by these activating agents is usually insignificant. A small amount of mercuric chloride will amalgamate the surface of the metal, enhancing its reactivity. Addition of preformed Grignard reagent is often used as the initiator.
Specially activated magnesium, such as Rieke magnesium , circumvents this problem. [ 5 ] The oxide layer can also be broken up using ultrasound, using a stirring rod to scratch the oxidized layer off, [ 6 ] or by adding a few drops of iodine or 1,2-Diiodoethane . Another option is to use sublimed magnesium or magnesium anthracene . [ 7 ]
"Rieke magnesium" is prepared by a reduction of an anhydrous magnesium chloride with a potassium :
In terms of mechanism, the reaction proceeds through single electron transfer : [ 8 ] [ 9 ] [ 10 ]
R − X + Mg ⟶ [ R − X ∙ ] − + [ Mg ∙ ] + [ R − X ∙ ] − ⟶ R ∙ + X − R ∙ + [ Mg ∙ ] + ⟶ R − Mg + R − Mg + + X − ⟶ R − MgX {\displaystyle {\begin{aligned}{\ce {R-X{}+Mg}}&\longrightarrow {\ce {[R-X^{\bullet }]^{-}{}+[Mg^{\bullet }]+}}\\{\ce {[R-X^{\bullet }]-}}&\longrightarrow {\ce {R^{\bullet }{}+X-}}\\{\ce {R^{\bullet }{}+[Mg^{\bullet }]+}}&\longrightarrow {\ce {R-Mg+}}\\{\ce {R-Mg+{}+X-}}&\longrightarrow {\ce {R-MgX}}\end{aligned}}}
An alternative preparation of Grignard reagents involves transfer of Mg from a preformed Grignard reagent to an organic halide. Other organomagnesium reagents are used as well. [ 11 ] This method offers the advantage that the Mg transfer tolerates many functional groups. An illustrative reaction involves isopropylmagnesium chloride and aryl bromide or iodides: [ 12 ]
A further method to synthesize Grignard reagents involves reaction of Mg with an organozinc compound . This method has been used to make adamantane -based Grignard reagents, which are, due to C-C coupling side reactions, difficult to make by the conventional method from the alkyl halide and Mg. The reductive transmetalation achieves: [ 13 ]
Because Grignard reagents are so sensitive to moisture and oxygen, many methods have been developed to test the quality of a batch. Typical tests involve titrations with weighable, anhydrous protic reagents, e.g. menthol in the presence of a color-indicator. The interaction of the Grignard reagent with phenanthroline or 2,2'-biquinoline causes a color change. [ 14 ]
Grignard reagents react with a variety of carbonyl derivatives. [ 15 ]
The most common application of Grignard reagents is the alkylation of aldehydes and ketones, i.e. the Grignard reaction : [ 16 ]
Note that the acetal functional group (a protected carbonyl) does not react.
Such reactions usually involve an aqueous acidic workup, though this step is rarely shown in reaction schemes. In cases where the Grignard reagent is adding to an aldehyde or a prochiral ketone, the Felkin-Anh model or Cram's Rule can usually predict which stereoisomer will be formed. With easily deprotonated 1,3- diketones and related acidic substrates, the Grignard reagent RMgX functions merely as a base, giving the enolate anion and liberating the alkane RH.
Grignard reagents also react with many "carbonyl-like" compounds and other electrophiles:
Grignard reagents are nucleophiles in nucleophilic aliphatic substitutions for instance with alkyl halides in a key step in industrial Naproxen production:
In the Bruylants reaction , a nitrile can be replaced by the Grignard nucleophile, rather than the Grignard attacking the nitrile to form an imino structure. [ 17 ]
Grignard reagents serve as a base for non-protic substrates (this scheme does not show workup conditions, which typically includes water). Grignard reagents are basic and react with alcohols, phenols, etc. to give alkoxides (ROMgBr). The phenoxide derivative is susceptible to formylation by paraformaldehyde to give salicylaldehyde . [ 18 ]
Like organolithium compounds , Grignard reagents are useful for forming carbon–heteroatom bonds.
R 4 B − Et 2 O ⋅ BF 3 or NaBF 4 ↑ Et 2 O ⋅ BF 3 or NaBF 4 Ph 2 PR ← Ph 2 PCl RMgX → Bu 3 SnCl Bu 3 SnR B ( OMe ) 3 ↓ B ( OMe ) 3 RB ( OMe ) 2 {\displaystyle {\begin{matrix}{\ce {R4B-}}\\{\color {White}\scriptstyle {\ce {Et2O.BF3\ or\ NaBF4}}}{\Bigg \uparrow }\scriptstyle {\ce {Et2O.BF3\ or\ NaBF4}}\\{\ce {Ph2PR<-[{\ce {Ph2PCl}}]RMgX->[{\ce {Bu3SnCl}}]Bu3SnR}}\\{\color {White}\scriptstyle {\ce {B(OMe)3}}}{\Bigg \downarrow }\scriptstyle {\ce {B(OMe)3}}\\{\ce {RB(OMe)2}}\end{matrix}}} Grignard reagents react with many metal-based electrophiles. For example, they undergo transmetallation with cadmium chloride (CdCl 2 ) to give dialkylcadmium : [ 19 ]
Most Grignard reactions are conducted in ethereal solvents, especially diethyl ether and THF . Grignard reagents react with 1,4-dioxane to give the diorganomagnesium compounds and insoluble coordination polymer MgX 2 (dioxane) 2 and (R = organic group, X = halide):
This reaction exploits the Schlenk equilibrium , driving it toward the right.
Grignard reagents react with organolithium compounds to give ate complexes (Bu = butyl): [ 20 ]
Grignard reagents do not typically react with organic halides, in contrast with their high reactivity with other main group halides. In the presence of metal catalysts, however, Grignard reagents participate in C-C coupling reactions . For example, nonylmagnesium bromide reacts with methyl p -chlorobenzoate to give p -nonylbenzoic acid, in the presence of Tris(acetylacetonato)iron(III) (Fe(acac) 3 ), after workup with NaOH to hydrolyze the ester , shown as follows. Without the Fe(acac) 3 , the Grignard reagent would attack the ester group over the aryl halide . [ 21 ]
For the coupling of aryl halides with aryl Grignard reagents, nickel chloride in tetrahydrofuran (THF) is also a good catalyst. Additionally, an effective catalyst for the couplings of alkyl halides is the Gilman catalyst lithium tetrachlorocuprate ( Li 2 CuCl 4 ), prepared by mixing lithium chloride (LiCl) and copper(II) chloride ( CuCl 2 ) in THF. The Kumada-Corriu coupling gives access to [substituted] styrenes .
Treatment of a Grignard reagent with oxygen gives the magnesium organoperoxide. Hydrolysis of this material yields hydroperoxides or alcohol. These reactions involve radical intermediates.
R − MgX + O 2 ⟶ R ∙ + [ O 2 ∙ ] − + MgX + ⟶ R − O − O − MgX + H 3 O + ⟶ R − O − O − H + HO − MgX + H + ↓ R − MgX R − O − MgX + H 3 O + ⟶ R − O − H + HO − MgX + H + {\displaystyle {\begin{array}{lcrll}{\ce {{R-MgX}+O2->}}\ {\color {Red}{\ce {{R^{\bullet }}+[O2^{\bullet }]-}}}+{\ce {MgX+->}}&{\ce {R-O-O-MgX}}&{\color {Gray}+\ {\ce {H3O+}}}&{\ce {->{R-O-O-H}}}&{\color {Gray}+\ {\ce {{HO-MgX}+H+}}}\\&{\Bigg \downarrow }{\ce {R-MgX}}\\&{\ce {R-O-MgX}}&{\color {Gray}+\ {\ce {H3O+}}}&{\ce {->{R-O-H}}}&{\color {Gray}+\ {\ce {{HO-MgX}+H+}}}\\\end{array}}}
The simple oxidation of Grignard reagents to give alcohols is of little practical importance as yields are generally poor. In contrast, two-step sequence via a borane ( vide supra ) that is subsequently oxidized to the alcohol with hydrogen peroxide is of synthetic utility.
The synthetic utility of Grignard oxidations can be increased by a reaction of Grignard reagents with oxygen in presence of an alkene to an ethylene extended alcohol . [ 22 ] This modification requires aryl or vinyl Grignards. Adding just the Grignard and the alkene does not result in a reaction demonstrating that the presence of oxygen is essential. The only drawback is the requirement of at least two equivalents of Grignard although this can partly be circumvented by the use of a dual Grignard system with a cheap reducing Grignard such as n-butylmagnesium bromide.
In the Boord olefin synthesis , the addition of magnesium to certain β-haloethers results in an elimination reaction to the alkene. This reaction can limit the utility of Grignard reactions.
An example of the Grignard reaction is a key step in the (non-stereoselective) industrial production of Tamoxifen [ 23 ] (currently used for the treatment of estrogen receptor positive breast cancer in women): [ 24 ] | https://en.wikipedia.org/wiki/Grignard_reagent |
Grigori Mints (June 7, 1939 – May 29, 2014) was a Russian philosopher and mathematician who worked in mathematical logic .
He was born in Leningrad , in the Soviet Union (now St. Petersburg , Russia), and received his Ph.D. in 1965 from the Leningrad State University under Nikolai Aleksandrovich Shanin with a thesis entitled "On Predicate and Operator Variants for Building Theories of Constructive Mathematics". In 1990 he received his D.Sc. from Leningrad State University with a thesis entitled "Proof Transformations and Synthesis of Programs". [ 1 ] He was a Stanford University professor . [ 2 ] Since 1991, Grigori "Grisha" Mints was a professor of philosophy and, by courtesy, of mathematics and of computer science at Stanford University. Before joining Stanford, Mints held research positions at the Steklov Mathematical Institute , Leningrad University , and the Estonian Academy of Sciences .
Considered one of the most distinguished logicians in the world, Mints was passionate about the applications of logic to philosophy. His expertise was in proof theory – the analysis of the structure of mathematical reasoning. Mints was elected to the Estonian Academy of Sciences in 2008 and to the American Academy of Arts and Sciences in 2010.
Mints was a very active member of the steering committee of the WoLLIC series of workshops on logic and language, after having been a member of the community in several capacities such as invited speaker, PC member, PC chair, Organising Committee chair, guest editor of proceedings and special issue, and steering committee member. | https://en.wikipedia.org/wiki/Grigori_Mints |
In philosophy , the Grim Reaper paradox is a paradox involving an infinite sequence of grim reapers, each tasked with killing a person if no reaper has already killed them. The paradox raises questions about the possibility of continuous time and the infinite past ( temporal finitism ). [ 1 ]
The paradox is inspired by J. A. Benardete 's paradoxes from the 1964 book Infinity: An Essay in Metaphysics . In fact, various formulations of paradoxes involving beginningless sets , whose members perform a function only if no previous member performs it, are all labelled Benardete Paradoxes . [ 2 ] They are examples of supertasks .
The paradox supposes there is an infinite sequence of Reapers, each assigned a time to kill a particular person. Each Reaper will only kill this person if no earlier Reaper has already killed them.
It is 12pm, the first Reaper is set to kill the person at 1pm. The second Reaper is set to kill them at 12:30pm, the third at 12:15pm, and so on.
As a consequence of these propositions, the person will certainly be killed by a Reaper before 1pm, however, no individual Reaper can kill them, as there is always an earlier Reaper who would do so first. Therefore, it is impossible that the person survives, but also impossible that any Reaper kills them. [ 3 ]
One solution to the paradox is supposing that time must be discrete rather than continuous. If so, an infinite number of Reapers cannot all have a separate time in which they will kill you, as there are only finitely many "moments" in each period of time. A possible issue with this solution is that the Reaper paradox can take different forms which do not rely upon continuous time. One such example appears in Benardete's book, in which a god throws up a wall if a man travels 1/2 mile, another god throws up a wall after 1/4 mile, another at 1/8 mile, ad infinitum. Discrete time would do nothing to prevent this paradox. [ 4 ] [ 5 ]
Another solution is the idea of Causal finitism, which asserts that there cannot be an infinite regress of causes. In other words, every causal chain must have a starting point. Thus, there cannot be an infinite number of Reapers whose actions depend on all previous Reapers. All Benardete paradoxes share this feature of an infinite causal chain, and so are all impossible.
Causal finitism could plausibly imply the discreteness of time, temporal finitism , infinitely large spatial regions, and continuously dense spatial regions, all of which are heavy metaphysical commitments. [ 6 ]
A third potential solution to the Grim Reaper paradox has been suggested, known as the Unsatisfiable Pair Diagnosis (UPD). The UPD asserts that Benardete paradoxes (including the Grim Reaper paradox) are simply logically impossible, and no metaphysical thesis needs to be adopted. In The Form of the Benardete Dichotomy Nickolas Shackel observes that all Benardete Paradoxes involve two conditions:
Shackel shows these statements to be formally inconsistent, they logically cannot both be true. The paradox assumes that some set of items could satisfy both statements, but no set can. [ 6 ] [ 7 ]
According to Pruss, the Grim Reaper paradox provides grounds for thinking that the past is finite, i.e. that there must be a first period of time. This would support the Kalam cosmological argument , backing up the premise that the universe began to exist. [ 1 ]
In 2018, Pruss provided a more thorough cosmological argument using causal finitism to motivate a necessary uncaused cause . The argument is as follows:
Pruss then adds the following Causal Principle: 5. Every contingent item has a cause. [ Note 2 ] From this the conclusion can be drawn that there is an uncaused cause which exists necessarily. Pruss states that it is still a major task to argue from a necessary first cause to theism.
Whilst The Kalam argument opposes sequences that go infinitely backwards in time, this argument denies all causally backwards-infinite sequences. [ 4 ] | https://en.wikipedia.org/wiki/Grim_Reaper_paradox |
The grimace scale ( GS ), sometimes called the grimace score , is a method of assessing the occurrence or severity of pain experienced by non-human animals according to objective and blinded scoring of facial expressions, as is done routinely for the measurement of pain in non-verbal humans. Observers score the presence or prominence of "facial action units" (FAU) , e.g. Orbital Tightening, Nose Bulge, Ear Position and Whisker Change. These are scored by observing the animal directly in real-time, or post hoc from photographs or screen-grabs from videos. The facial expression of the animals is sometimes referred to as the pain face .
The GS method of pain assessment is highly applicable to laboratory rodents as these are usually prey species which tend to inhibit the expression of pain to prevent appearing vulnerable to predators. For this reason, behavioural changes in these species are mainly observed with acute pain (hours) but are less pronounced in longer-lasting pain (days). [ 1 ]
For mice at least, the GS has been shown to be a highly accurate, repeatable and reliable means of assessing pain requiring only a short period of training for the observer. [ 2 ] [ 3 ] Across species, GS are proven to have high accuracy and reliability, and are considered useful for indicating both procedural and postoperative pain, and for assessing the efficacy of analgesics. [ 4 ] [ 5 ]
The overall accuracy of GS is reported as 97% for mice, 84% for rabbits, 82% for rats and 73.3% for horses. [ citation needed ]
Facial expressions have long been considered as indicators of emotion in both human and non-human animals. The biologist Charles Darwin considered that non-human animals exhibit similar facial expressions to emotional states as do humans. [ 6 ] The assessment of changes in human anatomy during facial expressions were successfully translated from humans to non-human primates, such as the chimpanzee (ChimpFACS) [ 7 ] and rhesus macaque (MaqFACS), [ 8 ] but were not originally applied to assess pain in these species. In 2010, a team of researchers successfully developed [ 9 ] the first method to assess pain using changes in facial expression in any non-human animal species. Broadly speaking, GS quantify spontaneous pain according to objective and blinded scoring of facial expressions, as is done routinely for the measurement of pain in non-verbal humans. Observers score the presence and extent of "facial action units" (FAU), e.g. Orbital Tightening, Nose Bulge, Ear Position and Whisker Change. These are scored in real-time by observing the animal directly, or, post hoc from photographs or screen-grabs from videos.
This method of pain assessment is highly applicable to prey animals which tend to inhibit the overt expression of pain to prevent appearing vulnerable to predators. For this reason, behavioural changes in these species are mainly observed with acute pain (hours) but are less pronounced in longer-lasting pain (days). [ 1 ]
GS offer advantages over other methods of pain assessment. For example, the analgesic morphine reduces pain but can affect other aspects of behaviour in pain-free animals, for example, excitement, increased activity or sedation, which can hamper traditional behavioural assessment of its action on pain. Morphine not only reduces the frequency of "pain faces" but has no effect on GS in baseline, pain-free mice. [ 10 ]
The GS for mice usually consists of five FAU, i.e. Orbital Tightening, Nose Bulge, Cheek Bulge, Ear position and Whisker Change. These are scored on a 0-2 scale where 0=the criterion is absent, 1=moderately present and 2=obviously present. In mice, the GS offers a means of assessing post-operative pain that is as effective as manual behavioural-based scoring, without the limitations of such approaches.
Facial grimacing by mice after undergoing laparotomy surgery indicates postoperative pain lasts for 36 to 48 h (and at relatively high levels for 8 to 12 h) with relative exacerbation during the early dark (active) photo-phase. Furthermore, the grimacing indicates that buprenorphine is fully efficacious at recommended doses against early postoperative pain, but carprofen and ketoprofen are efficacious only at doses much higher than currently recommended: acetaminophen is not efficacious. [ 11 ]
A study in 2014 examined postoperative pain in mice following surgical induction of myocardial infarction . The effectiveness of the GS at identifying pain was compared with a traditional welfare scoring system based on behavioural, clinical and procedure-specific criteria. It was reported that post hoc GS (but not real-time GS) indicated a significant proportion of the mice were in low-level pain at 24 h which were not identified as such by traditional assessment methods. Importantly, those mice identified as experiencing low-level pain responded to analgesic treatment, indicating the traditional methods of welfare assessment were insensitive in this aspect of pain recognition. [ 1 ]
Mice with induced sickle cell disease and their controls exhibited a "pain face" when tested on a cold plate, but sickle mice showed increased intensity compared to controls; this was confirmed using Von Frey filaments a traditional method of pain assessment. [ 12 ] GS have also been used to assess pain and methods of its alleviation in pancreatitis. [ 13 ] GS have also been used to test the degree of pain caused as a side-effect of therapeutic drugs and methods of mitigating the pain. [ 14 ]
The mouse GS has been shown to be a highly accurate, repeatable and reliable means of assessing pain, requiring only a short period of training for the observer. [ 2 ] Assessment approaches that train deep neural networks to detect pain and no-pain images of mice may further speed up MGS scoring, with an accuracy of 94%. [ 15 ]
It has been noted that DBA/2 strain mice, but not CBA strain mice, show an increase in GS score following only isoflurane anaesthesia, which should be taken into account when using the GS to assess pain. Administration of a common analgesic, buprenorphine , had no effect on the GS of either strain. [ 16 ]
There are interactions between the sex and strain of mice in their GS and also the method that is used to collect the data (i.e. real-time or post hoc), which indicates scorers need to consider these factors. [ 2 ]
It is important to establish whether methods of pain assessment in laboratory animals are influenced by other factors, especially those which are a normal part of routine procedures or husbandry. There is no difference in GS scores between mice handled using a tube compared with mice picked up by the tail, indicating these handling techniques are not confounding factors in GS assessment. [ 17 ] A similar study reported there was no difference between GS scores at baseline and immediately post ear notching (a method frequently used to identify laboratory mice), potentially indicating that the pain associated with ear notching is either too acute to assess using the GS tool or the practice is not painful. [ 18 ]
There are differences between the "pain face" of mice and rats. In mice, the nose and cheek at baseline have a smooth appearance, but in the presence of pain, change to distinct bulges in both the nose and cheek regions. By contrast, in rats at baseline, the nose and cheek regions show distinct bulging, and with pain, the bridge of the nose flattens and elongates, causing the whisker pads to flatten. As a consequence of these differences, the GS for rats sometimes use four FAU, i.e. Orbital Tightening, Nose/Cheek Flattening, Ear Changes and Whisker Changes. Nose/Cheek Flattening appears to show the highest correlation with the presence of pain in the rat. [ 3 ] [ 19 ]
GS for rats has been used to assess pain due to surgery, orthodontic tooth movement, osteoarthritis, acute chemotherapy-induced mucositis, and the efficacy of analgesics for these procedures and other painful conditions. [ 19 ] [ 20 ] [ 21 ] [ 22 ] [ 23 ] [ 24 ] [ 25 ] Furthermore, GS have been used to examine the effects of postoperative analgesia on the reduction of post-operative cognitive dysfunction in aged rats. [ 26 ]
As with mice, studies have examined the extent of agreement in assessing pain between rat GS and the use of von Frey filaments. Good agreement has been found between these [ 27 ] in relation to three models of pain (intra plantar carrageenan , intraplantar complete Freund's adjuvant and plantar incision). The GS score significantly increased in all pain models and the peak GS score also coincided with the development of paw hypersensitivity, although hypersensitivity persisted after GS scores returned to baseline. [ 28 ]
For rats, software (Rodent Face Finder) has been developed which successfully automates the most labour-intensive step in the process of quantifying the GS, i.e. frame-grabbing individual face-containing frames from digital video, which is hindered by animals not looking directly at the camera or poor images due to motion blurring. [ 29 ]
A GS for rabbits using four FAU, i.e. Orbital Tightening, Cheek Flattening, Nose Shape, Whisker Position (Ear Position is excluded from the analysis) has been developed (for exemplar images, see here [1] ) and used to assess the effectiveness of an analgesic cream for rabbits having undergone ear- tattooing . [ 30 ] Similarly, a GS has been used to evaluate wellness in the post-procedural monitoring of rabbits. [ 31 ]
Based on the identification of FAU in rodents and rabbits, a GS for horses has been developed from post-operative ( castration ) individuals. This is based on six FAU, i.e. Stiffly Backwards Ears, Orbital Tightening, Tension Above the Eye Area, Prominent Strained Chewing Muscles, Mouth Strained and Pronounced Chin, Strained Nostrils and Flattening of the Profile (for exemplar images, see here. [2] ) [ 32 ] The HGS has thereafter been used to evaluate pain behavior in the laminitic horse, where it was concluded that the grimace scale can be used to assess the degree of pain also here, when compared to the Obel scale. [ 33 ]
A related study [ 34 ] describes the equine "pain face" after pain induction by a tourniquet on the antebrachium or topical capsaicin. The pain face here involves similar facial expressions described for the HGS; low and/or asymmetrical ears, an angled appearance of the eyes, a withdrawn and/or tense stare, medio-laterally dilated nostrils and tension of the lips, chin and certain mimetic muscles and can potentially be incorporated to improve existing pain evaluation tools. From the described pain face, The Equine Pain Scale has been developed. [ 35 ] Another pain scale has been described (EQUUS-FAP) which also has proven to assess acute pain in horses in a significant way. [ 36 ]
To map and explain the different facial expressions seen in the equine face during acute pain, an equine facial action coding system (EquiFACS) has been developed. Seventeen FAU have been identified and the involved anatomical structures behind each facial expression are explained and compared to facial expressions seen in other species. [ 37 ]
A preliminary study based on landmarks and distances between ears and in the muzzle demonstrated that observers shown facial images from painful and pain-free cats had difficulty in identifying pain-free from painful cats, with only 13% of observers being able to discriminate more than 80% of painful cats. Accuracy (based on a dichotomous judgement - pain or no pain) ranged from 18 to 94%. [ 38 ]
A complete GS (Feline Grimace Scale - FGS) for cats was published in 2019 to detect naturally-occurring acute pain. Five FAU were identified: ear position, orbital tightening, muzzle tension, whiskers change and head position. Each FAU receives a score from 0 to 2 and a total pain score is calculated as the sum of the FAU's scores divided by the total possible score excluding those AU marked as "not possible to score" (i.e. 4/10 = 0.4 or 4/8 = 0.5). A training manual is available as "Supplementary information" within the original article. [ 39 ]
The FGS has been thoroughly validated and reported high discriminative ability, good overall inter-rater reliability, excellent intra-rater reliability, and excellent internal consistency. The FGS scores were higher in painful than in control cats; a very strong correlation with another validated instrument for pain assessment in cats was observed and the FGS detected response to analgesic treatment (scores after analgesia were lower than before). Additionally, an analgesic threshold was determined (total pain score > 0.39 out of 1.0). The FGS is an easy and quick-to-use tool for acute pain assessment in cats. [ 39 ]
The clinical applicability of the FGS in cats undergoing ovariohysterectomy has been explored by comparing the scores assigned in real-time by an experienced observer with those scores assigned to still images, and good agreement has been reported. [ 40 ] The FGS is also a reliable tool for pain assessment in cats undergoing dental extractions and the caregiver's presence did not affect FGS scores. [ 41 ]
A GS for sheep has been developed to detect pain caused by naturally occurring diseases such as footrot and mastitis . [ 42 ] A GS has been used to assess pain due to the routine husbandry procedure of tail-docking in lambs. There was high reliability between and within the observers, and high accuracy. Restraint of the lambs during the tail-docking caused changes in facial expression, which needs to be taken into account in use of the GS. [ 43 ]
Facial musculature of ferrets and compared lateral photographs of ferret faces were studied before and after intraperitoneal telemetry probe implantation. The FAU orbital tightening , nose bulging , cheek bulging , ear changes and whisker retraction were identified as potential indicators of pain in ferrets. All AU-scores assigned to the photographs taken five hours after surgery were significantly higher compared to their time-matched baseline scores. Further analysis using weights that were obtained using a Linear Discriminant Analysis revealed that scoring orbital tightening alone was sufficient to make this distinction with high sensitivity, specificity and accuracy. [ 44 ] | https://en.wikipedia.org/wiki/Grimace_scale |
Grimm's Hydride Displacement Law is an early hypothesis, formulated in 1925, to describe bioisosterism , the ability of certain chemical groups to function as or mimic other chemical groups. [1] [2]
According to Grimm, each vertical column (of Table below) would represent a group of isosteres .
This article about medicinal chemistry is a stub . You can help Wikipedia by expanding it . | https://en.wikipedia.org/wiki/Grimm's_hydride_displacement_law |
In chemistry , the Grimm– Sommerfeld rule [ 1 ] predicts that binary compounds with covalent character that have an average of 4 electrons per atom will have structures where both atoms are tetrahedrally coordinated (e.g. have the wurtzite structure). Examples are silicon carbide , the III-V semiconductors indium phosphide and gallium arsenide , the II-VI semiconductors , cadmium sulfide , cadmium selenide .
Gorynova expanded the scope of the rules to include ternary compounds where the average number of valence electrons per atom was four. Examples of this are the I-IV 2 -V 3 CuGe 2 P 3 compound which has a zincblende structure. [ 2 ]
Compounds or phases that obey the Grimm–Sommerfeld rule are termed Grimm–Sommerfeld compounds or phases. [ 3 ]
The rule has also been extended to predict bond lengths in Grimm–Sommerfeld compounds. When the sum of the atomic numbers is the same the bond lengths are the same. [ 4 ] An example is the series of bond lengths ranging from 244.7 pm to 246 pm. for the Ge–Ge bond in elemental germanium , the Ga–As bond in gallium arsenide , the Zn–Se bond in zinc selenide and the Cu–Br bond in copper(I) bromide . [ 4 ]
This quantum chemistry -related article is a stub . You can help Wikipedia by expanding it . | https://en.wikipedia.org/wiki/Grimm–Sommerfeld_rule |
A Hegman gauge , sometimes referred to as a grind gauge , grind gage , or grindometer , is an instrument which indicates the fineness of grind or the presence of coarse particles and agglomeration in a dispersion . [ 1 ] It is commonly used to determine how finely ground the particles of pigment (or other solid) dispersed in a sample of paint (or other liquid) are. This is important because many types of solid materials must be ground into finer particles in order to be dispersed in liquids. [ 2 ] The resulting properties of the dispersion vary based on the size of individual particles and the degree which they are dispersed.
The Hegman gauge usually consists of a stainless steel block with a series of very small parallel grooves machined into it. The grooves decrease in depth from one end of the block to the other, according to a scale stamped next to them. A typical Hegman gauge is 170mm by 65mm by 15mm, with a channel of grooves running lengthwise, 12.5mm across and narrowing uniformly in depth from 100 μm to zero and used to determine particle size . [ 3 ]
A Hegman gauge is used by placing a sample of paint at the deep end of the gauge and drawing the paint down with a flat edge along the grooves. Grind gages are sold with machined flat 'drawdown bars' specifically for this purpose. The paint fills the grooves, and the location where a regular, significant "pepperyness" in the appearance of the coating appears, marks the coarsest-ground dispersed particles. [ 3 ] This is the point where oversized particles start to appear in high density and determines the rating for that material. [ 4 ] The reading is taken from the scale marked next to the grooves, in dimensionless "Hegman units" (or National Standard units; NS) and/or mils or micrometres . [ 5 ] Hegman units are defined in terms of an inverted size scale as shown below: [ 6 ]
A lesser-used scale, North (or PCU), is also occasionally employed in the paint industry. Like the Hegman scale, this is also inverted compared to the value in microns:
Determining the fineness of a paint's grind is important, because too coarse a grind may reduce the paint's color uniformity, gloss , and opacity. [ 7 ] The Hegman gauge is widely used for this purpose because it requires minimal skill and only a few seconds' work. [ 3 ]
Grind gauges are used in a variety of fields, including; food, pharmaceutical, plastic and many others. In all of these fields, grind gauges are utilized to produce, store, and apply dispersion products.
Hegman gauges are commonly available in the following ranges: 0 to 100 micrometres, 0 to 50 micrometres, 0 to 25 micrometres, 0 to 15 micrometres, and 0 to 10 micrometres.
[ 1 ] | https://en.wikipedia.org/wiki/Grind_gage |
Grindhouse Wetware is an open source biotechnology startup company based in Pittsburgh, Pennsylvania . Grindhouse applies the biohacker ethic to create technology that augments human capabilities. The company is most well known for their Circadia device, a wireless biometric sensor that was implanted into co-founder Tim Cannon on the 22 October 2013. [ 1 ] Grindhouse has been featured in television shows such as Taboo on National Geographic Channel , [ 2 ] Joe Rogan Questions Everything on Syfy , [ 3 ] The Big Picture with Kal Penn , [ 4 ] [ 5 ] as well as podcasts including Future Grind [ 6 ] and Roderick Russell's Remarkably Human. [ 7 ]
In November 2015, Grindhouse members Tim Cannon, Shawn Sarver, Justin Worst, Jessica Waldrip, Michael Seeler, and Marlo Webber had prototypes of Grindhouse's Northstar device implanted into their hands during simultaneous procedures occurring at the "Cyborg Fair" in Düsseldorf, Germany , and at a studio in Pittsburgh. The implantation procedure was featured in an episode of the MTV documentary series True Life . Also featured in the episode was the public debut of the morse code functionality of Grindhouse's Bottlenose device at the first Pittsburgh Maker Faire . [ 8 ] | https://en.wikipedia.org/wiki/Grindhouse_Wetware |
A grindometer is a device used to measure the particle size of suspensions , typically inks such as those used in printing, or paints. It consists of a steel block with a channel of varying depth machined into it, starting at a convenient depth for the type of suspension to be measured, and becoming shallower until it ends flush with the block's surface. The depth of the groove is marked off on a graduated scale next to it. The suspension to be tested is poured into the deep end of the groove, and scraped towards the shallow end with a flat metal scraper. At the point where the depth of the groove equals the largest particles in the suspension, irregularities (for example pinholes in an ink sample) will become visible.
The advantages of this method are that it uses a small sample and gives a very quick indication of the high end of the particle size distribution, allowing production processes to be followed in real time.
The following standards are relevant on conjunction with the use of grindometers: ASTM D 1210, ASTM D 1316, JIS K 5600-2-5, ISO 1524, EN ISO 1524, BS 3900-C6
This chemistry -related article is a stub . You can help Wikipedia by expanding it . | https://en.wikipedia.org/wiki/Grindometer |
In percussion , grip refers to the manner in which the player holds the sticks or mallets , whether drum sticks or other mallets.
For some instruments, such as triangles and gongs , only a single mallet or beater is normally used, held either in one hand or both for larger beaters. For others, such as snare drums , two beaters are often used, one in each hand. More rarely, more than one beater may be held in one hand. For example, when four mallets are used on a vibraphone , or when a kit drummer performs a cymbal roll by holding two soft sticks in one hand while keeping a rhythm with the other.
When two identical beaters are used, one in each hand, there are two main varieties of grip:
Traditional grip (also known as orthodox grip or conventional grip , fundamental grip and, to a lesser extent, the jazz grip ) is a technique used to hold drum sticks while playing percussion instruments . Unlike matched grip , each hand holds the stick differently. Commonly, the right hand uses an overhand grip and the left hand uses an underhand grip . Traditional grip is almost exclusively used to play the snare drum , especially the marching snare drum , and the drum kit . Traditional grip is more popular in jazz drumming than in other drum kit styles due to the early jazz drummers evolving their style from marching and military styles and instrumentation, [ 1 ] although it is also used by several rock drummers. Virgil Donati and Nick Pierce are two examples of drummers who play metal using the traditional grip.
This grip is called traditional because it originates from military marching drummers who carried a snare drum on a sling hung from the neck or one shoulder, with the drum riding closer to one hip than the other and tilted slightly for easier reach. This allowed the drummer to play the drum and march without banging their knees or thighs into the drum. Due to that drum position, using an overhand grip on the high (left) side of the drum would force the elbow into a very awkward position while an underhand grip is much more comfortable. Even when the drum is on a stand, many drummers will tilt their drum when using traditional grip. Although tilting is not required, it helps align the shoulders and spine thus being more ergonomic. Many drummers use traditional grip on drums that are perfectly horizontal, especially in marching percussion .
With the underhand grip, there are several different techniques employed which involve slight variations in finger positioning and usage. Common with all techniques is the usage of the wrist in rotating (a motion like turning a door knob) as the fundamental motion of the stick. Once the stick has started moving, more involved techniques require the exclusive use of the thumb for bouncing the stick when playing at a faster tempo. The stick then rests in the space between the thumb and index finger, and the two fingers close around the stick with the thumb atop the index at the first knuckle. The middle finger then rests slightly on the top side of the stick (typically the side fingertip is the only contact made). The stick then rests on the cuticle of the ring finger with the little finger supporting the ring finger from below.
Sanford A. Moeller (whose book discusses the Moeller method or Moeller technique) suggests that one should learn the traditional grip 'ancient style', as well ... where the overhand grip should hold or grip the drumstick almost entirely with the little finger. [ 2 ]
Some Scottish pipe band players have a variation on the traditional left hand grip in which the underhand grip is played entirely with the thumb [ 3 ] on top of the stick, [ 4 ] utilizing no other fingers for downward pressure. [ 5 ] This suits the pipe band's light and snappy style of playing well, but is not as suitable for American style drum corps playing or jazz drumming on a full kit.
Traditional grip can also be useful when playing with brushes in a stirring motion. Normally this style is used in a jazz context. The underhand grip naturally angles the left hand further away from the right hand than would matched grip and allows more room for crossovers and sweeping maneuvers across the surface of the drum.
Physiologically, the traditional left underhand grip uses fewer muscles than the right overhand grip and this causes each muscle to do a larger percentage of the work. [ 6 ] Matched grip is therefore technically easier to play, though for reasons stated above, it is not always the superior choice for every application.
Matched grip (also known as parallel grip ) is a method of holding drum sticks and mallets to play percussion instruments . In the matched grip, each hand holds the stick in the same way, whereas in the traditional grip , each hand holds the stick differently. Almost all commonly used matched grips are overhand grips. Specific forms of the grip are French grip, German grip, and American grip.
The matched grip is performed by gripping the drum sticks with one's index finger and middle finger curling around the bottom of the stick and the thumb on the top. This allows the stick to move freely and bounce after striking a percussion instrument. Any of the major grips below can be played with an index finger fulcrum, a middle finger fulcrum, or a combination of both. The fulcrum can also be placed on the first or second knuckles of the primary fulcrum finger. These options lead to many technical variations in playing position. All of the grips, with all of the fulcrum variations, apply to the right hand of traditional grip as well.
In French grip, the palms of the hands face directly toward each other and the stick is moved primarily with the fingers rather than the wrist as in German grip. This allows a greater degree of finesse and the addition of forearm rotation to the stroke, which is why many timpanists prefer French grip. This grip uses the smaller and faster finger muscles. It also comes in handy for playing fast tempos, including for swing or jazz on the ride cymbal. For louder strokes, the wrist rotates much in the same way as when hammering a nail.
In German grip, the palms of the hands are parallel to the drumhead or other playing surface, and the stick is moved primarily with the wrist. German grip provides a large amount of power, but sacrifices the speed provided by the use of the fingers as in French grip. It is used when power is the primary concern, such as when playing a bass drum . This is also the primary grip for the Moeller method . German grip provides a wide dynamic range, achieving the control necessary for pianissimo passages without the need for much rebound from the drum and also allowing for very loud fortissimo strokes from the arm.
American grip is a hybrid of the French grip and German grip. The palms of the hands typically are at about a 45-degree angle to the drum and both the fingers and wrist are used to move the stick. This grip is considered a general-purpose grip by percussionists because it combines the power and larger wrist motion of the German grip with the quick finger strokes of the French grip. Each element of the stroke, finger or wrist motion, can be isolated as needed. It is widely used on membranophone instruments ( drums ).
Single-beater grips are common for:
Unmatched grips are common for:
Matched grips are common for: | https://en.wikipedia.org/wiki/Grip_(percussion) |
The grit, not grass hypothesis is an evolutionary hypothesis that explains the evolution of high-crowned teeth , particularly in New World mammals. The hypothesis is that the ingestion of gritty soil is the primary driver of hypsodont tooth development, not the silica -rich composition of grass , as was previously thought. [ 1 ]
Since the morphology of the hypsodont tooth is suited to a more abrasive diet, hypsodonty was thought to have evolved concurrently with the spread of grasslands . During the Cretaceous Period (145-66 million years ago), the Great Plains were covered by a shallow inland sea called the Western Interior Seaway which began to recede during the Late Cretaceous to the Paleocene (65-55 million years ago), leaving behind thick marine deposits and relatively flat terrain. During the Miocene and Pliocene epochs (25 million years), the continental climate became favorable to the evolution of grasslands. Existing forest biomes declined and grasslands became much more widespread. The grasslands provided a new niche for mammals, including many ungulates that switched from browsing diets to grazing diets. Grass contains silica-rich phytoliths (abrasive granules), which wear away dental tissue more quickly. So the spread of grasslands was linked to the development of high-crowned (hypsodont) teeth in grazers. [ 2 ]
In 2006 Strömberg examined the independent acquisition of high-crowned cheek teeth (hypsodonty) in several ungulate lineages (e.g., camelids , equids , rhinoceroses) from the early to middle Miocene of North America, which had been classically linked to the spread of grasslands. She showed habitats dominated by C3 grasses (cool-season grasses) were established in the Central Great Plains by early late Arikareean (≥21.9 Million years ago), at least 4 million years prior to the emergence of hypsodonty in Equidae . [ 3 ] In 2008 Mendoza and Palmqvist determined the relative importance of grass consumption and open habitat foraging in the development of hypsodont teeth using a dataset of 134 species of artiodactyls and perissodactyls . The results suggested that high-crowned teeth represent are adapted for a particular feeding environment, not diet preference. [ 4 ]
More recent examination of mammalian teeth suggests that it is the open, gritty habitat and not the grass itself which is linked to diet changes. [ 1 ] [ 4 ] [ 5 ] Analysis of dental microwear patterns of hypsodont notoungulates from the Late Oligocene Salla Beds of Bolivia showed shearing movements are associated with a diet rich in tough plants, not necessarily grasses. Hence the relationship between high-crowned mammals and the source of tooth wear in the fossil record may not be straightforward and the spread of grasslands in South America, traditionally linked with the development of notoungulate hypsodonty, was called into question. [ 5 ]
Most importantly, evidence has shown, that the development of hypsodonty in Cenozoic mammals is out of sync with the flourishing of grasslands both in North America and South America, where grasslands spread 10 million years earlier. [ 1 ] [ 5 ] Observations of this temporal discontinuity between the spread grasslands and the development of hypsodonty in mammals is also supported by earlier evidence of hypsodonty in dinosaurs. For example, hadrosaurs , a group of herbivorous dinosaurs , likely grazed on low-lying vegetation and microwear patterns show that their diet contained an abrasive material, such as grit or silica. Grasses had evolved by the Late Cretaceous, but were not particularly common, so this study concluded that grass probably did not play a major component in the hadrosaur's diet. [ 6 ]
Hypsodonty is observed both in the fossil record and the modern world. It is a characteristic of large clades (equids) as well as subspecies level specialization. For example, the Sumatran rhinoceros and the Javan rhinoceros both have brachydont , lophodont cheek teeth whereas the Indian rhinoceros , has hypsodont dentition. A mammal may have exclusively hypsodont molars or have a mix of dentitions. Hypsodont dentition is characterized by: [ 7 ] [ 8 ] | https://en.wikipedia.org/wiki/Grit,_not_grass_hypothesis |
Grizzly 399 (1996 – October 22, 2024) [ 1 ] was a grizzly bear living in Grand Teton National Park and Bridger-Teton National Forest in Wyoming , United States. [ 2 ] She was followed by as many as 40 wildlife photographers, [ 3 ] [ 4 ] and millions of tourists came to the Greater Yellowstone Ecosystem to see her and other grizzly bears. [ 5 ] [ 6 ] There are official Facebook, Twitter, and Instagram accounts for Grizzly 399. [ 7 ] [ 8 ]
Grizzly bears ( Ursus arctos horribilis ) are a subspecies of the North American Brown bear species U. arctos . [ 9 ] [ 10 ] Several decades ago, grizzlies were assessed as being at risk of rapid extinction due to the rate at which the population was declining. Protection under the Endangered Species Act of 1973 has resulted in a population rebound: there are now approximately 2,000 grizzly bears in the contiguous United States, [ 11 ] of which about half are estimated to live in the Greater Yellowstone Ecosystem. Grizzlies are stereotyped as ferocious, but the typical bear avoids contact with humans, living away from settlements and attacking only to protect themselves when startled by a human. [ 5 ] [ 12 ]
Grizzly 399 was a grizzly bear who resided on federal land in a range of hundreds of miles throughout the Grand Teton National Park and the Bridger-Teton National Forest . She was born in a den in Pilgrim Creek, Wyoming, in the winter of 1996. [ 2 ] She was captured in 2001 and fitted with a radio collar by the Interagency Grizzly Bear Study Team. She was the 399th bear to be tracked with this method as part of the long-term research project. In 2018, monitoring of 399 via radio telemetry ceased, with the research continuing as she resided in an area where she was easily observable. [ 13 ]
399 reached age 28, becoming older than is usual for a grizzly bear, as "more than 85 percent of them are killed because of some kind of human activity before they reach old age". [ 5 ] She weighed almost 400 pounds (180 kg). When standing upright on her hind legs, she was 7 ft 0 in (2.13 m). [ 2 ] Unlike the typical grizzly, she lived in close proximity to humans, although she was not particularly concerned with their presence; scientists have speculated that this was in response to a death of a cub in a more remote area, so she wanted to avoid that area. She never killed a human despite at least two known close encounters. [ 5 ]
Grizzly 399 successfully reared many progeny, including 22 cubs and grandcubs. [ 2 ] In mid-May 2020 she was observed with four new cubs born the previous winter. [ 14 ] She taught her offspring habits to benefit from rather than be harmed by human proximity, such as loitering during the fall elk hunt to consume abandoned elk innards and looking both ways before crossing roadways to avoid being struck by vehicles, a common cause of death among bears. [ 5 ]
Despite this, at least three of her cubs were killed due to human encounters, including Grizzly 399's only 2016 cub, nicknamed "Snowy" because of his whitish-blonde facial coloration. [ 15 ] In June of that year, Snowy was struck and killed by a car in Grand Teton National Park, an incident investigated as a potential hit-and-run accident. [ 16 ] [ 17 ] In all, she lost half of her descendants due to encounters with people or male bears. [ 18 ]
On May 21, 2020, a wildlife photographer saw Grizzly 399 coming out of hibernation in Pilgrim Creek with four cubs. This was her largest brood to date. [ 19 ] On May 16, 2023, Grizzly 399 emerged from hibernation and appeared in the area of Pilgrim Creek in Grand Teton National Park. She was seen with a single cub. [ 20 ] At age 26 or 27, this made her the oldest female bear known to have reproduced in the Greater Yellowstone Ecosystem. [ 20 ]
Unlike the typical bear, Grizzly 399 regularly gave birth to triplets rather than twins. This typically has a paradoxical effect on the bear population. A mother bear with three cubs expends significantly more energy in caring for them, which can potentially decrease rather than increase the survival rate. Grizzly 399, conversely, typically handled triplets well. [ 5 ] One of her triplet cubs also grew to be a prolific mother and was tagged for research as Grizzly 610. [ 2 ] [ 15 ] In 2011, Grizzly 610 had twins, while Grizzly 399 had another set of triplets. The scientists observing the bears were concerned due to 399's advanced age, but to their surprise Grizzly 610 amicably adopted one of her mother's triplet cubs. [ 5 ]
One of 399's 2017 twin cubs, numbered 964, was relocated to Yellowstone in 2019. She was spotted with twins in 2023. Grizzly 610's daughter, numbered 926, released twins in 2023. [ citation needed ]
Grizzly 399 was known to be habituated to people when near roads and lightly developed areas. A researcher determined that she sought out these roadside areas rather than backcountry because it was safer for her cubs, which male bears often attempted to kill. [ 15 ] [ 18 ] The fact that she spent so much time near roads also contributed to her popularity. In 2011, the sight of a mother grizzly bear and her three cubs near a road in central Grand Teton National Park was enough to cause traffic to come to a halt in both directions for miles. In Willow Flats, Grizzly 399 taught each set of cubs to hunt elk calves, within view of the guests at Jackson Lake Lodge . [ 21 ] [ 18 ]
Grizzly 399 was usually found along the roadside near the Oxbow Bend of the Snake River . The number of photographers following her grew to approximately 40–50 by of 2015. 399 was considered the "grand matriarch of the park's roadside bears." [ 3 ]
In 2016, Grizzly 399 was feared dead after a hunter claimed to have killed her; however, she emerged from hibernation on May 10, 2016, with one cub in tow. She emerged from the Bridger-Teton National Forest into the Grand Teton National Park with a white-faced cub at her side. [ 22 ] In 2017, now older than the age beyond which most brown bears usually breed, she was spotted in a spring snowstorm with two cubs following her. [ 23 ] [ 10 ]
On the evening of October 22, 2024, Grizzly 399 was fatally struck by a vehicle on Highway 26/89 in Snake River Canyon , south of Jackson, Wyoming . The bear's identity was confirmed through ear tags and a microchip. [ 24 ] Most of her ashes were scattered in Grand Teton National Park. [ 25 ] Static Peak was renamed to Peak 399 in her honor as of May 1, 2025. [ 26 ]
Created in 2007 in response to the magnitude of visitors coming to Grand Teton to view Grizzly 399 and her cubs, the Grand Teton Wildlife Brigade keeps animals and people apart and safe. In 2011, ranger Kate Wilmot, whose official title is "bear management specialist", said that that year things had become "completely chaotic". The real duty was managing the behavior of park visitors. This was partly due to social media increasing the popularity of the bears, and drawing more people to seek them out. [ 21 ]
Wilmot directs 16 volunteers in the brigade throughout the summer until snowfall. If not for the brigaders, "wildlife watching would be a mess". The brigaders carry bear spray, but their primary role is to persuade tourists to respect the 100-yard viewing guideline established after incidents with Grizzly 610, 399's daughter. [ 21 ]
Feeding the bears is illegal, so the brigadiers prevent this. If bears receive food from people, they can become habituated to people and more aggressive toward them. The brigadiers remind tourists of their role in respecting bears' space. The brigadiers' success can be measured in the rarity of major incidents and bear removals from the park. When bears become too habituated to human presence and aggressive in their pursuit of human food, or when a bear attacks a human, the "problem bear" is typically euthanized. Grizzly mothers are known for being aggressively protective of their progeny . In 2011, in Yellowstone National Park , a mother bear fatally mauled a hiker who got too close. Grizzly 610, 399's daughter, twice "charged" tourists who got too close. No injuries were reported. [ 21 ]
In 2017, the United States Fish and Wildlife Service officials removed grizzly bears from the endangered species list and turned management of grizzlies outside Yellowstone and Grand Teton National Parks over to Wyoming, Montana , and Idaho . [ 27 ] Grizzlies live in ranges covering hundreds of miles, which can place them outside of parks, where they become the targets of hunters. [ 28 ] Grizzly 399 lived outside of the parks. [ 29 ]
Hunters in the area targeted 399 because she was the biggest and most famous trophy. [ 7 ] Daryl Hunter, a wildlife photographer who followed Grizzly 399, related a conversation with an outfitter who said, "I met a guy who wants Grizzly 399's rug on his wall, stating that because she is famous, she makes a better trophy". [ 22 ] Grizzly 399 spent part of the year in Grand Teton National Park, but also hibernated in the national forest where hunting is allowed. [ 30 ]
For the 2018 hunting season, Montana decided against a hunt. Idaho, with the fewest grizzlies, decided to allow hunting of only one bear. On May 23, 2018, the Wyoming wildlife commission voted unanimously to approve a grizzly bear hunt. [ 29 ] The Wyoming Game and Fish Department let a vote decide the number of grizzlies to be killed. The tally came to 22 grizzlies in a unanimous vote of 7–0. [ 31 ] The hunting season was planned for September 15 to November 15. This was to be the first authorized hunt in Wyoming in 44 years - since 1975 - a time when they were first listed as endangered, when no hunting was allowed inside the national parks or on the connecting road between them, [ 29 ] and when the grizzly population had fallen to around 136 individuals. [ 31 ]
Wyoming's planned hunt met with a public outcry. Five women in Jackson Hole quickly organized "Shoot'em With A Camera-Not A Gun", which encouraged opponents of trophy hunting to join the tag lottery in hopes of preventing hunters from winning tags. [ 29 ] Approximately 7,000 people applied for Wyoming bear tags, including wildlife photographer Thomas D. Mangelsen , Jane Goodall , and other conservationists. [ 31 ]
In July 2018, Mangelsen learned he was positioned high enough on a hunting lottery to actually receive a hunting tag, as he held slot number 8 in the queue. [ 31 ] In September, just weeks before hunting season was to begin, a federal judge in Montana restored protection to all of the bears in the Greater Yellowstone Ecosystem. The judge ruled that the United States Fish and Wildlife Service officials were "arbitrary and capricious" when they removed protection from the bears under the Endangered Species Act of 1973 . [ 32 ] In July 2020, the Ninth Circuit Court of Appeals upheld the Montana judge's ruling. [ 33 ]
In March 2021, the U.S. Fish and Wildlife Service recommended no change to the protection status of the grizzly bear in the lower-48 states . According to the ESA after a five-year status review, they remained threatened . [ 34 ]
Grizzly 399: The Story of a Remarkable Bear is a children's book published in May 2020 in Idaho Falls. [ 35 ] The book is written by Sylvia M. Medina, illustrated by Morgan Spicer and includes photographs by American nature and wildlife photographer, Thomas D. Mangelsen. [ 36 ] The publisher published a subsequent book with the same author, illustrator and photographer in April 2021 to include Grizzly 399's new cubs, titled, "Grizzly 399's Hibernation Pandemonium" after the 24-year-old mother bear surprised the world with the birth of four more cubs in the spring of 2020. [ 37 ] [ 38 ]
Grizzlies of Pilgrim Creek In 2015, Thomas D. Mangelsen collaborated with Wilkinson to create the book about Grizzly 399 and her progeny . [ 39 ] Mangelsen made it one of his priorities for over ten years to record her life, including her hibernation schedule, feeding, and mothering; he recorded the birth of three sets of triplets and a set of twins. His photographs, especially the one he dubbed, "An Icon of Motherhood", helped make her the most famous mother grizzly, maybe the most famous grizzly, in the world. [ 7 ] Millions of people visit the Greater Yellowstone Ecosystem just to see these grizzly bears. [ 5 ] [ 40 ]
By 2015, Grizzly 399 had a full social media presence, although it was a mystery who is running the accounts. She had her own Facebook page, Instagram account, and a Twitter handle. [ 18 ] "These aren't just any bears", explained Thomas D. Mangelsen, a global wildlife photographer who lives in Jackson Hole, Wyoming , "They might be the most famous grizzlies alive today on the planet. For all these people, catching a glimpse of them is the thrill of a lifetime." [ 21 ] Mangelsen followed her movements for over ten years. Grizzly 399 dispelled the stereotype that all grizzlies are agents of terror, wrote Bozeman author Todd Wilkinson: "She's more well-behaved a lot of times than people around her. But she's wild." [ 18 ] | https://en.wikipedia.org/wiki/Grizzly_399 |
The Grob-Werke GmbH & Co. KG (stylized as GROB-WERKE) is the parent company of the Grob Group. The family-owned business operates in the field of universal machines, assembly lines, system solutions [ clarification needed ] , electromobility, additive manufacturing, automation, and digitalization. Grob-Werke is headquartered in Mindelheim , located in Upper Swabia in the Unterallgäu district.
Grob-Werke was founded in 1926 by Ernst Grob in Munich . [ 1 ] Initially, the company focused on producing machine tools. From 1935 onwards, it shifted its attention to internal combustion engines, [ 2 ] [ 3 ] which made the operation "crucial" during World War II. [ 4 ] Between 50 and 100 prisoners of war were employed as forced labourers , housed in a separate camp on the company premises at Hofmannstraße 50. [ 5 ] In 1944, the factory was extensively destroyed during Allied air raids on Munich, but it was rebuilt in 1945. [ 6 ] Grob-Werke is one of the contributors to the Foundation Remembrance, Responsibility and Future , which compensates forced labourers.
Burkhart Grob (* 26 March 1926; † 20 May 2016) led the company from 1952. [ 7 ] [ 8 ] In 1956, the first overseas plant was established in São Paulo , Brazil. [ 9 ]
The main factory was established in Mindelheim in 1968, where production of transfer lines began. [ 1 ] [ 10 ] By 1976, the Munich facility had been shut down, and the company headquarters were relocated to Mindelheim. [ 6 ] In 1971, Burkhart Grob Luft- und Raumfahrt GmbH & Co. KG was founded, and Grob began manufacturing gliders . [ 11 ] Grob received its largest order to date in 1998 from the Royal Air Force to deliver a total of 85 G-115 D training sports planes within two years. [ 12 ] That same year, Grob also secured a major contract from General Motors and BMV Rover. Grob was awarded contracts worth DM 320 million for the machining and assembly of cylinder heads and engine blocks. [ 13 ]
In subsequent years, Grob invested in new facilities and its own branches in Brazil and the United States. [ 13 ] In 2003, Grob began establishing service and sales branches in Beijing and Shanghai. [ 1 ] Until 2006, Grob-Werke consisted of independent business units of machine tools and aerospace . The aerospace division was sold in 2006 and operated as Grob Aerospace AG afterwards. In 2009, it was renamed Grob Aircraft . [ 1 ] [ 11 ] Burghardt Grob passed away in 2016, and Christian Grob took leadership of the group in the third generation as the new chairman of the supervisory board. [ 11 ] In 2017, Grob acquired the Italian electromotor machine and equipment manufacturer DMG meccanica in Turin , marking its entry into electromobility. [ 6 ]
In 2021, Grob entered into a strategic partnership with Manz AG in the field of lithium-ion battery systems. [ 14 ] Due to the shift towards electromobility, there was an increased demand for battery production technology, and the collaboration enabled Grob to offer the entire production process of lithium-ion battery cells and modules. [ 15 ] [ 16 ] In 2022, Dürr AG joined the partnership. [ 16 ] [ 17 ] [ 18 ]
Grob also initiated an expansion at its headquarters in Mindelheim. With an investment of €19 million, construction of a new hall began, [ 19 ] [ 20 ] while existing roofs have been increasingly reinforced and equipped with photovoltaic modules since 2023. [ 21 ] Grob cites the focus on renewable energies as one of the reasons for these expansions at the headquarters, for which a third energy center based on heat pumps and biomass for sustainable heat supply was also planned in 2023. [ 19 ] Internationally, Grob expanded its presence with new locations in Bangalore , India, and Bluffton, Ohio . [ 22 ] [ 23 ]
The Grob-Werke GmbH & Co. KG is the parent company of the Grob Group. In the fiscal year 2022/23, Grob employed 8,085 people and generated a revenue of over €1.37 billion. [ 24 ] Nearly half of the turnover is generated by machinery for electric drives and battery storage technologies. [ 25 ] The Mindelheim site has been the main plant of Grob-Werke since 1976 and serves as its headquarters. The Mindelheim facility has a production area of more than 199,000 square meters. [ 26 ] In addition to the main plant in Mindelheim, the company has production facilities in Bluffton ( Ohio ), Dalian (China), São Paulo (Brazil), Pianezza (Italy), [ 1 ] [ 10 ] and Bangalore (India). [ 27 ]
Grob-Werke operates in the field of universal machines, assembly lines , system applications, electromobility , additive manufacturing, automation , and digitalization. The group primarily supplies its machine tools to the automotive industry , [ 1 ] which is why Grob's products and services are aligned with developments in the automotive sector. [ 28 ] [ 29 ] [ 30 ]
Grob-Werke designs and manufactures machining centers and assembly facilities, digitalization and automation applications, as well as linking systems for production and assembly lines. [ 13 ] [ 31 ] Here, Grob is particularly active in the planning and construction of complete manufacturing systems for engines, vehicle transmissions, injection pumps, and similar components. [ 32 ] The universal machines are used in various sectors, including the automotive industry, energy and medical technology, aerospace, as well as machinery, tooling, and mould making. The 4- and 5-axis universal machines, for example, can machine or manufacture delicate components. [ 33 ] Another mainstay is the construction of special machines for transfer lines. [ 32 ] Among the assembly systems distributed by Grob-Werke are systems for stator production using hairpin technology, rotor production, and battery module assembly, which enable fully automated manufacturing for vehicle drives. [ 34 ]
Grob-Werke offers various machining concepts used by the automotive industry. [ 9 ] These systems consist of individually modular machining centers as well as custom machines and are customisable. [ 35 ] For example, the G520F of the F-Series, introduced in 2023, is designed for the machining of battery housings and lightweight components such as frame structures and chassis parts. [ 36 ]
Grob entered into the development of electromobility and adapted the company structure accordingly. [ 25 ] [ 34 ] [ 37 ] This division primarily develops and produces products for battery technology and electric drive trains, including battery pack systems, battery module systems and large-scale systems for battery cell production. [ 38 ] [ 28 ] Other areas include the complete assembly of electric motors as well as stator and rotor assembly with permanent magnet technology. [ 28 ] The company also develops individual processes and systems, for example for processing structural and chassis parts or coating engine components. [ 39 ]
The electric mobility sector accounts for over 60% of Grob's business. Customers include European, American, and Asian car manufacturers, with a predominant presence of emerging Chinese manufacturers in the Asian market. [ 22 ]
In the field of additive manufacturing, Grob developed the liquid metal printing (LMP) manufacturing process, [ 40 ] which can be used to produce customised near-net-shape aluminium components. This is intended to compensate for the disadvantages of powder-based additive manufacturing, which was often used previously. [ 28 ] [ 41 ] [ 42 ]
The digitalization area includes software developed by Grob-Werke that digitally links all areas of Grob machine tool production and allows the individual modules to be organised. [ 36 ] | https://en.wikipedia.org/wiki/Grob-Werke |
A Grob fragmentation is an elimination reaction that breaks a neutral aliphatic chain into three fragments: a positive ion spanning atoms 1 and 2 (the " electrofuge "), an unsaturated neutral fragment spanning positions 3 and 4, and a negative ion (the " nucleofuge ") comprising the rest of the chain. [ 1 ] [ 2 ] [ 3 ]
For example, the positive ion may be a carbenium , carbonium or acylium ion ; the neutral fragment could be an alkene , alkyne , or imine ; and the negative fragment could be a tosyl or hydroxyl ion:
The reaction is named for the Swiss chemist Cyril A. Grob [ de ] .
Alternately, atom 1 could begin as an anion, in which case it becomes neutral rather than going from neutral to cationic.
An early instance of fragmentation is the dehydration of di( tert -butyl)methanol yielding 2-methyl-2-butene and isobutene , a reaction described in 1933 by Frank C. Whitmore . [ 4 ] This reaction proceeds by formation of a secondary carbocation followed by a rearrangement reaction to a more stable tertiary carbocation and elimination of a t -butyl cation:
Albert Eschenmoser in 1952 investigated the base catalysed fragmentation of certain beta hydroxy ketones : [ 5 ]
The original work by Grob (1955) concerns the formation of 1,5-hexadiene from cis - or trans -1,4-dibromocyclohexane by sodium metal: [ 1 ]
According to reviewers Prantz and Mulzer (2010), the name Grob fragmentation was chosen "in more or less glaring disregard of the earlier contributions". [ 6 ]
The reaction mechanism varies with reactant and reaction conditions with the fragmentation taking place in a concerted reaction or taking place in two steps with a carbocationic intermediate when the nucleofuge leaves first or taking place in two steps with an anionic intermediate when the electrofuge leaves first. The carbanionic pathway is more common and is facilitated by the stability of the cation formed and the leaving group ability of the nucleofuge. With cyclic substrates, the preferred geometry of elimination is for the sigma bond that drives out the leaving group to being anti to it, analogous to the conformational orientation in the E2 mechanism of elimination reactions .
An example of a Grob-like fragmentation in organic synthesis is the expansion of the Wieland–Miescher ketone to thapsigargin : [ 7 ]
In this reaction, diastereoselective reduction of the ketone 1 with sodium borohydride yields alcohol 2 , which is functionalized to the mesylate 3 with mesyl chloride in pyridine . The selectivity of the initial reduction of ketone 1 is a result of borohyride approaching from the bottom face to avoid steric clash with the axial methyl group. Then reduction of the enone to allyl alcohol 4 with tri- tert -butoxyaluminium hydride in tetrahydrofuran followed by hydroboration with borane in THF yields the borane 5 (only one substituent displayed for clarity). The diastereoselectivity of the hydroboration is a result of two factors: avoidance of the axial methyl group as well as axial hydride addition to avoid a twist-boat conformation in the transition state . The Grob fragmentation to 6 takes place with sodium methoxide in methanol at reflux . A methoxide group attacks the boron atom giving a borate complex which fragments. As each boron atom can hold three substrate molecules (R), the ultimate boron byproduct is trimethyl borate . As seen in 6 , the mesylate being in the equatorial position allows its sigma star orbital to align ideally with the sigma bond drawn, allowing for the correct olefin geometry seen in 7 .
Another example is an epoxy alcohol fragmentation reaction as part of the Holton Taxol total synthesis .
3-aza-Grob fragmentation is variation which takes place when an electrofuge and nucleofuge are situated at positions 1 and 5 on a secondary or tertiary amine chain with the nitrogen at the 3 position. [ 8 ] [ 9 ] The reaction products are an electrofugal fragment, an imine , and a nucleofugal fragment (such as an alcohol).
3-aza-Grob fragmentation can proceed with several different nucleofuges. The reaction mechanism has been reported to begin with the reduction of an ether protected amide to form a secondary alcohol. Fragmentation then takes place in a concerted step to form the reaction products.
The scope of the reaction has been found to cover THF and tetrahydrothiophene protecting groups using various hydride agents. [ 10 ] | https://en.wikipedia.org/wiki/Grob_fragmentation |
Grodno Azot ( Belarusian «Гро́дна Азо́т») is an open joint-stock company, Belarusian state-run producer of nitrogen compounds and fertilizers located in Grodno, Belarus .
The construction of temporary auxiliary facilities started in October 1960. In January 1965, the first lines of Ammiak-1 and Karbamid-1 workshops were put in operation. In October 1970, Grodno Nitrogen and Fertiliser Plant was transformed into Grodno Chemicals Plant named after Siarhei Prytytski. In May 1975, it was transformed into Grodno Production Association Azot named after Siarhei Prytytski.
In August 2000, the association was changed into a unitary enterprise and in 2002 it became OJSC Grodno Azot. [ 2 ]
In 2006, the United States imposed sanctions against nine Belarusian companies including Grodno Azot and its affiliate [ 3 ] Grodno Khimvolokno for "undermining the democratic process”. In October 2015, the sanctions were partially lifted. [ 4 ]
After the falsified Belarusian Presidential elections on August 9, 2020, Grodno Azot workers joined the opposition protests and national strike; however, many were detained and beaten by the police on multiple occasions. [ 5 ] [ 6 ]
In 2021, the United States reported that the sanctions against Grodno Azot can be renewed. [ 7 ] On 30 March 2021, Grodno Azot's subsidiary announced a tender for the shipment of its goods. One of the terms of the tender was the possibility of not marking the affiliation of the cargo with the Grodno Azot. It was caused by the threat of sanctions, according to the tender documentation and media. [ 8 ]
In April 2021, full-scale US sanctions against Grodno Azot and Grodno Khimvolokno were renewed. [ 9 ] [ 10 ] [ 11 ] On 9 August 2021, the US has added Grodno Azot CEO Igor Lyashenko to the SDN list . [ 12 ]
In September 2021, several Grodno Azot workers were detained. [ 13 ] New arrests were associated with the threat of Alexander Lukashenko that workers who reveal the ways of bypassing the sanctions would be put in jail for a long time. [ 13 ]
In December 2021, European Union sanctioned Grodno Azot and Grodno Khimvolokno. [ 14 ] Switzerland joined the EU sanctions on December 20. [ 15 ]
In 2022, Japan [ 16 ] [ 17 ] and Ukraine [ 18 ] joined the sanctions against Grodno Azot.
In 2023, several sanctions circumvention schemes involving companies registered in Kyrgyzstan , Uzbekistan , Serbia and Lithuania were identified as a result of journalistic investigations by the Belarusian Investigative Center and Siena. [ 19 ] [ 20 ] [ 21 ] [ 22 ] [ 23 ] In October 2024, it was reported that Grodno Azot products were being supplied to Ukraine under the guise of being produced in Turkmenistan through a company registered in the United Arab Emirates . [ 24 ]
On February 21, 2024, the Court of Justice of the European Union in Luxembourg rejected the claim of Grodno Azot and its subsidiary Khimvolokno, which had demanded the lifting of the European sanctions. [ 25 ] | https://en.wikipedia.org/wiki/Grodno_Azot |
Grome is an environmental modeling package developed by Quad Software dedicated for procedural and manual generation of large virtual outdoor worlds suitable for games and other 3D real-time simulation applications.
After more than two years of internal developing, the program was first launched on March 28, 2007 as version 1.0. [ 1 ] It immediately started to be used by various professional game studios and independent developers. Version 1.1 was launched on August 22, 2007 adding various optimizations, pen tablet support and more flexibility by introducing user defined data sets. [ 2 ] Following this was version 1.2 on March 12, 2008 which optimized existing fractal tools and added new ones, extended import and export with new data formats and introduced the Grome scripting language for automated tasks. [ 3 ]
Major release, 2.0, was introduced on June 18, 2009 with many new additions. With this new version, Grome becomes a more complete outdoor editing platform by introducing editing of water surfaces, vegetation and other decoration layers. 64bit processing was introduced to take advantage of large amounts of RAM and 64bit CPUs. With this version, Quad Software announced the availability for customization work for professional studios that need a specific Grome version for their projects.
The Grome 3.0 version was launched on April 4, 2011 bringing road networks editing, support for per-pixel materials, optimization tools for mobiles and low-end hardware and other new tools. [ 4 ] A new updated, version 3.1, was launched on October 12, 2011 with focus on optimization. A further update, version 3.11, was made on September 12, 2012, fixing various issues and improving the SDK.
Easy to use user interface, with shortcuts available for every operation. Real-time preview on multiple customizable hardware accelerated viewports. Presets system for tool parameters.
Supports unlimited terrain size using custom data swapping mechanism to hard disk space. Terrain made of multiple terrain zones supporting variable resolution and automatic border stitching.
Procedural heightmap generation using fractal, terracing and erosion tools. Possibility to create and assign multiple layers of heightmaps for each zone.
Manual editing of heightmap layers using selection layers and brushes . Flexible brush system that allows custom orientation, mask, strength, directions and pen tablet pressure. Elevation, terracing and fractals brushes.
Novel texture layering system that allows different shading methods, variable resolutions and images per terrain zones. Creation of images based on brushes or procedural generation based on slope, direction, altitude, external shape files and erosion flowmaps.
User defined format system using Grome SDK plugins.
Spawn objects individually or using brushes with custom orientation and area of effect. Tag, categorized, search and replace tools for objects.
Gizmo and numeric object transformation tools for translation, scaling and rotation. Possibility to group object for transformation around common pivot. Groups can be saved to disk and later restored.
Object-terrain linking mechanism that allows snapping objects to terrain and moving objects along ground surface.
Starting with version 3.0 road placement and manipulation was introduced. Control of road geometry generation, resolution, banking and intersections is now possible while real-time road/terrain interaction computation is done using the GPU.
Real-time water rendering using pixel lighting, bump mapping and masking for smooth shoreline transitions. Generation of water layers masks with brushes or based on terrain heights. Coloring and lightmapping of water based on terrain texture layers.
Allow rendering of massive population of decoration objects and grass blades. Animated grass effects to simulate wind and characters walking through. Generation with brushes or automatically based on terrain slope, orientation and altitude.
Offered for every Grome client to allow integration with their engine/application pipeline. Allows saving and loading from custom formats, define new mesh formats and automatic responses to various editing events. Comes with source code for all default plugins and documentation. Out-of-the-box, Grome has the default plugins to include support for COLLADA, XML, DTED, shape files, 16bit RAW, BT and all major images formats.
Grome 3.0 is currently licensed under two different prices, the same base version being offered to individuals and small developers under a reduced price. Professional companies benefit from premium support and dongle based protection. Normal builds offer file based activation per computer.
Starting with version 2.0, Quad Software publicly announced the availability of customization work for companies. In the way professional studios can obtain customized builds as part of under-contract projects. Examples of previous developed customized modules were presented, among them being road editing, AI navigation mesh generation and a more advanced lightmapper.
Graphite is a real-time outdoor rendering library created by Grome developers to allow easy integration of Grome scenes into any 3D engine and graphical application. The renderer is designed to be flexible and supports DirectX rendering API . It is offered for free in binary form for non-commercial usage for all Grome clients, while paid licenses are offered to professional studios allowing source access and commercial usage.
List of supported engines: | https://en.wikipedia.org/wiki/Grome |
In the mathematical field of symplectic topology , Gromov's compactness theorem states that a sequence of pseudoholomorphic curves in an almost complex manifold with a uniform energy bound must have a subsequence which limits to a pseudoholomorphic curve which may have nodes or (a finite tree of) "bubbles". A bubble is a holomorphic sphere which has a transverse intersection with the rest of the curve. This theorem, and its generalizations to punctured pseudoholomorphic curves, underlies the compactness results for flow lines in Floer homology and symplectic field theory .
If the complex structures on the curves in the sequence do not vary, only bubbles can occur; nodes can occur only if the complex structures on the domain are allowed to vary. Usually, the energy bound is achieved by considering a symplectic manifold with compatible almost-complex structure as the target, and assuming that curves to lie in a fixed homology class in the target. This is because the energy of such a pseudoholomorphic curve is given by the integral of the target symplectic form over the curve, and thus by evaluating the cohomology class of that symplectic form on the homology class of the curve. The finiteness of the bubble tree follows from (positive) lower bounds on the energy contributed by a holomorphic sphere.
This topology-related article is a stub . You can help Wikipedia by expanding it . | https://en.wikipedia.org/wiki/Gromov's_compactness_theorem_(topology) |
In mathematics, a smooth compact manifold M is called almost flat if for any ε > 0 {\displaystyle \varepsilon >0} there is a Riemannian metric g ε {\displaystyle g_{\varepsilon }} on M such that diam ( M , g ε ) ≤ 1 {\displaystyle {\mbox{diam}}(M,g_{\varepsilon })\leq 1} and g ε {\displaystyle g_{\varepsilon }} is ε {\displaystyle \varepsilon } -flat, i.e. for the sectional curvature of K g ε {\displaystyle K_{g_{\varepsilon }}} we have | K g ϵ | < ε {\displaystyle |K_{g_{\epsilon }}|<\varepsilon } .
Given n {\displaystyle n} , there is a positive number ε n > 0 {\displaystyle \varepsilon _{n}>0} such that if an n {\displaystyle n} -dimensional manifold admits an ε n {\displaystyle \varepsilon _{n}} -flat metric with diameter ≤ 1 {\displaystyle \leq 1} then it is almost flat. On the other hand, one can fix the bound of sectional curvature and get the diameter going to zero, so the almost-flat manifold is a special case of a collapsing manifold , which is collapsing along all directions.
According to the Gromov–Ruh theorem , M {\displaystyle M} is almost flat if and only if it is infranil . In particular, it is a finite factor of a nilmanifold , which is the total space of a principal torus bundle over a principal torus bundle over a torus.
This Riemannian geometry -related article is a stub . You can help Wikipedia by expanding it . | https://en.wikipedia.org/wiki/Gromov-Ruh_theorem |
Groombridge Transit Circle was a meridian transit circle made by Edward Troughton for the English astronomer Stephen Groombridge in 1806, which Groombridge used to compile data for the star catalogue , Catalogue of Circumpolar Stars . [ 1 ] The advantage of a transit circle over a mural circle (which can measure polar distances) is that it allows measuring right ascension and declination at the same time. [ 2 ]
It had an aperture of 3.5 inches and a 5-foot focal length, mounted inside two 4 foot circles on stone piers. [ 2 ] Groombridge used the instrument to determine the positions of over 4000 circumpolar stars . [ 2 ]
It was eventually bought by James South , and it remained at his observatory at Kensington until 1870. [ 2 ] | https://en.wikipedia.org/wiki/Groombridge_Transit_Circle |
Gross anatomy is the study of anatomy at the visible or macroscopic level. [ 1 ] [ 2 ] The counterpart to gross anatomy is the field of histology , which studies microscopic anatomy. [ 1 ] [ 2 ] Gross anatomy of the human body or other animals seeks to understand the relationship between components of an organism in order to gain a greater appreciation of the roles of those components and their relationships in maintaining the functions of life. The study of gross anatomy can be performed on deceased organisms using dissection or on living organisms using medical imaging . Education in the gross anatomy of humans is included training for most health professionals .
Gross anatomy is studied using both invasive and noninvasive methods with the goal of obtaining information about the macroscopic structure and organisation of organs and organ systems. Among the most common methods of study is dissection , in which the corpse of an animal or a human cadaver is surgically opened and its organs studied. Endoscopy , in which a video camera-equipped instrument is inserted through a small incision in the subject, may be used to explore the internal organs and other structures of living animals. The anatomy of the circulatory system in a living animal may be studied noninvasively via angiography , a technique in which blood vessels are visualised after being injected with an opaque dye. Other means of study include radiological techniques of imaging , such as X-ray and MRI .
Most health profession schools, such as medical, physician assistant , and dental schools, require that students complete a practical (dissection) course in gross human anatomy. Such courses aim to educate students in advanced funadmental human anatomy and seek to establish anatomical landmarks used to aid medical diagnosis . Most schools provide students with cadavers for investigation by dissection, aided by dissection manuals, as well as cadaveric atlases (e.g. Netter 's, Rohen 's).
Working intimately with a cadaver during a gross anatomy course has been shown to capture the essence of the patient-provider relationship. [ 3 ] However, the expense of maintaining cadaveric dissection facilities has limited the time and resources available for gross anatomy teaching in medical schools that are less funded, with some adopting alternative prosection-based or simulated teaching. [ 4 ] This, coupled with decreasing time dedicated to gross anatomical courses within the growing greater medical school curriculum, has caused controversy surrounding the sufficiency of anatomical teaching with nearly half of newly qualified doctors believing they received insufficient anatomy teaching due to the course often being condensed into one semester. [ 5 ]
Medical schools have implemented on-screen anatomical lessons and tutorials to teach students surgical procedures. The use of technological visual aids accompanied with gross dissection have been shown to be more effective than learning via one modality alone. [ 6 ] Online and physically made flashcards and quizzes [ 7 ] have been long used. | https://en.wikipedia.org/wiki/Gross_anatomy |
Gross processing, "grossing" or " gross pathology " is the process by which pathology specimens undergo examination with the bare eye to obtain diagnostic information, as well as cutting and tissue sampling in order to prepare material for subsequent microscopic examination. [ 1 ]
Gross examination of surgical specimens is typically performed by a pathologist , or by a pathologists' assistant working within a pathology practice. Individuals trained in these fields are often able to gather diagnostically critical information in this stage of processing, including the stage and margin status of surgically removed tumors. [ 1 ]
The initial step in any examination of a clinical specimen is confirmation of the identity of the patient and the anatomical site from which the specimen was obtained. Sufficient clinical data should be communicated by the clinical team to the pathology team in order to guide the appropriate diagnostic examination and interpretation of the specimen - if such information is not provided, it must be obtained by the examiner prior to processing the specimen.
There are usually two end products of the gross processing of a surgical specimen. The first is the gross description, a document which serves as the written record of the examiner's findings, and is included in the final pathology report. The second product is a set of tissue blocks, typically postage stamp-sized portions of tissue sealed in plastic cassettes, which will be processed into slides for microscopic examination. Since only a minority of the tissue from a large specimen can reasonably be subject to microscopic examination, the success of the final histological diagnosis is highly dependent on the skill of the professional performing the gross examination. The gross examiner may sample portions of the specimen for other types of ancillary tests as diagnostically indicated; these include microbiological culture , flow cytometry , cytogenetics , or electron microscopy .
Two major types of sections in gross processing are perpendicular and en face sections: | https://en.wikipedia.org/wiki/Gross_processing |
In mathematics , in the representation theory of algebraic groups , a Grosshans subgroup , named after Frank Grosshans , is an algebraic subgroup of an algebraic group that is an observable subgroup for which the ring of functions on the quotient variety is finitely generated . [ 1 ]
This algebra -related article is a stub . You can help Wikipedia by expanding it . | https://en.wikipedia.org/wiki/Grosshans_subgroup |
Grote–Hynes theory is a theory of reaction rate in a solution phase. This rate theory was developed by James T. Hynes with his graduate student Richard F. Grote in 1980. [ 1 ]
The theory is based on the generalized Langevin equation (GLE) . This theory introduced the concept of frequency dependent friction for chemical rate processes in solution phase. Because of inclusion of the frequency dependent friction instead of constant friction, the theory successfully predicts the rate constant including where the reaction barrier is large and of high frequency, where the diffusion over the barrier starts decoupling from viscosity of the medium. This was the weakness of Kramer's rate theory, [ 2 ] which underestimated the reaction rate having large barrier with high frequency.
This physical chemistry -related article is a stub . You can help Wikipedia by expanding it . | https://en.wikipedia.org/wiki/Grote–Hynes_theory |
In mathematics , the Grothendieck inequality states that there is a universal constant K G {\displaystyle K_{G}} with the following property. If M ij is an n × n ( real or complex ) matrix with
for all (real or complex) numbers s i , t j of absolute value at most 1, then
for all vectors S i , T j in the unit ball B ( H ) of a (real or complex) Hilbert space H , the constant K G {\displaystyle K_{G}} being independent of n . For a fixed Hilbert space of dimension d , the smallest constant that satisfies this property for all n × n matrices is called a Grothendieck constant and denoted K G ( d ) {\displaystyle K_{G}(d)} . In fact, there are two Grothendieck constants, K G R ( d ) {\displaystyle K_{G}^{\mathbb {R} }(d)} and K G C ( d ) {\displaystyle K_{G}^{\mathbb {C} }(d)} , depending on whether one works with real or complex numbers, respectively. [ 1 ]
The Grothendieck inequality and Grothendieck constants are named after Alexander Grothendieck , who proved the existence of the constants in a paper published in 1953. [ 2 ]
Let A = ( a i j ) {\displaystyle A=(a_{ij})} be an m × n {\displaystyle m\times n} matrix. Then A {\displaystyle A} defines a linear operator between the normed spaces ( R m , ‖ ⋅ ‖ p ) {\displaystyle (\mathbb {R} ^{m},\|\cdot \|_{p})} and ( R n , ‖ ⋅ ‖ q ) {\displaystyle (\mathbb {R} ^{n},\|\cdot \|_{q})} for 1 ≤ p , q ≤ ∞ {\displaystyle 1\leq p,q\leq \infty } . The ( p → q ) {\displaystyle (p\to q)} -norm of A {\displaystyle A} is the quantity
‖ A ‖ p → q = max x ∈ R n : ‖ x ‖ p = 1 ‖ A x ‖ q . {\displaystyle \|A\|_{p\to q}=\max _{x\in \mathbb {R} ^{n}:\|x\|_{p}=1}\|Ax\|_{q}.}
If p = q {\displaystyle p=q} , we denote the norm by ‖ A ‖ p {\displaystyle \|A\|_{p}} .
One can consider the following question: For what value of p {\displaystyle p} and q {\displaystyle q} is ‖ A ‖ p → q {\displaystyle \|A\|_{p\to q}} maximized? Since A {\displaystyle A} is linear, then it suffices to consider p {\displaystyle p} such that { x ∈ R n : ‖ x ‖ p ≤ 1 } {\displaystyle \{x\in \mathbb {R} ^{n}:\|x\|_{p}\leq 1\}} contains as many points as possible, and also q {\displaystyle q} such that ‖ A x ‖ q {\displaystyle \|Ax\|_{q}} is as large as possible. By comparing ‖ x ‖ p {\displaystyle \|x\|_{p}} for p = 1 , 2 , … , ∞ {\displaystyle p=1,2,\ldots ,\infty } , one sees that ‖ A ‖ ∞ → 1 ≥ ‖ A ‖ p → q {\displaystyle \|A\|_{\infty \to 1}\geq \|A\|_{p\to q}} for all 1 ≤ p , q ≤ ∞ {\displaystyle 1\leq p,q\leq \infty } .
One way to compute ‖ A ‖ ∞ → 1 {\displaystyle \|A\|_{\infty \to 1}} is by solving the following quadratic integer program :
max ∑ i , j A i j x i y j s.t. ( x , y ) ∈ { − 1 , 1 } m + n {\displaystyle {\begin{aligned}\max &\qquad \sum _{i,j}A_{ij}x_{i}y_{j}\\{\text{s.t.}}&\qquad (x,y)\in \{-1,1\}^{m+n}\end{aligned}}}
To see this, note that ∑ i , j A i j x i y j = ∑ i ( A y ) i x i {\displaystyle \sum _{i,j}A_{ij}x_{i}y_{j}=\sum _{i}(Ay)_{i}x_{i}} , and taking the maximum over x ∈ { − 1 , 1 } m {\displaystyle x\in \{-1,1\}^{m}} gives ‖ A y ‖ 1 {\displaystyle \|Ay\|_{1}} . Then taking the maximum over y ∈ { − 1 , 1 } n {\displaystyle y\in \{-1,1\}^{n}} gives ‖ A ‖ ∞ → 1 {\displaystyle \|A\|_{\infty \to 1}} by the convexity of { x ∈ R m : ‖ x ‖ ∞ = 1 } {\displaystyle \{x\in \mathbb {R} ^{m}:\|x\|_{\infty }=1\}} and by the triangle inequality. This quadratic integer program can be relaxed to the following semidefinite program :
max ∑ i , j A i j ⟨ x ( i ) , y ( j ) ⟩ s.t. x ( 1 ) , … , x ( m ) , y ( 1 ) , … , y ( n ) are unit vectors in ( R d , ‖ ⋅ ‖ 2 ) {\displaystyle {\begin{aligned}\max &\qquad \sum _{i,j}A_{ij}\langle x^{(i)},y^{(j)}\rangle \\{\text{s.t.}}&\qquad x^{(1)},\ldots ,x^{(m)},y^{(1)},\ldots ,y^{(n)}{\text{ are unit vectors in }}(\mathbb {R} ^{d},\|\cdot \|_{2})\end{aligned}}}
It is known that exactly computing ‖ A ‖ p → q {\displaystyle \|A\|_{p\to q}} for 1 ≤ q < p ≤ ∞ {\displaystyle 1\leq q<p\leq \infty } is NP-hard , while exacting computing ‖ A ‖ p {\displaystyle \|A\|_{p}} is NP-hard for p ∉ { 1 , 2 , ∞ } {\displaystyle p\not \in \{1,2,\infty \}} .
One can then ask the following natural question: How well does an optimal solution to the semidefinite program approximate ‖ A ‖ ∞ → 1 {\displaystyle \|A\|_{\infty \to 1}} ? The Grothendieck inequality provides an answer to this question: There exists a fixed constant C > 0 {\displaystyle C>0} such that, for any m , n ≥ 1 {\displaystyle m,n\geq 1} , for any m × n {\displaystyle m\times n} matrix A {\displaystyle A} , and for any Hilbert space H {\displaystyle H} ,
max x ( i ) , y ( i ) ∈ H unit vectors ∑ i , j A i j ⟨ x ( i ) , y ( j ) ⟩ H ≤ C ‖ A ‖ ∞ → 1 . {\displaystyle \max _{x^{(i)},y^{(i)}\in H{\text{ unit vectors}}}\sum _{i,j}A_{ij}\left\langle x^{(i)},y^{(j)}\right\rangle _{H}\leq C\|A\|_{\infty \to 1}.}
The sequences K G R ( d ) {\displaystyle K_{G}^{\mathbb {R} }(d)} and K G C ( d ) {\displaystyle K_{G}^{\mathbb {C} }(d)} are easily seen to be increasing, and Grothendieck's result states that they are bounded , [ 2 ] [ 3 ] so they have limits .
Grothendieck proved that 1.57 ≈ π 2 ≤ K G R ≤ sinh π 2 ≈ 2.3 , {\displaystyle 1.57\approx {\frac {\pi }{2}}\leq K_{G}^{\mathbb {R} }\leq \operatorname {sinh} {\frac {\pi }{2}}\approx 2.3,} where K G R {\displaystyle K_{G}^{\mathbb {R} }} is defined to be sup d K G R ( d ) {\displaystyle \sup _{d}K_{G}^{\mathbb {R} }(d)} . [ 4 ]
Krivine (1979) [ 5 ] improved the result by proving that K G R ≤ π 2 ln ( 1 + 2 ) ≈ 1.7822 {\displaystyle K_{G}^{\mathbb {R} }\leq {\frac {\pi }{2\ln(1+{\sqrt {2}})}}\approx 1.7822} , conjecturing that the upper bound is tight. However, this conjecture was disproved by Braverman et al. (2011) . [ 6 ]
Boris Tsirelson showed that the Grothendieck constants K G R ( d ) {\displaystyle K_{G}^{\mathbb {R} }(d)} play an essential role in the problem of quantum nonlocality : the Tsirelson bound of any full correlation bipartite Bell inequality for a quantum system of dimension d is upperbounded by K G R ( 2 d 2 ) {\displaystyle K_{G}^{\mathbb {R} }(2d^{2})} . [ 7 ] [ 8 ]
Some historical data on best known lower bounds of K G R ( d ) {\displaystyle K_{G}^{\mathbb {R} }(d)} is summarized in the following table.
Some historical data on best known upper bounds of K G R ( d ) {\displaystyle K_{G}^{\mathbb {R} }(d)} :
Given an m × n {\displaystyle m\times n} real matrix A = ( a i j ) {\displaystyle A=(a_{ij})} , the cut norm of A {\displaystyle A} is defined by
The notion of cut norm is essential in designing efficient approximation algorithms for dense graphs and matrices. More generally, the definition of cut norm can be generalized for symmetric measurable functions W : [ 0 , 1 ] 2 → R {\displaystyle W:[0,1]^{2}\to \mathbb {R} } so that the cut norm of W {\displaystyle W} is defined by
This generalized definition of cut norm is crucial in the study of the space of graphons , and the two definitions of cut norm can be linked via the adjacency matrix of a graph .
An application of the Grothendieck inequality is to give an efficient algorithm for approximating the cut norm of a given real matrix A {\displaystyle A} ; specifically, given an m × n {\displaystyle m\times n} real matrix, one can find a number α {\displaystyle \alpha } such that
‖ A ‖ ◻ ≤ α ≤ C ‖ A ‖ ◻ , {\displaystyle \|A\|_{\square }\leq \alpha \leq C\|A\|_{\square },}
where C {\displaystyle C} is an absolute constant. [ 19 ] This approximation algorithm uses semidefinite programming .
We give a sketch of this approximation algorithm. Let B = ( b i j ) {\displaystyle B=(b_{ij})} be ( m + 1 ) × ( n + 1 ) {\displaystyle (m+1)\times (n+1)} matrix defined by
( a 11 a 12 … a 1 n − ∑ k = 1 n a 1 k a 21 a 22 … a 2 n − ∑ k = 1 n a 2 k ⋮ ⋮ ⋱ ⋮ ⋮ a m 1 a m 2 … a m n − ∑ k = 1 n a m k − ∑ ℓ = 1 m a ℓ 1 − ∑ ℓ = 1 m a ℓ 2 … − ∑ ℓ = 1 m a ℓ n ∑ k = 1 n ∑ ℓ = 1 m a ℓ k ) . {\displaystyle {\begin{pmatrix}a_{11}&a_{12}&\ldots &a_{1n}&-\sum _{k=1}^{n}a_{1k}\\a_{21}&a_{22}&\ldots &a_{2n}&-\sum _{k=1}^{n}a_{2k}\\\vdots &\vdots &\ddots &\vdots &\vdots \\a_{m1}&a_{m2}&\ldots &a_{mn}&-\sum _{k=1}^{n}a_{mk}\\-\sum _{\ell =1}^{m}a_{\ell 1}&-\sum _{\ell =1}^{m}a_{\ell 2}&\ldots &-\sum _{\ell =1}^{m}a_{\ell n}&\sum _{k=1}^{n}\sum _{\ell =1}^{m}a_{\ell k}\end{pmatrix}}.}
One can verify that ‖ A ‖ ◻ = ‖ B ‖ ◻ {\displaystyle \|A\|_{\square }=\|B\|_{\square }} by observing, if S ∈ [ m + 1 ] , T ∈ [ n + 1 ] {\displaystyle S\in [m+1],T\in [n+1]} form a maximizer for the cut norm of B {\displaystyle B} , then
S ∗ = { S , if m + 1 ∉ S , [ m ] ∖ S , otherwise , T ∗ = { T , if n + 1 ∉ T , [ n ] ∖ S , otherwise , {\displaystyle S^{*}={\begin{cases}S,&{\text{if }}m+1\not \in S,\\{[m]}\setminus S,&{\text{otherwise}},\end{cases}}\qquad T^{*}={\begin{cases}T,&{\text{if }}n+1\not \in T,\\{[n]}\setminus S,&{\text{otherwise}},\end{cases}}\qquad }
form a maximizer for the cut norm of A {\displaystyle A} . Next, one can verify that ‖ B ‖ ◻ = ‖ B ‖ ∞ → 1 / 4 {\displaystyle \|B\|_{\square }=\|B\|_{\infty \to 1}/4} , where
‖ B ‖ ∞ → 1 = max { ∑ i = 1 m + 1 ∑ j = 1 n + 1 b i j ε i δ j : ε 1 , … , ε m + 1 ∈ { − 1 , 1 } , δ 1 , … , δ n + 1 ∈ { − 1 , 1 } } . {\displaystyle \|B\|_{\infty \to 1}=\max \left\{\sum _{i=1}^{m+1}\sum _{j=1}^{n+1}b_{ij}\varepsilon _{i}\delta _{j}:\varepsilon _{1},\ldots ,\varepsilon _{m+1}\in \{-1,1\},\delta _{1},\ldots ,\delta _{n+1}\in \{-1,1\}\right\}.} [ 20 ]
Although not important in this proof, ‖ B ‖ ∞ → 1 {\displaystyle \|B\|_{\infty \to 1}} can be interpreted to be the norm of B {\displaystyle B} when viewed as a linear operator from ℓ ∞ m {\displaystyle \ell _{\infty }^{m}} to ℓ 1 m {\displaystyle \ell _{1}^{m}} .
Now it suffices to design an efficient algorithm for approximating ‖ A ‖ ∞ → 1 {\displaystyle \|A\|_{\infty \to 1}} . We consider the following semidefinite program :
SDP ( A ) = max { ∑ i = 1 m ∑ j = 1 n a i j ⟨ x i , y j ⟩ : x 1 , … , x m , y 1 , … , y n ∈ S n + m − 1 } . {\displaystyle {\text{SDP}}(A)=\max \left\{\sum _{i=1}^{m}\sum _{j=1}^{n}a_{ij}\left\langle x_{i},y_{j}\right\rangle :x_{1},\ldots ,x_{m},y_{1},\ldots ,y_{n}\in S^{n+m-1}\right\}.}
Then SDP ( A ) ≥ ‖ A ‖ ∞ → 1 {\displaystyle {\text{SDP}}(A)\geq \|A\|_{\infty \to 1}} . The Grothedieck inequality implies that SDP ( A ) ≤ K G R ‖ A ‖ ∞ → 1 {\displaystyle {\text{SDP}}(A)\leq K_{G}^{\mathbb {R} }\|A\|_{\infty \to 1}} . Many algorithms (such as interior-point methods , first-order methods, the bundle method, the augmented Lagrangian method ) are known to output the value of a semidefinite program up to an additive error ε {\displaystyle \varepsilon } in time that is polynomial in the program description size and log ( 1 / ε ) {\displaystyle \log(1/\varepsilon )} . [ 21 ] Therefore, one can output α = SDP ( B ) {\displaystyle \alpha ={\text{SDP}}(B)} which satisfies
‖ A ‖ ◻ ≤ α ≤ C ‖ A ‖ ◻ with C = K G R . {\displaystyle \|A\|_{\square }\leq \alpha \leq C\|A\|_{\square }\qquad {\text{with}}\qquad C=K_{G}^{\mathbb {R} }.}
Szemerédi's regularity lemma is a useful tool in graph theory, asserting (informally) that any graph can be partitioned into a controlled number of pieces that interact with each other in a pseudorandom way. Another application of the Grothendieck inequality is to produce a partition of the vertex set that satisfies the conclusion of Szemerédi's regularity lemma , via the cut norm estimation algorithm, in time that is polynomial in the upper bound of Szemerédi's regular partition size but independent of the number of vertices in the graph. [ 20 ]
It turns out that the main "bottleneck" of constructing a Szemeredi's regular partition in polynomial time is to determine in polynomial time whether or not a given pair ( X , Y ) {\displaystyle (X,Y)} is close to being ε {\displaystyle \varepsilon } -regular , meaning that for all S ⊂ X , T ⊂ Y {\displaystyle S\subset X,T\subset Y} with | S | ≥ ε | X | , | T | ≥ ε | Y | {\displaystyle |S|\geq \varepsilon |X|,|T|\geq \varepsilon |Y|} , we have
| e ( S , T ) | S | | T | − e ( X , Y ) | X | | Y | | ≤ ε , {\displaystyle \left|{\frac {e(S,T)}{|S||T|}}-{\frac {e(X,Y)}{|X||Y|}}\right|\leq \varepsilon ,}
where e ( X ′ , Y ′ ) = | { ( u , v ) ∈ X ′ × Y ′ : u v ∈ E } | {\displaystyle e(X',Y')=|\{(u,v)\in X'\times Y':uv\in E\}|} for all X ′ , Y ′ ⊂ V {\displaystyle X',Y'\subset V} and V , E {\displaystyle V,E} are the vertex and edge sets of the graph, respectively. To that end, we construct an n × n {\displaystyle n\times n} matrix A = ( a x y ) ( x , y ) ∈ X × Y {\displaystyle A=(a_{xy})_{(x,y)\in X\times Y}} , where n = | V | {\displaystyle n=|V|} , defined by
a x y = { 1 − e ( X , Y ) | X | | Y | , if x y ∈ E , − e ( X , Y ) | X | | Y | , otherwise . {\displaystyle a_{xy}={\begin{cases}1-{\frac {e(X,Y)}{|X||Y|}},&{\text{if }}xy\in E,\\-{\frac {e(X,Y)}{|X||Y|}},&{\text{otherwise}}.\end{cases}}}
Then for all S ⊂ X , T ⊂ Y {\displaystyle S\subset X,T\subset Y} ,
| ∑ x ∈ S , y ∈ T a x y | = | S | | T | | e ( S , T ) | S | | T | − e ( X , Y ) | X | | Y | | . {\displaystyle \left|\sum _{x\in S,y\in T}a_{xy}\right|=|S||T|\left|{\frac {e(S,T)}{|S||T|}}-{\frac {e(X,Y)}{|X||Y|}}\right|.}
Hence, if ( X , Y ) {\displaystyle (X,Y)} is not ε {\displaystyle \varepsilon } -regular, then ‖ A ‖ ◻ ≥ ε 3 n 2 {\displaystyle \|A\|_{\square }\geq \varepsilon ^{3}n^{2}} . It follows that using the cut norm approximation algorithm together with the rounding technique, one can find in polynomial time S ⊂ X , T ⊂ Y {\displaystyle S\subset X,T\subset Y} such that
min { n | S | , n | T | , n 2 | e ( S , T ) | S | | T | − e ( X , Y ) | X | | Y | | } ≥ | ∑ x ∈ S , y ∈ T a x y | ≥ 1 K G R ε 3 n 2 ≥ 1 2 ε 3 n 2 . {\displaystyle \min \left\{n|S|,n|T|,n^{2}\left|{\frac {e(S,T)}{|S||T|}}-{\frac {e(X,Y)}{|X||Y|}}\right|\right\}\geq \left|\sum _{x\in S,y\in T}a_{xy}\right|\geq {\frac {1}{K_{G}^{\mathbb {R} }}}\varepsilon ^{3}n^{2}\geq {\frac {1}{2}}\varepsilon ^{3}n^{2}.}
Then the algorithm for producing a Szemerédi's regular partition follows from the constructive argument of Alon et al. [ 22 ]
The Grothendieck inequality of a graph states that for each n ∈ N {\displaystyle n\in \mathbb {N} } and for each graph G = ( { 1 , … , n } , E ) {\displaystyle G=(\{1,\ldots ,n\},E)} without self loops, there exists a universal constant K > 0 {\displaystyle K>0} such that every n × n {\displaystyle n\times n} matrix A = ( a i j ) {\displaystyle A=(a_{ij})} satisfies that
max x 1 , … , x n ∈ S n − 1 ∑ i j ∈ E a i j ⟨ x i , x j ⟩ ≤ K max ε 1 , … , ε n ∈ { − 1 , 1 } ∑ i j ∈ E a i j ε i ε j . {\displaystyle \max _{x_{1},\ldots ,x_{n}\in S^{n-1}}\sum _{ij\in E}a_{ij}\left\langle x_{i},x_{j}\right\rangle \leq K\max _{\varepsilon _{1},\ldots ,\varepsilon _{n}\in \{-1,1\}}\sum _{ij\in E}a_{ij}\varepsilon _{i}\varepsilon _{j}.} [ 23 ]
The Grothendieck constant of a graph G {\displaystyle G} , denoted K ( G ) {\displaystyle K(G)} , is defined to be the smallest constant K {\displaystyle K} that satisfies the above property.
The Grothendieck inequality of a graph is an extension of the Grothendieck inequality because the former inequality is the special case of the latter inequality when G {\displaystyle G} is a bipartite graph with two copies of { 1 , … , n } {\displaystyle \{1,\ldots ,n\}} as its bipartition classes. Thus,
K G = sup n ∈ N { K ( G ) : G is an n -vertex bipartite graph } . {\displaystyle K_{G}=\sup _{n\in \mathbb {N} }\{K(G):G{\text{ is an }}n{\text{-vertex bipartite graph}}\}.}
For G = K n {\displaystyle G=K_{n}} , the n {\displaystyle n} -vertex complete graph, the Grothendieck inequality of G {\displaystyle G} becomes
max x 1 , … , x n ∈ S n − 1 ∑ i , j ∈ { 1 , … , n } , i ≠ j a i j ⟨ x i , x j ⟩ ≤ K ( K n ) max ε 1 , … , ε n ∈ { − 1 , 1 } ∑ i , j ∈ { 1 , … , n } , i ≠ j a i j ε i ε j . {\displaystyle \max _{x_{1},\ldots ,x_{n}\in S^{n-1}}\sum _{i,j\in \{1,\ldots ,n\},i\neq j}a_{ij}\left\langle x_{i},x_{j}\right\rangle \leq K(K_{n})\max _{\varepsilon _{1},\ldots ,\varepsilon _{n}\in \{-1,1\}}\sum _{i,j\in \{1,\ldots ,n\},i\neq j}a_{ij}\varepsilon _{i}\varepsilon _{j}.}
It turns out that K ( K n ) ≍ log n {\displaystyle K(K_{n})\asymp \log n} . On one hand, we have K ( K n ) ≲ log n {\displaystyle K(K_{n})\lesssim \log n} . [ 24 ] [ 25 ] [ 26 ] Indeed, the following inequality is true for any n × n {\displaystyle n\times n} matrix A = ( a i j ) {\displaystyle A=(a_{ij})} , which implies that K ( K n ) ≲ log n {\displaystyle K(K_{n})\lesssim \log n} by the Cauchy-Schwarz inequality : [ 23 ]
max x 1 , … , x n ∈ S n − 1 ∑ i , j ∈ { 1 , … , n } , i ≠ j a i j ⟨ x i , x j ⟩ ≤ log ( ∑ i ∈ { 1 , … , n } ∑ j ∈ { 1 , … , n } ∖ { i } | a i j | ∑ i ∈ { 1 , … , n } ∑ j ∈ { 1 , … , n } ∖ { i } a i j 2 ) max ε 1 , … , ε n ∈ { − 1 , 1 } ∑ i , j ∈ { 1 , … , n } , i ≠ j a i j ε 1 ε n . {\displaystyle \max _{x_{1},\ldots ,x_{n}\in S^{n-1}}\sum _{i,j\in \{1,\ldots ,n\},i\neq j}a_{ij}\left\langle x_{i},x_{j}\right\rangle \leq \log \left({\frac {\sum _{i\in \{1,\ldots ,n\}}\sum _{j\in \{1,\ldots ,n\}\setminus \{i\}}|a_{ij}|}{\sqrt {\sum _{i\in \{1,\ldots ,n\}}\sum _{j\in \{1,\ldots ,n\}\setminus \{i\}}a_{ij}^{2}}}}\right)\max _{\varepsilon _{1},\ldots ,\varepsilon _{n}\in \{-1,1\}}\sum _{i,j\in \{1,\ldots ,n\},i\neq j}a_{ij}\varepsilon _{1}\varepsilon _{n}.}
On the other hand, the matching lower bound K ( K n ) ≳ log n {\displaystyle K(K_{n})\gtrsim \log n} is due to Alon , Makarychev, Makarychev and Naor in 2006. [ 23 ]
The Grothendieck inequality K ( G ) {\displaystyle K(G)} of a graph G {\displaystyle G} depends upon the structure of G {\displaystyle G} . It is known that
log ω ≲ K ( G ) ≲ log ϑ , {\displaystyle \log \omega \lesssim K(G)\lesssim \log \vartheta ,} [ 23 ]
and
K ( G ) ≤ π 2 log ( 1 + ( ϑ − 1 ) 2 + 1 ϑ − 1 ) , {\displaystyle K(G)\leq {\frac {\pi }{2\log \left({\frac {1+{\sqrt {(\vartheta -1)^{2}+1}}}{\vartheta -1}}\right)}},} [ 27 ]
where ω {\displaystyle \omega } is the clique number of G {\displaystyle G} , i.e., the largest k ∈ { 2 , … , n } {\displaystyle k\in \{2,\ldots ,n\}} such that there exists S ⊂ { 1 , … , n } {\displaystyle S\subset \{1,\ldots ,n\}} with | S | = k {\displaystyle |S|=k} such that i j ∈ E {\displaystyle ij\in E} for all distinct i , j ∈ S {\displaystyle i,j\in S} , and
ϑ = min { max i ∈ { 1 , … , n } 1 ⟨ x i , y ⟩ : x 1 , … , x n , y ∈ S n , ⟨ x i , x j ⟩ = 0 ∀ i j ∈ E } . {\displaystyle \vartheta =\min \left\{\max _{i\in \{1,\ldots ,n\}}{\frac {1}{\langle x_{i},y\rangle }}:x_{1},\ldots ,x_{n},y\in S^{n},\left\langle x_{i},x_{j}\right\rangle =0\;\forall ij\in E\right\}.}
The parameter ϑ {\displaystyle \vartheta } is known as the Lovász theta function of the complement of G {\displaystyle G} . [ 28 ] [ 29 ] [ 23 ]
In the application of the Grothendieck inequality for approximating the cut norm, we have seen that the Grothendieck inequality answers the following question: How well does an optimal solution to the semidefinite program SDP ( A ) {\displaystyle {\text{SDP}}(A)} approximate ‖ A ‖ ∞ → 1 {\displaystyle \|A\|_{\infty \to 1}} , which can be viewed as an optimization problem over the unit cube? More generally, we can ask similar questions over convex bodies other than the unit cube.
For instance, the following inequality is due to Naor and Schechtman [ 30 ] and independently due to Guruswami et al: [ 31 ] For every n × n {\displaystyle n\times n} matrix A = ( a i j ) {\displaystyle A=(a_{ij})} and every p ≥ 2 {\displaystyle p\geq 2} ,
max x 1 , … , x n ∈ R n , ∑ k = 1 n ‖ x k ‖ 2 p ≤ 1 ∑ i = 1 n ∑ j = 1 n a i j ⟨ x i , x j ⟩ ≤ γ p 2 max t 1 , … , t n ∈ R , ∑ k = 1 n | t k | p ≤ 1 ∑ i = 1 n ∑ j = 1 n a i j t i t j , {\displaystyle \max _{x_{1},\ldots ,x_{n}\in \mathbb {R} ^{n},\sum _{k=1}^{n}\|x_{k}\|_{2}^{p}\leq 1}\sum _{i=1}^{n}\sum _{j=1}^{n}a_{ij}\left\langle x_{i},x_{j}\right\rangle \leq \gamma _{p}^{2}\max _{t_{1},\ldots ,t_{n}\in \mathbb {R} ,\sum _{k=1}^{n}|t_{k}|^{p}\leq 1}\sum _{i=1}^{n}\sum _{j=1}^{n}a_{ij}t_{i}t_{j},}
where
γ p = 2 ( Γ ( ( p + 1 ) / 2 ) π ) 1 / p . {\displaystyle \gamma _{p}={\sqrt {2}}\left({\frac {\Gamma ((p+1)/2)}{\sqrt {\pi }}}\right)^{1/p}.}
The constant γ p 2 {\displaystyle \gamma _{p}^{2}} is sharp in the inequality. Stirling's formula implies that γ p 2 = p / e + O ( 1 ) {\displaystyle \gamma _{p}^{2}=p/e+O(1)} as p → ∞ {\displaystyle p\to \infty } . | https://en.wikipedia.org/wiki/Grothendieck_inequality |
In commutative algebra , Grothendieck local duality is a duality theorem for cohomology of modules over local rings , analogous to Serre duality of coherent sheaves .
Suppose that R is a Cohen–Macaulay local ring of dimension d with maximal ideal m and residue field k = R / m . Let E ( k ) be a Matlis module , an injective hull of k , and let Ω be the completion of its dualizing module . Then for any R -module M there is an isomorphism of modules over the completion of R :
where H m is a local cohomology group.
There is a generalization to Noetherian local rings that are not Cohen–Macaulay, that replaces the dualizing module with a dualizing complex .
This commutative algebra -related article is a stub . You can help Wikipedia by expanding it . | https://en.wikipedia.org/wiki/Grothendieck_local_duality |
In mathematics , the Grothendieck–Teichmüller group GT is a group closely related to (and possibly equal to) the absolute Galois group of the rational numbers . It was introduced by Vladimir Drinfeld ( 1990 ) and named after Alexander Grothendieck and Oswald Teichmüller , based on Grothendieck's suggestion in his 1984 essay Esquisse d'un Programme to study the absolute Galois group of the rationals by relating it to its action on the Teichmüller tower of Teichmüller groupoids T g , n , the fundamental groupoids of moduli stacks of genus g curves with n points removed. There are several minor variations of the group: a discrete version, a pro- l version, a k -pro-unipotent version, and a profinite version; the first three versions were defined by Drinfeld, and the version most often used is the profinite version. | https://en.wikipedia.org/wiki/Grothendieck–Teichmüller_group |
A Grotrian diagram , or term diagram , shows the allowed electronic transitions between the energy levels of atoms. They can be used for one-electron and multi-electron atoms. They take into account the specific selection rules related to changes in angular momentum of the electron. The diagrams are named after Walter Grotrian , who introduced them in his 1928 book Graphische Darstellung der Spektren von Atomen und Ionen mit ein, zwei und drei Valenzelektronen [ 1 ] ("Graphical representation of the spectra of atoms and ions with one, two and three valence electrons").
Atomic energy-level and Grotrian diagrams by Stanley Bashkin and John O. Stoner Jr.
This spectroscopy -related article is a stub . You can help Wikipedia by expanding it . | https://en.wikipedia.org/wiki/Grotrian_diagram |
The Grotthuss mechanism (also known as proton jumping ) is a model for the process by which an 'excess' proton diffuses through the hydrogen bond network of water molecules or other hydrogen-bonded liquids through the formation and concomitant cleavage of covalent bonds involving neighboring molecules.
In his 1806 publication “Theory of decomposition of liquids by electrical currents”, Theodor Grotthuss proposed a theory of water conductivity. [ 1 ] Grotthuss envisioned the electrolytic reaction as a sort of ‘bucket line’ where each oxygen atom simultaneously passes and receives a single hydrogen ion.
It was an astonishing theory to propose at the time, since the water molecule was thought to be OH, not H 2 O, and the existence of ions was not fully understood.
On its 200th anniversary, his article was reviewed by Cukierman. [ 2 ]
Although Grotthuss was using an incorrect empirical formula of water, his description of the passing of protons through the cooperation of neighboring water molecules proved prescient.
Lemont Kier suggested that proton hopping may be an important mechanism for nerve transduction. [ 3 ]
The Grotthuss mechanism is now a general name for the proton-hopping mechanism. In liquid water the solvation of the excess proton is idealized by two forms: the H 9 O 4 + ( Eigen cation ) or H 5 O 2 + ( Zundel cation ). While the transport mechanism is believed to involve the inter-conversion between these two solvation structures, the details of the hopping and transport mechanism is still debated.
Currently there are two plausible mechanisms:
The calculated energetics of the hydronium solvation shells were reported in 2007 and it was suggested that the activation energies of the two proposed mechanisms do not agree with their calculated hydrogen bond strengths, but mechanism 1 might be the better candidate of the two. [ 5 ]
By use of conditional and time-dependent radial distribution functions (RDF), it was shown that the hydronium RDF can be decomposed into contributions from two distinct structures, Eigen and Zundel. The first peak in g(r) (the RDF) of the Eigen structure is similar to the equilibrium, standard RDF, only slightly more ordered, while the first peak of the Zundel structure is actually split into two peaks. The actual proton transfer (PT) event was then traced (after synchronizing all PT events so that t=0 is the actual event time), revealing that the hydronium indeed starts from an Eigen state, and quickly transforms into the Zundel state as the proton is being transferred, with the first peak of g(r) splitting into two. [ 6 ]
For a number of important gas phase reactions, like the hydration of carbon dioxide , a Grotthuss-like mechanism involving concerted proton hopping over several water molecules at the same time has been shown to describe the reaction kinetics. [ 7 ] [ 8 ] This Grotthuss-like concerted proton transfer seems to be especially important for atmospheric chemistry reactions, like the hydration of sulfur oxides , [ 9 ] [ 10 ] the hydrolysis of chlorine nitrate [ 11 ] and other reactions important for ozone depletion . [ 12 ] [ 13 ] [ 14 ]
The Grotthuss mechanism, along with the relative lightness and small size ( ionic radius ) of the proton, explains the unusually high diffusion rate of the proton in an electric field , relative to that of other common cations whose movement is due simply to acceleration by the field. Random thermal motion opposes the movement of both protons and other cations. Quantum tunnelling becomes more probable the smaller the mass of the cation is, and the proton is the lightest possible stable cation. Thus there is a minor effect from quantum tunnelling also, although it dominates at low temperatures only.
Some evidence from theoretical calculations, supported by recent X-ray absorption spectroscopy findings, has suggested an alternative mechanism in which the proton is attached to a "train" of three water molecules as it moves through the liquid. [ 15 ] | https://en.wikipedia.org/wiki/Grotthuss_mechanism |
A ground-coupled heat exchanger is an underground heat exchanger that can capture heat from and/or dissipate heat to the ground. They use the Earth's near constant subterranean temperature to warm or cool air or other fluids for residential, agricultural or industrial uses. If building air is blown through the heat exchanger for heat recovery ventilation , they are called earth tubes (or Canadian well, Provençal well, Solar chimney , also termed earth cooling tubes, earth warming tubes, earth-air heat exchangers (EAHE or EAHX), air-to-soil heat exchanger, earth channels, earth canals, earth-air tunnel systems, ground tube heat exchanger, hypocausts , subsoil heat exchangers, thermal labyrinths, underground air pipes, and others).
Earth tubes are often a viable and economical alternative or supplement to conventional central heating or air conditioning systems since there are no compressors, chemicals or burners and only blowers are required to move the air. These are used for either partial or full cooling and/or heating of facility ventilation air. Their use can help buildings meet Passive House standards or LEED certification.
Earth-air heat exchangers have been used in agricultural facilities (animal buildings) and horticultural facilities (greenhouses) in the United States of America over the past several decades and have been used in conjunction with solar chimneys in hot arid areas for thousands of years, probably beginning in the Persian Empire. Implementation of these systems in India as well as in the cooler climates of Austria, Denmark and Germany to preheat the air for home ventilation systems has become fairly common since the mid-1990s, and is slowly being adopted in North America.
Ground-coupled heat exchanger may also use water or antifreeze as a heat transfer fluid, often in conjunction with a geothermal heat pump . See, for example downhole heat exchangers . The rest of this article deals primarily with earth-air heat exchangers or earth tubes.
Passive ground-coupled heat exchange is a common traditional technique. It drives circulation using pressure differences caused by wind, rain, and buoyancy-driven convection (from selectively engineering areas of solar heating and evaporative, radiative, or conductive cooling).
Earth-air heat exchangers can be analyzed for performance with several software applications using weather gage data. These software applications include GAEA, AWADUKT Thermo, EnergyPlus, L-EWTSim, WKM, and others. However, numerous earth-air heat exchanger systems have been designed and constructed improperly, and failed to meet design expectations. Earth-air heat exchangers appear best suited for air pretreatment rather than for full heating or cooling. Pretreatment of air for an air source heat pump or ground-source heat pump often provides the best economic return on investment , with simple payback often achieved within one year after installation.
Most systems are usually constructed from 100 to 600 mm (3.9 to 23.6 in) diameter, smooth-walled (so they do not easily trap condensation moisture and mold), rigid or semi-rigid plastic, plastic-coated metal pipes or plastic pipes coated with inner antimicrobial layers, buried 1.5 to 3 m (4.9 to 9.8 ft) underground where the ambient earth temperature is typically 10 to 23 °C (50 to 73 °F) all year round in the temperate latitudes where most humans live. Ground temperature becomes more stable with depth. Smaller diameter tubes require more energy to move the air and have less earth contact surface area. Larger tubes permit a slower airflow, which also yields more efficient energy transfer and permits much higher volumes to be transferred, permitting more air exchanges in a shorter time period, when, for example, you want to clear the building of objectionable odors or smoke but suffer from poorer heat transfer from the pipe wall to the air due to increased distances.
Some consider that it is more efficient to pull air through a long tube than to push it with a fan. A solar chimney can use natural convection (warm air rising) to create a vacuum to draw filtered passive cooling tube air through the largest diameter cooling tubes. Natural convection may be slower than using a solar-powered fan. Sharp 90-degree angles should be avoided in the construction of the tube – two 45-degree bends produce less-turbulent, more efficient air flow. While smooth-wall tubes are more efficient in moving the air, they are less efficient in transferring energy.
There are three configurations, a closed loop design, an open 'fresh air' system or a combination:
Single-pass earth air heat exchangers offer the potential for indoor air quality improvement over conventional systems by providing an increased supply of outdoor air. In some configurations of single-pass systems, a continuous supply of outdoor air is provided. This type of system would usually include one or more ventilation heat recovery units.
A shared ground array comprises connected ground heat exchangers for use by more than one home. [ 1 ] They can deliver low-carbon heating where individual ground-coupled heat exchangers are not viable, such as in terraced housing with little outside space. They can also provide opportunities to decarbonise heating for groups of homes away from dense urban centres where traditional district heating is unlikely to be economically viable. [ 1 ] Other benefits include higher efficiency and lower capital cost, greater resident control to choose their own electricity supplier, and reduction in the number of exchangers required due to the variance in peak load times between different households. [ 1 ]
A thermal labyrinth performs the same function as an earth tube, but they are usually formed from a larger volume rectilinear space, sometimes incorporated into building basements or under ground floors, and which are in turn divided by numerous internal walls to form a labyrinthine air path. Maximising the length of the air path ensures a better heat transfer effect. The construction of the labyrinth walls, floors, and dividing walls is normally of high thermal mass cast concrete and concrete block, with the exterior walls and floors in direct contact with the surrounding earth. [ 2 ]
If humidity and associated mold colonization is not addressed in system design, occupants may face health risks. At some sites, the humidity in the earth tubes may be controlled simply by passive drainage if the water table is sufficiently deep and the soil has relatively high permeability. In situations where passive drainage is not feasible or needs to be augmented for further moisture reduction, active ( dehumidifier ) or passive ( desiccant ) systems may treat the air stream.
Formal research indicates that earth-air heat exchangers reduce building ventilation air pollution. Rabindra (2004) states, “The tunnel [earth-Air heat exchanger] is found not to support the growth of bacteria and fungi; rather it is found to reduce the quantity of bacteria and fungi thus making the air safer for humans to inhale. It is therefore clear that the use of EAT [Earth Air Tunnel] not only helps save the energy but also helps reduce the air pollution by reducing bacteria and fungi.” [ 3 ] Likewise, Flueckiger (1999) in a study of twelve earth-air heat exchangers varying in design, pipe material, size and age, stated, “This study was performed because of concerns of potential microbial growth in the buried pipes of ground-coupled air systems. The results however demonstrate, that no harmful growth occurs and that the airborne concentrations of viable spores and bacteria, with few exceptions, even decreases after passage through the pipe-system”, and further stated, “Based on these investigations the operation of ground-coupled earth-to-air heat exchangers is acceptable as long as regular controls are undertaken and if appropriate cleaning facilities are available”. [ 4 ]
Whether using earth tubes with or without antimicrobial material, it is extremely important that the underground cooling tubes have an excellent condensation drain and be installed at a 2-3 degree grade to ensure the constant removal of condensed water from the tubes. When implementing in a house without a basement on a flat lot, an external condensation tower can be installed at a depth lower than where the tube enters into the house and at a point close to the wall entry. The condensation tower installation requires the added use of a condensate pump in which to remove the water from the tower. For installations in houses with basements, the pipes are graded so that the condensation drain located within the house is at the lowest point. In either installation, the tube must continually slope towards either the condensation tower or the condensation drain. The inner surface of the tube, including all joints must be smooth to aid in the flow and removal of condensate. Corrugated or ribbed tubes and rough interior joints must not be used. Joints connecting the tubes together must be tight enough to prevent water or gas infiltration. In certain geographic areas, it is important that the joints prevent Radon gas infiltration. Porous materials like uncoated concrete tubes cannot be used. Ideally, Earth Tubes with antimicrobial inner layers should be used in installations to inhibit the potential growth of molds and bacteria within the tubes.
Implementations of earth-air heat exchangers for either partial or full cooling and/or heating of facility ventilation air have had mixed success. The literature is, unfortunately, well populated with over-generalizations about the applicability of these systems – both in favor of, and against. A key aspect of earth-air heat exchangers is the passive nature of operation and consideration of the wide variability of conditions in natural systems.
Earth-air heat exchangers can be very cost effective in both up-front/capital costs as well as long-term operation and maintenance costs. However, this varies widely depending on the location’s latitude, altitude, ambient Earth temperature, climatic temperature-and-relative-humidity extremes, solar radiation, water table, soil type ( thermal conductivity ), soil moisture content and the efficiency of the building's exterior envelope design / insulation. Generally, dry-and-low-density soil with little or no ground shade will yield the least benefit, while dense damp soil with considerable shade should perform well. A slow drip watering system may improve thermal performance. Damp soil in contact with the cooling tube conducts heat more efficiently than dry soil.
Earth cooling tubes are much less effective in hot humid climates (like Florida) where the ambient temperature of the earth approaches human comfort temperature. The higher the ambient temperature of the earth, the less effective it is for cooling and dehumidification. However, the earth can be used to partially cool and dehumidify the replacement fresh air intake for passive-solar thermal buffer zone [ 5 ] areas like the laundry room, or a solarium / greenhouse, especially those with a hot tub, swim spa, or indoor swimming pool, where warm humid air is exhausted in the summer, and a supply of cooler drier replacement air is desired.
Not all regions and sites are suitable for earth-air heat exchangers. Conditions which may hinder or preclude proper implementation include shallow bedrock, high water table, and insufficient space, among others. In some areas, only cooling or heating may be afforded by earth-air heat exchangers. In these areas, provision for thermal recharge of the ground must especially be considered. In dual function systems (both heating and cooling), the warm season provides ground thermal recharge for the cool season and the cool season provides ground thermal recharge for the warm season, though overtaxing the thermal reservoir must be considered even with dual function systems.
In the context of today's diminishing fossil fuel reserves, increasing electrical costs, air pollution and global warming , properly designed earth cooling tubes offer a sustainable alternative to reduce or eliminate the need for conventional compressor-based air conditioning systems, in non-tropical climates. They can also help to balance the electricity grid to support fluctuating supply from other renewable energy sources. [ 1 ] They also provide the added benefit of controlled, filtered, temperate fresh air intake, which is especially valuable in tight, well-weatherized, efficient building envelopes.
An alternative to the earth-to-air heat exchanger is the "water" to earth heat exchanger. This is typically similar to a geothermal heat pump tubing embedded horizontally in the soil (or could be a vertical sonde ) to a similar depth of the earth-air heat exchanger. It uses approximately double the length of pipe of 35 mm diameter, e.g., around 80 m compared to an EAHX of 40 m. A heat exchanger coil is placed before the air inlet of the heat recovery ventilator. Typically a brine liquid (heavily salted water) is used as the heat exchanger fluid.
Many European installations are now using this setup due to the ease of installation. No fall or drainage point is required and it is safe because of the reduced risk from mold. | https://en.wikipedia.org/wiki/Ground-coupled_heat_exchanger |
In electronic engineering , ground bounce is a phenomenon associated with transistor switching where the gate voltage can appear to be less than the local ground potential , causing the unstable operation of a logic gate .
Ground bounce is usually seen on high density VLSI where insufficient precautions have been taken to supply a logic gate with a sufficiently low impedance connection to ground (or sufficiently high bypass capacitance ). In this phenomenon, when the base of an NPN transistor is turned on, enough current flows through the emitter - collector circuit that the silicon in the immediate vicinity of the emitter-ground connection is pulled partially high, sometimes by several volts, thus raising the local ground, as perceived at the gate, to a value significantly above true ground. Relative to this local ground, the base voltage can go negative, thus shutting off the transistor. As the excess local charge dissipates, the transistor turns back on, possibly causing a repeat of the phenomenon, sometimes up to a half-dozen bounces.
Ground bounce is one of the leading causes of "hung" or metastable gates in modern digital circuit design. This happens because the ground bounce puts the input of a flip flop effectively at voltage level that is neither a one nor a zero at clock time, or causes untoward effects in the clock itself. A similar voltage sag phenomenon may be seen on the collector side, called supply voltage sag (or V CC sag ), where V CC is pulled unnaturally low. As a whole, ground bounce is a major issue in nanometer range technologies in VLSI.
Ground bounce can also occur when the circuit board has poorly designed ground paths. Improper ground or V CC can lead to local variations in the ground level between various components. This is most commonly seen in circuit boards that have ground and V CC paths on the surfaces of the board.
Ground bounce may be reduced by placing a 10–30-ohm resistor in series to each of the switching outputs to limit the current flow during the gate switch. [ 1 ] | https://en.wikipedia.org/wiki/Ground_bounce |
An aircraft ground carriage (also " ground power assisted takeoff and landing concept ") is a landing gear system connected to the ground, on which aircraft can take off and land without their aircraft-installed landing gear. [ 1 ] The technical feasibility of the ground carriage is being investigated by two research groups. In 2013, IATA included the technology into their "Technology Roadmap"; [ 2 ] Airbus pursues the concept as part of its "Future by Airbus” strategy. [ 3 ]
The aircraft-installed landing gear and related structures and systems account for 6 to 15 per cent of the empty weight of an aircraft, but it is only required on the ground for takeoff and landing as well as for taxiing and parking. During cruise flight , it is carried along as unused ballast . An aircraft without landing gear could therefore require 8 to 20 per cent less fuel in flight. Furthermore, landing gears are one of the most expensive aircraft systems and complex in operation and maintenance. [ 4 ] Finally, less noise is emitted when the drag of the undercarriage is omitted during approach and the engines are switched off while taxiing on ground. [ 5 ]
A ground carriage provides the means for an aircraft to takeoff and land without carrying an own aircraft-installed landing gear. Instead, the aircraft is equipped with much lighter interfaces, which connect to the ground carriage. [ 5 ]
Every airport approached by aircraft without landing gear must operate at least one ground carriage. In addition, alternate airports must be available if an airport is closed due to bad weather or a system failure. For emergency landings outside of runways , unsuitable flooring or unpaved ground cannot absorb the high wheel loads. Therefore, the landing gear of heavy long-haul aircraft in emergency landing on unsuitable ground is often not extended, since it would otherwise sink into ground first and then bend or break off. [ 4 ] [ 6 ]
The precursor of the aircraft ground carriage is the jettisonable or detachable landing gear , wherein the aircraft takes-off from a cart, which is then released and eventually lands on skids (IE: SNCASE Baroudeur ). It was used on all operational examples of the Messerschmitt Me 163B Komet with its jettisonable twin-wheel "dolly" main gear — its conventional arrangement included a semi-retractable tailwheel on the Komet's rear fuselage — and the first eight prototypes of the Arado Ar 234 "Blitz" , which all used a jettisonable tricycle-gear arrangement "trolley" design. The glider Schleicher Ka 1 , which was built in the 1950s, also had a droppable landing gear. A Sea Vampire Mk.21 landed with retracted landing gear on an aircraft carrier with flexible rubber decks for testing purposes. [ 7 ] [ 8 ] The Rockwell HiMAT unmanned aerial vehicle used skids for landing.
The idea of the aircraft ground carriage is finally related to the aircraft catapult , especially with the Electromagnetic Aircraft Launch System , which is currently under development. [ 9 ]
"GroLaS" (Ground-based Landing gear System) is an aircraft ground carriage system which is being developed since 2008 by a Hamburg -based company in cooperation with the Technical University of Hamburg-Harburg and the German Aerospace Center . [ 10 ] [ 11 ]
Currently, the setup of a small scale demonstrator is envisioned, the full scale system is planned to be ready for market entry in 2035. The focus of the GroLaS study starts with long-haul cargo aircraft . In a first implementation of the system, the world's major cargo airports and corresponding alternate airports have to be equipped. The costs for an airport are expected to be 500 million euros. [ 5 ] GroLaS is patented in Europe, the USA and China. A model in 1:87 scale, which was built in 2013, was exhibited at the Berlin Air Show in 2014. [ 12 ]
GroLaS consists of a slide which speeds up and slows down by means of a maglev system installed on both sides of the runway. Thus, the conventional runway remains and enables a dual usability of conventional aircraft and of aircraft without installed landing gear. Upon landing, the slide automatically accelerates the mounted ground carriage to the approaching speed of the aircraft before touchdown and adjusts its position longitudinally and laterally to the aircraft. Pins located on the ground carriage couple into corresponding aircraft installed interfaces. Takeoff and landing are less susceptible to side winds due to a yaw angle adjustment. The braking energy is converted into electrical energy, which can be used to support the aircraft engines during takeoff. The braking distance is shortened, and there is no reverse thrust required. For taxiing, the ground carriage can be decoupled from the slide to remain underneath the aircraft. [ 5 ]
"GABRIEL" ("Integrated Ground and on-board system for support of the Aircraft Safe Takeoff and Landing") is a research project to develop an aircraft ground carriage started in 2011 by a consortium of several European universities, companies and institutions.
The proposed aircraft ground carriage moves on its own electromagnetic rail system and not on a conventional runway. The pins for attaching to the ground carriage are installed on the aircraft and the aircraft needs to synchronize laterally to the position of the ground carriage, which is a different approach compared to the GroLaS-concept. [ 9 ] Parallels are the longitudinal and yaw angle synchronisation and, that the electrically driven ground carriage is designed demountable from the slide for taxiing. | https://en.wikipedia.org/wiki/Ground_carriage |
In telecommunications , ground constants are the electrical parameters of earth : electrical conductivity , σ, electrical permittivity , ε, and magnetic permeability , μ.
The values of these parameters vary with the local chemical composition and density of the Earth. For a propagating electromagnetic wave , such as a surface wave propagating along the surface of the Earth, these parameters vary with frequency and direction.
This article related to telecommunications is a stub . You can help Wikipedia by expanding it . | https://en.wikipedia.org/wiki/Ground_constants |
In mathematical logic , a ground term of a formal system is a term that does not contain any variables . Similarly, a ground formula is a formula that does not contain any variables.
In first-order logic with identity with constant symbols a {\displaystyle a} and b {\displaystyle b} , the sentence Q ( a ) ∨ P ( b ) {\displaystyle Q(a)\lor P(b)} is a ground formula. A ground expression is a ground term or ground formula.
Consider the following expressions in first order logic over a signature containing the constant symbols 0 {\displaystyle 0} and 1 {\displaystyle 1} for the numbers 0 and 1, respectively, a unary function symbol s {\displaystyle s} for the successor function and a binary function symbol + {\displaystyle +} for addition.
What follows is a formal definition for first-order languages . Let a first-order language be given, with C {\displaystyle C} the set of constant symbols, F {\displaystyle F} the set of functional operators, and P {\displaystyle P} the set of predicate symbols .
A ground term is a term that contains no variables. Ground terms may be defined by logical recursion (formula-recursion):
Roughly speaking, the Herbrand universe is the set of all ground terms.
A ground predicate , ground atom or ground literal is an atomic formula all of whose argument terms are ground terms.
If p ∈ P {\displaystyle p\in P} is an n {\displaystyle n} -ary predicate symbol and α 1 , α 2 , … , α n {\displaystyle \alpha _{1},\alpha _{2},\ldots ,\alpha _{n}} are ground terms, then p ( α 1 , α 2 , … , α n ) {\displaystyle p\left(\alpha _{1},\alpha _{2},\ldots ,\alpha _{n}\right)} is a ground predicate or ground atom.
Roughly speaking, the Herbrand base is the set of all ground atoms, [ 1 ] while a Herbrand interpretation assigns a truth value to each ground atom in the base.
A ground formula or ground clause is a formula without variables.
Ground formulas may be defined by syntactic recursion as follows:
Ground formulas are a particular kind of closed formulas . | https://en.wikipedia.org/wiki/Ground_expression |
Ground pressure is the pressure exerted on the ground by the tires or tracks of a motorized vehicle, and is one measure of its potential mobility, [ 1 ] especially over soft ground. It also applies to the feet of a walking person or machine . Pressure is measured in the SI unit of pascals (Pa). Average ground pressure can be calculated using the standard formula for average pressure: P = F / A . [ 2 ] In an idealised case, i.e. a static , uniform net force normal to level ground, this is simply the object's weight divided by contact area. The ground pressure of motorized vehicles is often compared with the ground pressure of a human foot , which can be 60 – 80 kPa while walking or as much as 13 MPa for a person in spike heels. [ 3 ]
Increasing the size of the contact area on the ground (the footprint ) in relation to the weight decreases the unit ground pressure. Ground pressure of 14 kPa (2 psi) or less is recommended for fragile ecosystems like marshes. [ 4 ] Decreasing the ground pressure increases the flotation, allowing easier passage of the body over soft terrain. This is exemplified by use of equipment such as snowshoes .
All examples are approximate, and will vary based on conditions
Note: The pressures for average human and horse are for standing still position. [ 7 ] A walking human will exert more than double his standing pressure. A galloping horse will exert up to 3.5 MPa (500 psi). The ground pressure for a pneumatic tire is roughly equal to its inflation pressure.
This physics -related article is a stub . You can help Wikipedia by expanding it . | https://en.wikipedia.org/wiki/Ground_pressure |
A Ground Proximity Warning System ( GPWS ) is a system designed to alert pilots if their aircraft is in immediate danger of flying into the ground or an obstacle. The United States Federal Aviation Administration (FAA) defines GPWS as a type of terrain awareness and warning system (TAWS). [ 1 ] More advanced systems, introduced in 1996, [ 2 ] are known as enhanced ground proximity warning systems ( EGPWS ), a modern type of TAWS.
In the late 1960s, a series of controlled flight into terrain (CFIT) accidents took the lives of hundreds of people. A CFIT accident is one where a properly functioning airplane under the control of a fully qualified and certified crew is flown into terrain, water or obstacles with no apparent awareness on the part of the crew. [ 3 ]
Beginning in the early 1970s, a number of studies examined the occurrence of CFIT accidents. [ 3 ] Findings from these studies indicated that many such accidents could have been avoided if a warning device called a ground proximity warning system (GPWS) had been used. As a result of these studies and recommendations from the U.S. National Transportation Safety Board (NTSB), in 1974, the FAA required all large turbine and turbojet airplanes to install TSO -approved GPWS equipment. [ 3 ] [ 4 ]
The UN International Civil Aviation Organization (ICAO), recommended the installation of GPWS in 1979. [ 5 ]
C. Donald Bateman , a Canadian-born engineer, developed and is credited with the invention of GPWS. [ 6 ]
In March 2000, the U.S. FAA amended operating rules to require that all U.S. registered turbine-powered airplanes with six or more passenger seats (exclusive of pilot and copilot seating) be equipped with an FAA-approved TAWS. [ 3 ] The mandate affects aircraft manufactured after March 29, 2002. [ 7 ]
Prior to the development of GPWS, large passenger aircraft were involved in 3.5 fatal CFIT accidents per year, falling to 2 per year in the mid-1970s. A 2006 report stated that from 1974, when the U.S. FAA made it a requirement for large aircraft to carry such equipment, until the time of the report, there had not been a single passenger fatality in a CFIT crash by a large jet in U.S. airspace. [ 8 ]
After 1974, there were still some CFIT accidents that GPWS was unable to help prevent, due to the "blind spot" of those early GPWS systems. More advanced systems were developed.
Older TAWS, or deactivation of the EGPWS, or ignoring its warnings when an airport is not in its database, [ 9 ] still leave aircraft vulnerable to possible CFIT incidents. In April 2010, a Polish Air Force Tupolev Tu-154M aircraft crashed near Smolensk, Russia, in a possible CFIT accident [ 10 ] killing all passengers and crew, including the President of Poland Lech Kaczyński . [ 11 ] [ 12 ] [ 13 ] [ 14 ] The aircraft was equipped with TAWS made by Universal Avionics Systems of Tucson. [ 11 ] According to the Russian Interstate Aviation Committee, the TAWS was turned on. [ 15 ] However, the airport where the aircraft was going to land (Smolensk (XUBS)) is not in the TAWS database. [ 16 ] [ 17 ] In January 2008 a Polish Air Force Casa C-295M crashed in a CFIT accident near Mirosławiec, Poland, despite being equipped with EGPWS; the EGPWS warning sounds had been disabled, and the pilot-in-command was not properly trained with EGPWS. [ 18 ]
The system monitors an aircraft's height above ground as determined by a radar altimeter . A computer then keeps track of these readings, calculates trends, and will warn the flight crew with visual and audio messages if the aircraft is in certain defined flying configurations ("modes").
The modes are:
The traditional GPWS does have a blind spot. Since it can only gather data from directly below the aircraft, it must predict future terrain features. If there is a dramatic change in terrain, such as a steep slope, GPWS will not detect the aircraft closure rate until it is too late for evasive action.
In the late 1990s, improvements were developed and the system is now named "Enhanced Ground Proximity Warning System" (EGPWS/ TAWS ). The system is combined with a worldwide digital terrain database and relies on Global Positioning System (GPS) technology. On-board computers compare current location with a database of the Earth's terrain. The Terrain Display gives pilots a visual orientation to high and low points near the aircraft.
EGPWS software improvements are focused on solving two common problems: no warning at all, and late or improper response.
The primary cause of CFIT occurrences with no GPWS warning is landing short. When the landing gear is down and landing flaps are deployed, the GPWS expects the airplane to land and therefore, issues no warning. However, the GPWS can also malfunction because of a short circuit. On September 26, 1997, Garuda Indonesia Flight 152 crashed into a hilly area, killing all 222 passengers and 12 crew on board. Despite the fact that the plane was nearing terrain, the GPWS did not activate, even though the landing gear and landing flaps were not deployed. EGPWS introduces the Terrain Clearance Floor (TCF) function, which provides GPWS protection even in the landing configuration.
The occurrence of a GPWS alert typically happens at a time of high workload and nearly always surprises the flight crew. Almost certainly, the aircraft is not where the pilot thinks it should be, and the response to a GPWS warning can be late in these circumstances. Warning time can also be short if the aircraft is flying into steep terrain since the downward-looking radio altimeter is the primary sensor used for the warning calculation. The EGPWS improves terrain awareness and warning times by introducing the Terrain Display and the Terrain Data Base Look Ahead protection. [ citation needed ]
In commercial and airline operations, there are legally mandated procedures that must be followed should an EGPWS caution or warning occur. Both pilots must respond and act accordingly once the alert has been issued. In Indonesia, the captain of Garuda Indonesia Flight 200 was charged with manslaughter for not adhering to these procedures. [ 20 ]
In 2015, Air France Flight 953 (a Boeing 777-200ER aircraft) avoided controlled flight into terrain after the EGPWS detected Mount Cameroon in the aircraft's flight path. The pilot flying immediately responded to the initial warning from the EGPWS. [ 21 ]
TAWS equipment is not required by the U.S. FAA in piston-engined aircraft, but optional equipment categorised as TAWS Type C may be installed. Depending on the type of operation, TAWS is only required to be installed into turbine-powered aircraft with six or more passenger seats.
A smaller and less expensive version of EGPWS was developed by AlliedSignal (now merged with Honeywell ) for general aviation and private aircraft. [ 22 ]
For fast military aircraft, the high speed and low altitude that may frequently be flown make traditional GPWS systems unsuitable, as the blind spot becomes the critical part. Thus, an enhanced system is required, taking inputs not only from the radar altimeter , but also from inertial navigation system (INS), Global Positioning System (GPS), and flight control system (FCS), using these to accurately predict the flight path of the aircraft up to 5 miles (8.0 km) ahead. Digital maps of terrain and obstacle features are then used to determine whether a collision is likely if the aircraft does not pull up at a given pre-set g-level . If a collision is predicted, a cockpit warning may be provided. This is the type of system deployed on aircraft such as the Eurofighter Typhoon . [ 23 ] The U.S. FAA has also conducted a study about adapting 3-D military thrust vectoring to recover civil jetliners from catastrophes. [ 24 ]
On May 5, 2016, a military GPWS called Automatic Ground Collision Avoidance System (Auto-GCAS) equipped aboard an F-16 was activated after a trainee pilot lost consciousness from excessive G forces during basic fighter manoeuvre training. In an approximately 55 degree nose down attitude at 8,760 ft (2,670 m) and a speed of 750 mph (1,210 km/h), the Auto-GCAS detected that the aircraft was going to strike the terrain and executed an automatic recovery, saving the pilot's life. [ 25 ] | https://en.wikipedia.org/wiki/Ground_proximity_warning_system |
In physics , and in particular in biomechanics , the ground reaction force ( GRF ) is the force exerted by the ground on a body in contact with it. [ 1 ] For example, a person standing motionless on the ground exerts a contact force on it (equal to the person's weight ) and at the same time an equal and opposite ground reaction force is exerted by the ground on the person.
In the above example, the ground reaction force coincides with the notion of a normal force . However, in a more general case, the GRF will also have a component parallel to the ground, for example when the person is walking – a motion that requires the exchange of horizontal ( frictional ) forces with the ground. [ 2 ]
The use of the word reaction derives from Newton's third law , which essentially states that if a force, called action , acts upon a body, then an equal and opposite force, called reaction , must act upon another body. The force exerted by the ground is conventionally referred to as the reaction, although, since the distinction between action and reaction is completely arbitrary, the expression ground action would be, in principle, equally acceptable.
The component of the GRF parallel to the surface is the frictional force. When slippage occurs the ratio of the magnitude of the frictional force to the normal force yields the coefficient of static friction. [ 3 ]
GRF is often observed to evaluate force production in various groups within the community. One of these groups studied often are athletes to help evaluate a subject's ability to exert force and power. This can help create baseline parameters when creating strength and conditioning regimens from a rehabilitation and coaching standpoint. Plyometric jumps such as a drop-jump is an activity often used to build greater power and force which can lead to overall better ability on the playing field. When landing from a safe height in a bilateral comparisons on GRF in relation to landing with the dominant foot first followed by the non-dominant limb, literature has shown there were no significances in bilateral components with landing with the dominant foot first faster than the non-dominant foot on the GRF of the drop-jump or landing on vertical GRF output. [ 4 ] | https://en.wikipedia.org/wiki/Ground_reaction_force |
Ground reconnaissance (also terrestrial reconnaissance , ground recon ), is a type of reconnaissance that is employed along the elements of ground warfare . [ 1 ] It is the collection of intelligence that strictly involves routes, areas, zones (terrain-oriented); and the enemy (force-oriented). Ground reconnaissance is considered to be the most effective type of reconnaissance but also the slowest method in obtaining information about the terrain and enemy. [ 1 ]
Those units in contact with the enemy, especially patrols, are among the most reliable sources of information. Combat engineers are also good sources of information. These engineer units conduct engineer reconnaissance of an area and can provide detailed reporting on lines of communications ; i.e., roads, rivers, railroad lines, bridges, and obstacles to maneuver. [ citation needed ] | https://en.wikipedia.org/wiki/Ground_reconnaissance |
A ground source heat pump (also geothermal heat pump ) is a heating/cooling system for buildings that use a type of heat pump to transfer heat to or from the ground, taking advantage of the relative constancy of temperatures of the earth through the seasons. Ground-source heat pumps (GSHPs)—or geothermal heat pumps (GHP), as they are commonly termed in North America—are among the most energy-efficient technologies for providing HVAC and water heating , using less energy than can be achieved by use of resistive electric heaters .
Efficiency is given as a coefficient of performance (CoP) which is typically in the range 3-6, meaning that the devices provide 3-6 units of heat for each unit of electricity used. Setup costs are higher than for other heating systems, due to the requirement of installing ground loops over large areas or of drilling bore holes, hence ground source is often installed when new blocks of flats are built. [ 1 ] Air-source heat pumps have lower set-up costs but have a lower CoP in very cold or hot weather.
Ground-source heat pumps take advantage of the difference between the ambient temperature and the temperature at various depths in the ground.
The thermal properties of the ground near the surface [ 2 ] [ 3 ] can be described as follows:
The "penetration depth" [ 3 ] is defined as the depth at which the temperature variable is less than 0.01 of the variation at the surface. This also depends on the type of soil:
The heat pump was described by Lord Kelvin in 1853 and developed by Peter Ritter von Rittinger in 1855. Heinrich Zoelly had patented the idea of using it to draw heat from the ground in 1912. [ 4 ]
After experimentation with a freezer, Robert C. Webber built the first direct exchange ground source heat pump in the late 1940s; sources disagree, however, as to the exact timeline of his invention [ 4 ] [ 5 ] The first successful commercial project was installed in the Commonwealth Building (Portland, Oregon) in 1948, and has been designated a National Historic Mechanical Engineering Landmark by ASME . [ 6 ] Professor Carl Nielsen of Ohio State University built the first residential open loop version in his home in 1948. [ 7 ]
As a result of the 1973 oil crisis , ground source heat pumps became popular in Sweden and have since grown slowly in worldwide popularity as the technology has improved. Open loop systems dominated the market until the development of polybutylene pipe in 1979 made closed loop systems economically viable. [ 6 ]
As of 2004, there are over a million units installed worldwide, providing 12 GW of thermal capacity with a growth rate of 10% per year. [ 8 ] Each year (as of 2011/2004, respectively), about 80,000 units are installed in the US [ 9 ] and 27,000 in Sweden. [ 8 ] In Finland, a geothermal heat pump was the most common heating system choice for new detached houses between 2006 and 2011 with market share exceeding 40%. [ 10 ]
A heat pump is the central unit for the building's heating and cooling. It usually comes in two main variants:
Liquid-to-water heat pumps (also called water-to-water ) are hydronic systems that carry heating or cooling through the building through pipes to conventional radiators , underfloor heating , baseboard radiators and hot water tanks . These heat pumps are also preferred for pool heating. Heat pumps typically only heat water to about 55 °C (131 °F) efficiently, whereas boilers typically operate at 65–95 °C (149–203 °F) [ citation needed ] . The size of radiators designed for the higher temperatures achieved by boilers may be too small for use with heat pumps, requiring replacement with larger radiators when retrofitting a home from boiler to heat pump. When used for cooling, the temperature of the circulating water must normally be kept above the dew point to ensure that atmospheric humidity does not condense on the radiator.
Liquid-to-air heat pumps (also called water-to-air ) output forced air, and are most commonly used to replace legacy forced air furnaces and central air conditioning systems. There are variations that allow for split systems, high-velocity systems, and ductless systems. Heat pumps cannot achieve as high a fluid temperature as a conventional furnace, so they require a higher volume flow rate of air to compensate. When retrofitting a residence, the existing ductwork may have to be enlarged to reduce the noise from the higher air flow.
Ground source heat pumps employ a ground heat exchanger in contact with the ground or groundwater to extract or dissipate heat. Incorrect design can result in the system freezing after a number of years or very inefficient system performance; thus accurate system design is critical to a successful system [ 11 ]
Pipework for the ground loop is typically made of high-density polyethylene pipe and contains a mixture of water and anti-freeze ( propylene glycol , denatured alcohol or methanol ). Monopropylene glycol has the least damaging potential when it might leak into the ground, and is, therefore, the only allowed anti-freeze in ground sources in an increasing number of European countries.
A horizontal closed loop field is composed of pipes that are arrayed in a plane in the ground. A long trench , deeper than the frost line , is dug and U-shaped or slinky coils are spread out inside the same trench. Shallow 3–8-foot (0.91–2.44 m) horizontal heat exchangers experience seasonal temperature cycles due to solar gains and transmission losses to ambient air at ground level. These temperature cycles lag behind the seasons because of thermal inertia, so the heat exchanger will harvest heat deposited by the sun several months earlier, while being weighed down in late winter and spring, due to accumulated winter cold. Systems in wet ground or in water are generally more efficient than drier ground loops since water conducts and stores heat better than solids in sand or soil. If the ground is naturally dry, soaker hoses may be buried with the ground loop to keep it wet.
A vertical system consists of a number of boreholes some 50 to 400 feet (15–122 m) deep fitted with U-shaped pipes through which a heat-carrying fluid that absorbs (or discharges) heat from (or to) the ground is circulated. [ 12 ] [ 13 ] Bore holes are spaced at least 5–6 m apart and the depth depends on ground and building characteristics. Alternatively, pipes may be integrated with the foundation piles used to support the building. Vertical systems rely on migration of heat from surrounding geology, unless recharged during the summer and at other times when surplus heat is available. Vertical systems are typically used where there is insufficient available land for a horizontal system.
Pipe pairs in the hole are joined with a U-shaped cross connector at the bottom of the hole or comprises two small-diameter high-density polyethylene (HDPE) tubes thermally fused to form a U-shaped bend at the bottom. [ 14 ] The space between the wall of the borehole and the U-shaped tubes is usually grouted completely with grouting material or, in some cases, partially filled with groundwater. [ 15 ] For illustration, a detached house needing 10 kW (3 ton ) of heating capacity might need three boreholes 80 to 110 m (260 to 360 ft) deep. [ 16 ]
As an alternative to trenching, loops may be laid by mini horizontal directional drilling (mini-HDD). This technique can lay piping under yards, driveways, gardens or other structures without disturbing them, with a cost between those of trenching and vertical drilling. This system also differs from horizontal & vertical drilling as the loops are installed from one central chamber, further reducing the ground space needed. Radial drilling is often installed retroactively (after the property has been built) due to the small nature of the equipment used and the ability to bore beneath existing constructions.
In an open-loop system (also called a groundwater heat pump), the secondary loop pumps natural water from a well or body of water into a heat exchanger inside the heat pump. Since the water chemistry is not controlled, the appliance may need to be protected from corrosion by using different metals in the heat exchanger and pump. Limescale may foul the system over time and require periodic acid cleaning. This is much more of a problem with cooling systems than heating systems. [ 17 ] A standing column well system is a specialized type of open-loop system where water is drawn from the bottom of a deep rock well, passed through a heat pump, and returned to the top of the well. [ 18 ] A growing number of jurisdictions have outlawed open-loop systems that drain to the surface because these may drain aquifers or contaminate wells. This forces the use of more environmentally sound injection wells or a closed-loop system.
A closed pond loop consists of coils of pipe similar to a slinky loop attached to a frame and located at the bottom of an appropriately sized pond or water source. Artificial ponds are used as heat storage (up to 90% efficient) in some central solar heating plants, which later extract the heat (similar to ground storage) via a large heat pump to supply district heating . [ 19 ] [ 20 ]
The direct exchange geothermal heat pump (DX) is the oldest type of geothermal heat pump technology where the refrigerant itself is passed through the ground loop. Developed during the 1980s, this approach faced issues with the refrigerant and oil management system, especially after the ban of CFC refrigerants in 1989 and DX systems now are infrequently used. [ citation needed ]
Because of the technical knowledge and equipment needed to design and size the system properly (and install the piping if heat fusion is required), a GSHP system installation requires a professional's services. Several installers have published real-time views of system performance in an online community of recent residential installations. The International Ground Source Heat Pump Association ( IGSHPA ), [ 21 ] Geothermal Exchange Organization (GEO), [ 22 ] Canadian GeoExchange Coalition and Ground Source Heat Pump Association maintain listings of qualified installers in the US, Canada and the UK. [ 23 ] Furthermore, detailed analysis of soil thermal conductivity for horizontal systems and formation thermal conductivity for vertical systems will generally result in more accurately designed systems with a higher efficiency. [ 24 ]
Cooling performance is typically expressed in units of BTU/hr/watt as the energy efficiency ratio (EER), while heating performance is typically reduced to dimensionless units as the coefficient of performance (COP). The conversion factor is 3.41 BTU/hr/watt. Since a heat pump moves three to five times more heat energy than the electric energy it consumes, the total energy output is much greater than the electrical input. This results in net thermal efficiencies greater than 300% as compared to radiant electric heat being 100% efficient. Traditional combustion furnaces and electric heaters can never exceed 100% efficiency. Ground source heat pumps can reduce energy consumption – and corresponding air pollution emissions – up to 72% compared to electric resistance heating with standard air-conditioning equipment. [ 25 ]
Efficient compressors, variable speed compressors and larger heat exchangers all contribute to heat pump efficiency. Residential ground source heat pumps on the market today have standard COPs ranging from 2.4 to 5.0 and EERs ranging from 10.6 to 30. [ 26 ] [ 27 ] To qualify for an Energy Star label, heat pumps must meet certain minimum COP and EER ratings which depend on the ground heat exchanger type. For closed-loop systems, the ISO 13256-1 heating COP must be 3.3 or greater and the cooling EER must be 14.1 or greater. [ 28 ]
Standards ARI 210 and 240 define Seasonal Energy Efficiency Ratio (SEER) and Heating Seasonal Performance Factors (HSPF) to account for the impact of seasonal variations on air source heat pumps. These numbers are normally not applicable and should not be compared to ground source heat pump ratings. However, Natural Resources Canada has adapted this approach to calculate typical seasonally adjusted HSPFs for ground-source heat pumps in Canada. [ 16 ] The NRC HSPFs ranged from 8.7 to 12.8 BTU/hr/watt (2.6 to 3.8 in nondimensional factors, or 255% to 375% seasonal average electricity utilization efficiency) for the most populated regions of Canada.
For the sake of comparing heat pump appliances to each other, independently from other system components, a few standard test conditions have been established by the American Refrigerant Institute (ARI) and more recently by the International Organization for Standardization . Standard ARI 330 ratings were intended for closed-loop ground-source heat pumps, and assume secondary loop water temperatures of 25 °C (77 °F) for air conditioning and 0 °C (32 °F) for heating. These temperatures are typical of installations in the northern US. Standard ARI 325 ratings were intended for open-loop ground-source heat pumps, and include two sets of ratings for groundwater temperatures of 10 °C (50 °F) and 21 °C (70 °F). ARI 325 budgets more electricity for water pumping than ARI 330. Neither of these standards attempts to account for seasonal variations. Standard ARI 870 ratings are intended for direct exchange ground-source heat pumps. ASHRAE transitioned to ISO 13256–1 in 2001, which replaces ARI 320, 325 and 330. The new ISO standard produces slightly higher ratings because it no longer budgets any electricity for water pumps. [ 26 ]
Soil without artificial heat addition or subtraction and at depths of several metres or more remains at a relatively constant temperature year round. This temperature equates roughly to the average annual air temperature of the chosen location, usually 7–12 °C (45–54 °F) at a depth of 6 metres (20 ft) in the northern US. Because this temperature remains more constant than the air temperature throughout the seasons, ground source heat pumps perform with far greater efficiency during extreme air temperatures than air conditioners and air-source heat pumps.
A challenge in predicting the thermal response of a ground heat exchanger (GHE) [ 29 ] is the diversity of the time and space scales involved. Four space scales and eight time scales are involved in the heat transfer of GHEs. The first space scale having practical importance is the diameter of the borehole (~ 0.1 m) and the associated time is on the order of 1 hr, during which the effect of the heat capacity of the backfilling material is significant. The second important space dimension is the half distance between two adjacent boreholes, which is on the order of several meters. The corresponding time is on the order of a month, during which the thermal interaction between adjacent boreholes is important. The largest space scale can be tens of meters or more, such as the half-length of a borehole and the horizontal scale of a GHE cluster. The time scale involved is as long as the lifetime of a GHE (decades). [ 30 ]
The short-term hourly temperature response of the ground is vital for analyzing the energy of ground-source heat pump systems and for their optimum control and operation. By contrast, the long-term response determines the overall feasibility of a system from the standpoint of the life cycle.
The main questions that engineers may ask in the early stages of designing a GHE are (a) what the heat transfer rate of a GHE as a function of time is, given a particular temperature difference between the circulating fluid and the ground, and (b) what the temperature difference as a function of time is, given a required heat exchange rate. In the language of heat transfer, the two questions can probably be expressed as q l = [ T f ( t ) − T 0 ] / R ( t ) {\displaystyle q_{l}=[T_{f}(t)-T_{0}]/R(t)}
where T f is the average temperature of the circulating fluid, T 0 is the effective, undisturbed temperature of the ground, q l is the heat transfer rate of the GHE per unit time per unit length (W/m), and R is the total thermal resistance (m . K/W). R ( t ) is often an unknown variable that needs to be determined by heat transfer analysis. Despite R ( t ) being a function of time, analytical models exclusively decompose it into a time-independent part and a time-dependent part to simplify the analysis.
Various models for the time-independent and time-dependent R can be found in the references. [ 12 ] [ 13 ] Further, a thermal response test is often performed to make a deterministic analysis of ground thermal conductivity to optimize the loopfield size, especially for larger commercial sites (e.g., over 10 wells).
The efficiency of ground source heat pumps can be greatly improved by using seasonal thermal energy storage and interseasonal heat transfer. [ 31 ] Heat captured and stored in thermal banks in the summer can be retrieved efficiently in the winter. Heat storage efficiency increases with scale, so this advantage is most significant in commercial or district heating systems.
Geosolar combisystems have been used to heat and cool a greenhouse using an aquifer for thermal storage. [ 20 ] [ 32 ] In summer, the greenhouse is cooled with cold ground water. This heats the water in the aquifer which can become a warm source for heating in winter. [ 32 ] [ 33 ] The combination of cold and heat storage with heat pumps can be combined with water/humidity regulation. These principles are used to provide renewable heat and renewable cooling [ 34 ] to all kinds of buildings.
Also the efficiency of existing small heat pump installations can be improved by adding large, cheap, water-filled solar collectors. These may be integrated into a to-be-overhauled parking lot, or in walls or roof constructions by installing one-inch PE pipes into the outer layer.
The US Environmental Protection Agency (EPA) has called ground source heat pumps the most energy-efficient, environmentally clean, and cost-effective space conditioning systems available. [ 35 ] Heat pumps offer significant emission reductions potential where the electricity is produced from renewable resources.
GSHPs have unsurpassed thermal efficiencies and produce zero emissions locally, but their electricity supply includes components with high greenhouse gas emissions unless it is a 100% renewable energy supply. Their environmental impact, therefore, depends on the characteristics of the electricity supply and the available alternatives.
The GHG emissions savings from a heat pump over a conventional furnace can be calculated based on the following formula: [ 39 ]
GHG Savings = H L ( F I A F U E × 1000 k g t o n − E I C O P × 3600 s e c h r ) {\displaystyle {\text{GHG Savings}}=\mathrm {HL} \left({\frac {\mathrm {FI} }{\mathrm {AFUE} \times 1000{\frac {\mathrm {kg} }{\mathrm {ton} }}}}-{\frac {\mathrm {EI} }{\mathrm {COP} \times 3600{\frac {\mathrm {sec} }{\mathrm {hr} }}}}\right)}
Ground-source heat pumps always produce fewer greenhouse gases than air conditioners, oil furnaces, and electric heating, but natural gas furnaces may be competitive depending on the greenhouse gas intensity of the local electricity supply. In countries like Canada and Russia with low emitting electricity infrastructure, a residential heat pump may save 5 tons of carbon dioxide per year relative to an oil furnace, or about as much as taking an average passenger car off the road. But in cities like Beijing or Pittsburgh that are highly reliant on coal for electricity production, a heat pump may result in 1 or 2 tons more carbon dioxide emissions than a natural gas furnace. For areas not served by utility natural gas infrastructure, however, no better alternative exists.
The fluids used in closed loops may be designed to be biodegradable and non-toxic, but the refrigerant used in the heat pump cabinet and in direct exchange loops was, until recently, chlorodifluoromethane , which is an ozone-depleting substance. [ 26 ] Although harmless while contained, leaks and improper end-of-life disposal contribute to enlarging the ozone hole . For new construction, this refrigerant is being phased out in favor of the ozone-friendly but potent greenhouse gas R410A . Open-loop systems (i.e. those that draw ground water as opposed to closed-loop systems using a borehole heat exchanger) need to be balanced by reinjecting the spent water. This prevents aquifer depletion and the contamination of soil or surface water with brine or other compounds from underground. [ citation needed ]
Before drilling, the underground geology needs to be understood, and drillers need to be prepared to seal the borehole, including preventing penetration of water between strata. The unfortunate example is a geothermal heating project in Staufen im Breisgau , Germany which seems the cause of considerable damage to historical buildings there. In 2008, the city centre was reported to have risen 12 cm, [ 40 ] after initially sinking a few millimeters. [ 41 ] The boring tapped a naturally pressurized aquifer, and via the borehole this water entered a layer of anhydrite, which expands when wet as it forms gypsum. The swelling will stop when the anhydrite is fully reacted, and reconstruction of the city center "is not expedient until the uplift ceases". By 2010 sealing of the borehole had not been accomplished. [ 42 ] [ 43 ] [ 44 ] By 2010, some sections of town had risen by 30 cm. [ 45 ]
Ground source heat pumps are characterized by high capital costs and low operational costs compared to other HVAC systems. Their overall economic benefit depends primarily on the relative costs of electricity and fuels, which are highly variable over time and across the world. Based on recent prices, ground-source heat pumps currently have lower operational costs than any other conventional heating source almost everywhere in the world. Natural gas is the only fuel with competitive operational costs, and only in a handful of countries where it is exceptionally cheap, or where electricity is exceptionally expensive. [ 39 ] In general, a homeowner may save anywhere from 20% to 60% annually on utilities by switching from an ordinary system to a ground-source system. [ 46 ] [ 47 ]
Capital costs and system lifespan have received much less study until recently, and the return on investment is highly variable. The rapid escalation in system price has been accompanied by rapid improvements in efficiency and reliability. Capital costs are known to benefit from economies of scale , particularly for open-loop systems, so they are more cost-effective for larger commercial buildings and harsher climates. The initial cost can be two to five times that of a conventional heating system in most residential applications, new construction or existing. In retrofits, the cost of installation is affected by the size of the living area, the home's age, insulation characteristics, the geology of the area, and the location of the property. Proper duct system design and mechanical air exchange should be considered in the initial system cost.
Capital costs may be offset by government subsidies; for example, Ontario offered $7000 for residential systems installed in the 2009 fiscal year. Some electric companies offer special rates to customers who install a ground-source heat pump for heating or cooling their building. [ 48 ] Where electrical plants have larger loads during summer months and idle capacity in the winter, this increases electrical sales during the winter months. Heat pumps also lower the load peak during the summer due to the increased efficiency of heat pumps, thereby avoiding the costly construction of new power plants. For the same reasons, other utility companies have started to pay for the installation of ground-source heat pumps at customer residences. They lease the systems to their customers for a monthly fee, at a net overall saving to the customer.
The lifespan of the system is longer than conventional heating and cooling systems. Good data on system lifespan is not yet available because the technology is too recent, but many early systems are still operational today after 25–30 years with routine maintenance. Most loop fields have warranties for 25 to 50 years and are expected to last at least 50 to 200 years. [ 46 ] [ 49 ] Ground-source heat pumps use electricity for heating the house. The higher investment above conventional oil, propane or electric systems may be returned in energy savings in 2–10 years for residential systems in the US. [ 50 ] [ 47 ] [ 49 ] The payback period for larger commercial systems in the US is 1–5 years, even when compared to natural gas. [ 47 ] Additionally, because geothermal heat pumps usually have no outdoor compressors or cooling towers, the risk of vandalism is reduced or eliminated, potentially extending a system's lifespan. [ 51 ]
Ground source heat pumps are recognized as one of the most efficient heating and cooling systems on the market. They are often the second-most cost-effective solution in extreme climates (after co-generation ), despite reductions in thermal efficiency due to ground temperature. (The ground source is warmer in climates that need strong air conditioning, and cooler in climates that need strong heating.) The financial viability of these systems depends on the adequate sizing of ground heat exchangers (GHEs), which generally contribute the most to the overall capital costs of GSHP systems. [ 52 ]
Commercial systems maintenance costs in the US have historically been between $0.11 to $0.22 per m 2 per year in 1996 dollars, much less than the average $0.54 per m 2 per year for conventional HVAC systems. [ 6 ]
Governments that promote renewable energy will likely offer incentives for the consumer (residential), or industrial markets. For example, in the United States, incentives are offered both on the state and federal levels of government. [ 53 ] | https://en.wikipedia.org/wiki/Ground_source_heat_pump |
Ground substance is an amorphous gel-like substance in the extracellular space of animals that contains all components of the extracellular matrix (ECM) except for fibrous materials such as collagen and elastin . [ 1 ] Ground substance is active in the development, movement, and proliferation of tissues, as well as their metabolism. Additionally, cells use it for support, water storage, binding, and a medium for intercellular exchange (especially between blood cells and other types of cells). Ground substance provides lubrication for collagen fibers. [ 2 ]
The components of the ground substance vary depending on the tissue. Ground substance is primarily composed of water and large organic molecules, such as glycosaminoglycans (GAGs), proteoglycans , and glycoproteins . GAGs are polysaccharides that trap water, giving the ground substance a gel-like texture. Important GAGs found in ground substance include hyaluronic acid , heparan sulfate , dermatan sulfate , and chondroitin sulfate . With the exception of hyaluronic acid, GAGs are bound to proteins called proteoglycans. Glycoproteins are proteins that attach components of the ground substance to one another and to the surfaces of cells. [ 3 ] Components of the ground substance are secreted by fibroblasts . Usually it is not visible on slides, because it is lost during staining in the preparation process. [ 4 ]
Link proteins such as vinculin , spectrin and actomyosin stabilize the proteoglycans and organize elastic fibers in the ECM. [ 2 ] Changes in the density of ground substance can allow collagen fibers to form aberrant cross-links. [ 2 ] Loose connective tissue is characterized by few fibers and cells, and a relatively large amount of ground substance. Dense connective tissue has a smaller amount of ground substance compared to the fibrous material. [ 2 ]
The meaning of the term has evolved over time. [ 5 ] | https://en.wikipedia.org/wiki/Ground_substance |
A grounding resistance tester also called an earth tester is a soil resistance measuring instrument. It is used for sizing and projecting grounding grids . [ 1 ]
The first soil resistance measuring instrument was invented in the 1950s by Evershed & Vignoles Meggers who made the first insulation and earth resistance testers. [ 2 ] One of the most used analog grounding testers in USSR were М416. [ 3 ] From the 21st century several companies produced digital earth resistance meters and testers. The main purpose of the instrument [ 4 ] is to determine the adequacy of the grounding of an electrical system. By a standard of the National Electrical Code [ 5 ] the resistance of the soil should be less than 25 Ohms to reliably and efficiently ground the installation. [ 6 ]
The meter generates an electrical current and then supplies it to the measuring electrodes. [ 7 ] The potential difference between the two electrodes gives information about the value of soil resistance.
The analog grounding resistance tester is realized by four main blocks DC generator, rectifier , current and potential coil. The deflection of the pointer of the analog screen depends on the ratio of the voltage of pressure coil to the current of the current coil. [ 8 ]
The digital grounding resistance tester is realized by digital electronic blocks as Timers, voltage regulators , and digital display. The ranges are changed with multiturn trimpot . [ 9 ]
When measuring earth resistance with an instrument, it is important to know some of its basic characteristics in order to accurately measure the soil resistance and to properly size the grounding installation. Most importantly, the range of resistance the device measures. Usually the range is three or four degrees. The soil moisture at which the appliance operates is another important parameter. If the instrument cannot operate at a certain humidity, then the measurement may differ significantly from the real value of soil resistance.
Comparison analog and digital grounding resistance testers. [ 10 ] The main characteristics:
2(0.5 Ohm - 50 Ohm);
3(2 ohms - 200 ohms);
4(10 Ohm - 1000 ohms)
1(0 Ohm - 20 Ohm);
2(0 Ohm - 200 Ohm);
3(0 ohms - 2000 ohms);
1(0 Ohm - 20 Ohm);
2(0 Ohm - 200 Ohm);
3(0 ohms - 2000 ohms); | https://en.wikipedia.org/wiki/Grounding_resistance_tester |
Groundwater-Dependent Ecosystems (or GDE s) are ecosystems that rely upon groundwater for their continued existence. Groundwater is water that has seeped down beneath Earth's surface and has come to reside within the pore spaces in soil and fractures in rock, this process can create water tables and aquifers , which are large storehouses for groundwater . An ecosystem is a community of living organisms interacting with the nonliving aspects of their environment (such as air, soil, water, and even groundwater ). With a few exceptions, the interaction between various ecosystems and their respective groundwater is a vital yet poorly understood relationship, and their management is not nearly as advanced as in-stream ecosystems . [ 1 ]
Examining the composition of stable isotopes in the water found in soil, rivers, groundwater , and xylem (or vein systems) of vegetation, using mass spectroscopy , which measures and sort the masses in a sample, along with data on the changes in groundwater depth coupled with the time and vegetative rooting patterns, shows spatial changes over time in the use of groundwater by the vegetation in its respective ecosystem . [ 1 ]
A groundwater-dependent ecosystem can also be inferred through plant water use and growth. In areas with high rainfall groundwater reliance can be seen by monitoring the water use made by the plants of the ecosystem in relation to the water storage in the soil of the area. If the use of water in the vegetation exceeds that of the water being stored in the soil it is a strong indication of groundwater utilization. In areas of prolonged drought the continuation of water flow and plant growth are highly indicative of a groundwater reliant area. [ 1 ]
Remote Sensing is the scanning of Earth by satellite or aircraft to obtain information. [ 2 ] GIS is a system designed to capture, store, analyze and manage geographic data. [ 3 ] Together the data collected (such as elevation and bore holes measuring groundwater levels ) can very accurately predict where groundwater-dependent ecosystems are, how extensive they are, and can guide field expeditions to the right areas for further confirmation and data collection on the GDEs. [ 4 ] [ 5 ]
Due to the high variety of ecosystems and their individual fluctuation in dependency on groundwater there is some uncertainty when it comes to defining an ecosystem strictly as groundwater-dependent or merely groundwater-using. [ 6 ] Each ecosystem expresses a varying degree of dependency. An ecosystem can be directly or indirectly dependent, [ 7 ] as well as have a variation in groundwater use throughout the seasons. [ 1 ] There are a variety of methods for classifying types of groundwater-dependent ecosystems either by their geomorphological setting and/or by their respective groundwater flow mechanism (deep or shallow). [ 6 ]
Arid to humid terrestrial environments with no standing water but deeply rooted vegetation relies upon groundwater to support the producers of their ecosystem . The deeply rooted vegetation requires the groundwater to maintain a consistent or semi-consistent level to allow for their continued health and survival. [ 6 ]
Springs, arguably, rely the most heavily on the continued contribution of groundwater because they are a natural discharge from relatively deep groundwater flows rising to the surface. [ 6 ] Springs are often in association with uniquely adapted plants and animals. [ 7 ]
Wetlands require a shallow discharge of groundwater , it flows as a seepage into depressions in the land surface, [ 6 ] in some instances wetlands feed off of perched groundwater which is groundwater separated from the regular water table by an impermeable layer. [ 8 ] Marshes are a type of wetland and though not directly reliant on groundwater they use it as an area of recharge . [ 6 ] Bogs , are also a type of wetland that is not directly reliant on groundwater but uses the presence of groundwater to provide the area with recharge as well as buoyancy . [ 7 ]
Rivers collect groundwater discharge from aquifers . This can happen seasonally, intermittently or constantly, and can keep an area's water needs stable during a dry season. [ 6 ]
Lagoons and estuaries use groundwater flow to help dilute the salinity in the water and helps support their distinctly unique coastal ecosystems . [ 6 ]
The extraction of groundwater in both large and smaller amounts lowers the areas water table, and in too large of quantities can even collapse parts of the aquifer and permanently damage the quantity of water the aquifer can store. [ 9 ]
Due to the increase in populated areas estuaries and other aquatic ecosystems face a greater threat of pollution . In many cases groundwater can become polluted through toxins , or even just excessive amounts of certain nutrients seeping down to the water table . This polluting of the groundwater can have many different effects on the related ecosystems in the case of an estuary in Cape Cod it was noted that an influx of new nitrogen had come from septic tank fields in the groundwater's flow path . [ 10 ] Increased levels of nitrogen in aquatic ecosystems can cause eutrophication which is the process of excessive introduction of nutrients causing an abundance of plant growth which can result in the death of a variety of aquatic life. [ 11 ]
Urbanization of land has significant effects on groundwater recharge , deforestation and urbanizing limits the amount of surface area viable for water to actually infiltrate and contribute to the groundwater . [ 12 ] | https://en.wikipedia.org/wiki/Groundwater-dependent_ecosystems |
Groundwater banking is a water management mechanism designed to increase water supply reliability. [ 1 ] Groundwater can be created by using dewatered aquifer space to store water during the years when there is abundant rainfall . It can then be pumped and used during years that do not have a surplus of water. [ 1 ] People can manage the use of groundwater to benefit society through the purchasing and selling of these groundwater rights. The surface water should be used first, and then the groundwater will be used when there is not enough surface water to meet demand. [ 2 ] The groundwater will reduce the risk of relying on surface water and will maximize expected income. [ 2 ] There are regulatory storage-type aquifer recovery and storage systems which when water is injected into it gives the right to withdraw the water later on. [ 2 ] Groundwater banking has been implemented into semi-arid and arid southwestern United States because this is where there is the most need for extra water. [ 2 ] The overall goal is to transfer water from low-value to high-value uses by bringing buyers and sellers together. [ 2 ]
The bank is an aquifer used as an underground storage tank, and the recharge of water causes an increase in the volume of water stored in the aquifer to have increasing water levels. [ 2 ] In the case of a withdrawal there would be a decrease in water levels. [ 2 ] The amount of water depends also on a couple of other factors including groundwater pumping by other users, leakage, and natural recharge. [ 2 ] The recharge of water by land application or injection increases the volume of water, and then some of the water will be used at a future time. [ 2 ] It can be looked at as inputs of water minus outputs is equal to the change in water storage. [ 2 ]
Another aspect is the hydrology which is the difference between dynamic and static response to recharge and abstraction. [ 2 ] The water levels will rise or fall in the well during recharge and recovery. [ 2 ] Once recharge and recovery stops the water levels return to background levels, and one of the main issues is the change in static water levels after the dynamic response from recharge or recovery disappears. [ 2 ]
There can be some technical issues with the aquifer response to manage recharge and recovery. [ 2 ] If the aquifer is hydraulically connected to body of water on the surface there are increases in the water table elevation as the result of managed recharge. [ 2 ] This could increase the rate of discharge or decrease the amount of induced recharge, and both of these cause water to leave the basin. [ 2 ] The recharge and recovery could also affect the lateral and vertical groundwater flow into the aquifer. [ 2 ] There is not always a one-to-one correspondence between the volume of water and the change in storage. [ 2 ]
Groundwater banking is accomplished in two ways: through in-lieu and direct recharge. [ 1 ] In-lieu recharge is storing water by utilizing surface water "in-lieu" of pumping groundwater, thereby storing an equal amount in the groundwater basin. [ 1 ] In-lieu recharge is the renewable surface water used to irrigate the farmland in place of using regular groundwater. [ 3 ] This is helping to save more groundwater because the water stays in the aquifer to be used later. [ 3 ] Direct recharge is storing water by allowing it to percolate directly to storage in the groundwater basin. [ 1 ] With direct recharge it floods an area so that water seeps through the ground to get to the aquifers . [ 3 ] The water is then pumped out when there is more of a demand with the use of recovery wells. [ 3 ]
There are some disadvantages to retrieving this groundwater. As groundwater is withdrawn from below the surface, the ground above settles. [ 4 ] This settling of land, known as subsidence, can fracture roads and building foundations and can burst water, sewer, and gas lines. [ 4 ] This method hasn't been well tested yet, so there could be some negative impacts on the environment. Water is a public resource and this could make water become a private industry. [ 5 ]
Groundwater banking can be compared and contrasted to the use of surface reservoirs. Groundwater banking has many advantages over the use of surface reservoirs. The projects do not cost as much to construct and store the same amount, if not more, than the surface reservoirs. [ 6 ] The bank will have less of an impact on the environment than a surface reservoir. [ 6 ] The water that is in the bank will no longer be exposed to evaporation , but several feet per year of water is lost in the reservoirs. [ 6 ] It is more reliable to use when the climate is changing and can respond to seasonal changes better to manage the water than the surface reservoirs. [ 6 ]
There are also some disadvantages to groundwater banking over surface reservoirs. There are energy costs to recovering the water and these costs are usually more than the reservoirs. [ 6 ] There is also a pumping capacity and when the demands change during the year the productiveness can be limited. [ 6 ]
Not all groundwater is used when sold. Some groundwater is being studied for its benefits. Groundwater banking and aquifer storage systems are being explored to control flooding during times of high precipitation. [ 7 ]
The groundwater is being traded in many regions. There are trades even in the United States. The city of San Antonio , Texas is the largest city in the United States that relies solely on groundwater for its municipal supply. [ 7 ]
There have not been many successful trials of groundwater banking on agricultural land since the land is usually privately owned. [ 8 ] The owners have to be on board with the practice of groundwater banking knowing what the risks and best practices entail. [ 8 ] A study was done to find a Soil Agricultural Groundwater Banking Index (SAGBI) which evaluates soil suitability for the use of groundwater banking in California . [ 8 ] There are five factors that determine the feasibility of groundwater recharge on agricultural land: deep percolation, root zone residence time, topography , chemical limitations, and soil surface conditions. [ 8 ] The five factors were modeled using United States Department of Agriculture Natural Resources Conservation Service (USDA-NRCS) digital soil survey data. [ 8 ]
For the deep percolation factor a high rate of water transmission through the soil profile and into the aquifer below is the key to successful groundwater banking. [ 8 ] It becomes more important when there is flooding since it could be used as the main water source. [ 8 ] It is derived from the saturated hydraulic conductivity of the limiting layer. [ 8 ] Saturated hydraulic conductivity measures soil permeability when the soil is saturated. [ 8 ]
When looking at root zone residence time factor it was found that a prolonged duration of saturated conditions in the root zone has the possibility to cause damage to perennial crops. [ 8 ] If the soil causes a bud break it is more likely that the crop will become damaged. [ 8 ] Most crops are not able to withstand long periods of saturated conditions in the root zone. [ 8 ] The root zone residence time estimates the likelihood of having good enough drainage within the root zone once water is applied. [ 8 ]
The topography of land for spreading water across fields has the best outcome when there is level topography. [ 8 ] Level topography works the best because it holds water better on the landscape which allows infiltration across large areas. [ 8 ] Infiltration reduces ponding and minimizes erosion by runoff. [ 8 ]
The chemical limitations factor is related to the salinity which is a threat to the sustainability of agriculture and groundwater. [ 8 ] This factor was determined by electrical conductivity (EC) of the soil which measures the soil salinity . [ 8 ] The best soil has the lowest levels of salinity. [ 8 ] Soil also has pesticides and nitrate, but it is unable to be evaluated due to the dependency on management history. [ 8 ]
The surface condition factor is when banking by flood spreading can change the soil surfaces physical conditions. [ 8 ] Infiltrations limits can be caused by quality and depth of water that could lead to the destruction of aggregates, the formation of physical crusts, and compaction. [ 8 ] To determine soil condition two factors were examined: soil erosion factor and sodium absorption ratio (SAR). [ 8 ]
To determine the feasibility of groundwater banking each of the five factors were assigned a weight to how significant it was, and then a SAGBI score was calculated. [ 8 ] The weights were 27.5% deep percolation, 27.5% root zone residence time, 20% topography, 20% chemical limitations, and 5% surface conditions. [ 8 ] Of the 17.5 million acres of agricultural land examined only 5 million acres were considered soils with excellent, good, and moderately good suitability. [ 8 ]
Agricultural groundwater banking can be associated with financial risk which may cause crop loss, so in the end, the loss may exceed the benefits of water saving. [ 8 ] Adoption of this practice would require support to protect growers from risk of crop failure. [ 8 ]
The accounting system tracks the recharge and withdrawals of stored water and it can include a market system to reserve the storage of water. [ 2 ] Depositors could earn credits for the recharge of water which can be used later on for the water recovered from the bank. [ 2 ] The banking system could then be set up to allow trading of credits. [ 2 ] There are several objectives of the water bank accounting system: track water deposits, withdrawals, and to control the amount, timing, and location of withdrawals by the participants. [ 2 ] If groundwater is not regulated there is more of a chance for freeriding and overuse . [ 2 ] There needs to be sustainability of the system in order to continue with the operation of the system or there would be no point to using it. [ 2 ]
The accounting method that will be used is the double-entry accounting method, so every transaction is recorded as a debit and a credit in separate ledger accounts. [ 2 ] This also allows for tracking of inventory in asset accounts and claims to inventory in ownership accounts. [ 2 ] Deposits happen when more water is stored in an aquifer than there is supposed to be. [ 2 ] The recharge of water by a member would be a credit in a member's account and a liability in the bank's account. [ 2 ] For the bank to be successful then both ledgers have to be balanced, so the right to water in a member's accounts should be equal to the amount of water that can be recovered from the system. [ 2 ] If the right to water is greater than liabilities then the bank is insolvent, and this will become a problem when a drought occurs. [ 2 ]
There are some issues that could arise from using this water bank accounting system. These problems were evaluated for the Las Posas Basin groundwater bank and Fox Canyon Groundwater Management Agency (FCGMA) has jurisdiction over the project. [ 2 ] FCGMA reported that the accumulation of credits has been increasing for banks. [ 2 ] What can happen is the accumulated credits can become greater than the annual abstraction rate. [ 2 ] The volume of credits accumulating exceed the amount of water that can be taken out during a short-time period. [ 2 ] This will cause a threat to the regional groundwater resource or even depressions in groundwater elevations, land subsidence, and seawater intrusion. [ 2 ]
The banking systems need regulatory control over the basin to implement the withdrawal rates and to ensure that other participants will not extract too much stored water. [ 2 ] The best scenario would be that the bank owner or participants would be the main users to ensure that abstractions are controlled. [ 2 ] If this is not the case, then there must be another way to control the number and amount of abstractions happening. [ 2 ] It needs to be clear who has priority over stored water, so that when abstractions are constrained it is known who will get the water first. [ 2 ] There can be problems when multiple entities have jurisdiction over a project, and this can cause regulatory and organizational challenges. [ 2 ] There are some generally accepted rules and also many of the issues are handled through state-specific concepts. [ 2 ] One of the main requirements is an action needs to be in place so that the stored water is not being abstracted by other users who are not involved in the system. [ 2 ] These frameworks rely on the knowledge of hydrogeology to determine the success of a system, and the systems need to provide benefits to prove it was worth building. [ 2 ]
The different projects can become economically efficient by maximizing the benefits of the limited resource (water). [ 9 ] To maximize efficiency the users need to find where marginal cost is equal to marginal benefit . [ 9 ] It is important for supply to equal demand like in the figure below. The use of water becomes a negative externality when there is rivalry and the property rights are not well-defined. [ 9 ] The way to eliminate some of the negative externality is there can be a tax placed on the resource to increase the marginal cost. [ 9 ] When they tax the right amount the user will use the resource at the socially acceptable level. [ 9 ] The other way to affect the externality is to create a subsidy . A subsidy will increase the marginal benefit in order to get to the socially accepted level of use for the resource. [ 9 ]
Water is not a homogeneous commodity for several reasons which include sensitivity to location, time of use, form of the water, and administrative responses. [ 9 ] The use of groundwater banking can make water a more homogenous commodity. This can create a market value which will enhance private investment increasing the benefits. [ 9 ] It will also align marginal benefit with marginal cost causing the market to come to an economically efficient level. [ 9 ]
Water has high transaction costs and create market barriers which devalues the use to society restricting the reallocation of resources. [ 9 ] The demand does not change when there is a market barrier so there will be many unpleasant people if they do not get their share of water. [ 9 ] If a bank is in the process of being made the removal of market-access barriers can be part of the negotiations, but it is not necessarily the bank itself that is the cause. [ 9 ] Groundwater banking could reduce transaction costs because each individual won't have to analyze each transaction. [ 9 ] | https://en.wikipedia.org/wiki/Groundwater_banking |
Groundwater contamination by pharmaceuticals , which belong to the category of contaminants of emerging concern (CEC) or emerging organic pollutants (EOP), has been receiving increasing attention in the fields of environmental engineering , hydrology and hydrogeochemistry since the last decades of the twentieth century. [ 1 ]
Pharmaceuticals are suspected to provoke long-term effects in aquatic ecosystems even at low concentration ranges ( trace concentrations ) because of their bioactive and chemically stable nature, which leads to recalcitrant behaviours in the aqueous compartments , a feature that is typically associated with the difficulty in degrading these compounds to innocuous molecules, similarly with the behaviour exhibited by persistent organic pollutants . [ 1 ] [ 2 ] Furthermore, continuous release of medical products in the water cycle poses concerns about bioaccumulation and biomagnification phenomena. [ 3 ] As the vulnerability of groundwater systems is increasingly recognized even from the regulating authority (the European Medicines Agency , EMA), environmental risk assessment (ERA) procedures, which is required for pharmaceuticals appliance for marketing authorization and preventive actions urged to preserve these environments. [ 4 ] [ 5 ]
In the last decades of the twentieth century, scientific research efforts have been fostered towards deeper understanding of the interactions of groundwater transport and attenuation mechanisms with the chemical nature of polluting agents. [ 6 ] Amongst the multiple mechanisms governing solutes mobility in groundwater, biotransformation and biodegradation play a crucial role in determining the evolution of the system (as identified by developing concentration fields) in the presence of organic compounds , such as pharmaceuticals. [ 7 ] Other processes that might impact on pharmaceuticals fate in groundwater include classical advective - dispersive mass transfer , as well as geochemical reactions, such as adsorption onto soils and dissolution / precipitation . [ 7 ]
One major goal in the field of environmental protection and risk mitigation is the development of mathematical formulations yielding reliable predictions of the fate of pharmaceuticals in aquifer systems, eventually followed by an appropriate quantification of predictive uncertainty and estimation of the risks associated with this kind of contamination . [ 6 ]
Pharmaceuticals represent a serious threat to aquifer systems because of their bioactive nature, which makes them capable of interacting directly with therein residing living microorganisms and yielding bioaccumulation and biomagnification phenomena. [ 8 ] [ 9 ] Occurrence of xenobiotics in groundwater has been proven to harm the delicate equilibria of aquatic ecosystems in several ways, such as promoting the growth of antibiotic -resistant bacteria [ 10 ] [ 11 ] or causing hormones -related sexual disruption in living organisms in surface waters. [ 12 ] [ 13 ] [ 14 ] Considering then the role of groundwater systems as main worldwide drinking water resources, the capability of pharmaceuticals to interact with human tissues poses serious concerns also in terms of human health. Indeed, the majority of pharmaceuticals do not degrade in groundwater, where get accumulated due to their continuous release in the environment. [ 15 ] Then, these compounds reach subsurface systems through different sources, such as hospital effluents, wastewaters and landfill leachates , which clearly risk contaminating drinking water. [ 8 ] [ 9 ]
The main pharmaceutical classes detected in worldwide groundwater systems are listed below. [ 1 ] The following categorisation is based on a medical perspective and it is often referred to as therapeutic classification.
The chemical structure of pharmaceuticals affects the type of hydro-geochemical processes that mainly impacts on their fate in groundwater and it is strictly associated with their chemical properties. [ 1 ] [ 6 ] [ 7 ] [ 17 ] Therefore, a classification of pharmaceuticals based on chemical classes is a valid alternative to the purpose of understanding the role of molecular structures in determining the kind of physical and geochemical processes affecting their mobility in porous media .
With regard to the occurrence of medical drugs in subsurface aquatic systems, the following chemical properties are of major interest: [ 7 ]
Pharmaceuticals solubility in water affects the mobility of these compounds within aquifers. This feature depends on pharmaceuticals polarity , as polar substances are typically hydrophilic , thereby showing marked tendency to dissolve in the aqueous phase, where they become solutes . [ 7 ] This aspect impacts on dissolution / precipitation equilibrium, a phenomenon that is mathematically described in terms of the substance solubility product (addressed in many books with the notation K S {\displaystyle K_{S}} ). [ 7 ]
Large K O W {\displaystyle K_{OW}} values outline the non polar character of the chemical species, which shows instead particular affinity to dissolve into organic solvents. [ 7 ] Therefore, lipophilic pharmaceuticals are markedly subjected to the risk to bioaccumulate and biomagnificate in the environment, consistent with their preferential partition with the organic tissues of living organisms. Sufficiently large K O W {\displaystyle K_{OW}} pharmaceuticals are in fact subjected to specific tiers in the environmental risk assessment (ERA) procedure (to be supplied for the marketing authorisation application) and are highlighted as potential sources of bioaccumulation and biomagnification according to the EMA guidelines. [ 4 ] Lipophilic compounds are then insoluble in water, where they persist as a separated phase from the aqueous one. This renders their mobility in groundwater basically decoupled with dissolution / precipitation mechanisms and attributed to the mean flow transport (advection and dispersion) and soil-mediated mechanisms of reaction (adsorption). [ 7 ]
This feature is expressed in terms of the so-called organic carbon-water partition coefficient , that is usually referred to as K O C {\displaystyle K_{OC}} and is an intrinsic property of the molecule. [ 7 ]
Molecules behaviour in relation to aqueous dissociation reactions is typically related to their acid dissociation constants , that are typically outlined in terms of their P k a {\displaystyle Pk_{a}} coefficients. [ 18 ]
The molecular structure of xenobiotics typically outlines the existence of several possible reaction pathways, which are embedded in complex reaction networks and are typically referred to as transformation processes. [ 7 ] With reference to organic compounds, such as pharmaceuticals, innumerable kinds of chemical reactions exist, most of them involving common chemical mechanisms, such as functional groups elimination , addition and substitution . [ 19 ] These processes often involve further redox reactions accomplished on the substrates, which are here represented by pharmaceutical solutes and, eventually, their transformation products and metabolites . [ 19 ] [ 7 ] These processes can be then classified as either biotic or abiotic , depending on the presence or absence of bacterial communities acting as reaction mediators. [ 7 ] In the former case, these transformation pathways are typically addressed as biodegradation or biotransformation in the hydrogeochemical literature, depending on the extent of cleavage of the parent molecule into highly oxidized, innocuous species. [ 7 ] [ 20 ]
The fate of pharmaceuticals in groundwater is governed by different processes. The reference theoretical framework is that of reactive solute transport in porous media at the continuum scale, that is typically interpreted through the advective-dispersive-reactive equation (ADRE). [ 21 ] With reference to the saturated region of the aquifer, the ADRE is written as: [ 6 ]
∂ ∂ t ( ϕ C W ( x , t ) ) = − ∇ ( ϕ C W ( x , t ) v ( x , t ) ) ⏟ advection + ∇ ( ϕ D ( x , t ) ∇ C W ( x , t ) ) ⏟ hydrodynamic dispersion + R ⏟ accumulation {\displaystyle {\frac {\partial }{\partial t}}(\phi C_{W}({\boldsymbol {x}},t))=-\underbrace {\nabla (\phi C_{W}({\boldsymbol {x}},t){\boldsymbol {v}}({\boldsymbol {x}},t))} _{\text{advection}}+\underbrace {\nabla (\phi {\boldsymbol {D}}({\boldsymbol {x}},t)\nabla C_{W}({\boldsymbol {x}},t))} _{\text{hydrodynamic dispersion}}+\underbrace {R} _{\text{accumulation}}}
Where ϕ {\displaystyle \phi } represents the effective porosity of the medium, x {\displaystyle {\boldsymbol {x}}} and t {\displaystyle t} represent - respectively - the spatial coordinates vector and the time coordinate. ∇ {\displaystyle \nabla } represents the divergence operator, except for when it applies to C W {\displaystyle C_{W}} , where the nabla symbol stands for gradient of C W {\displaystyle C_{W}} . The term C W ( x , t ) {\displaystyle C_{W}({\boldsymbol {x}},t)} denotes then the pharmaceutical solute concentration field in the water phase (for unsaturated regions of the aquifer, the ADRE equation has a similar shape, but it includes additional terms accounting for volumetric contents and contaminants concentrations in other phases than water), while v ( x , t ) {\displaystyle {\boldsymbol {v}}({\boldsymbol {x}},t)} represents the velocity field. D ( x , t ) {\displaystyle D({\boldsymbol {x}},t)} is the hydrodynamic dispersion tensor and is typically function of the sole variable x {\displaystyle {\boldsymbol {x}}} . Lastly, the storage term R {\displaystyle R} includes the accumulation or removal contribution due to all possible reactive processes in the system, i.e., adsorption, dissolution / precipitation, acid dissociation and other transformation reactions, such as biodegradation. [ 6 ]
The main hydrological transport processes driving pharmaceuticals and organic contaminants migration in aquifer systems are: [ 7 ]
The most influential geochemical processes, also referred to as reactive processes and whose effect is embedded in the term R {\displaystyle R} of the ADRE, include: [ 7 ] [ 16 ]
Advective transport accounts for the contribution of solute mass transfer across the system that originates from bulk flow motion. At the continuum scale of analysis, the system is interpreted as a continuous medium rather than a collection of solid particles ( grains ) and empty spaces ( pores ) through which the fluid can flow. In this context, an average flow velocity can be typically estimated, which arises upscaling the pore scale velocities. Here, the fluid flow conditions ensure the validity of the Darcy's law , which governs the system evolution in terms of average fluid velocity, typically referred to as seepage or advective velocity . [ 22 ] Dissolved pharmaceuticals in groundwater are transferred within the domain along with the mean fluid flow and in agreement with the physical principles governing any other solute migration across the system.
Hydrodynamic dispersion identifies a process that arises as summation of two separate effects. First, it is associated with molecular diffusion , a phenomenon that is appreciated at the macroscale as consequence of microscale Brownian motions . Secondly, it includes a contribution (called mechanical dispersion ) arising as an effect of upscaling the fluid-dynamic transport problem from the pore to the continuum scale of investigation, due to the upscaling of local dishomogeneous velocities. The latter contribution is therefore not related to the occurrence of any physical process at the pore scale, but it is only a fictitious consequence of the modelling scale choice. Hydrodynamic dispersion is then embedded in the advective-dispersive-reactive equation (ADRE) assuming a Fickian closure model. Dispersion is felt at the macroscale as responsible of a spread effect of the contaminant plume around its center of mass. [ 22 ]
Sorption identifies a heterogeneous reaction that is often driven by instantaneous thermochemical equilibrium. [ 7 ] It describes the process for which a certain mass of solute dissolved in the aqueous phase adheres to a solid phase (such as the organic fraction of soil in the case of organic compounds), being therefore removed from the liquid phase. [ 7 ] In hydrogeochemistry, this phenomenon has been proved to cause a delayed effect in solute mobility with respect to the case in which solely advection and dispersion occur in the aquifer. [ 6 ] For pharmaceuticals, it can be typically interpreted using a linear adsorption model at equilibrium, which is fully applicable at low concentrations ranges. [ 23 ] The latter model relies upon assessment of a linear partition coefficient , usually denoted as k d {\displaystyle k_{d}} , that depends - for organic compounds - on both organic carbon-water partition coefficient K O C {\displaystyle K_{OC}} and organic carbon fraction f O C {\displaystyle f_{OC}} into soil. [ 7 ] While the former term is an intrinsic chemical property of the molecule, the latter one instead depends on the soil moisture of the analyzed aquifer. [ 7 ]
Sorption of trace elements like pharmaceuticals in groundwater is interpreted through the following linear isotherm model: [ 6 ]
C S = k d C W {\displaystyle C_{S}=k_{d}C_{W}}
Where C S {\displaystyle C_{S}} identifies the adsorbed concentration on the solid phase and k d ∝ K O C , f O C {\displaystyle k_{d}\propto K_{OC},f_{OC}} . [ 7 ]
The neutral form of the organic molecules dissolved in water is typically the sole responsible of sorptive mechanisms, that become as more important as the soils are rich in terms of organic carbon. [ 7 ] Anionic forms are instead insensitive to sorptive mechanisms, while cations can undergo adsorption only in very particular conditions. [ 7 ]
Dissolution represents the heterogeneous reaction during which a solid compound, such as an organic salt in the case of pharmaceuticals, gets dissolved into the aqueous phase. [ 7 ] Here, the original salt appears in the form of both aqueous cations and anions, depending on the stoichiometry of the dissolution reaction. [ 7 ] Precipitation represents the reverse reaction. This process is typically accomplished at thermochemical equilibrium, but in some applications of hydrogeochemical modelling it might be required to consider its kinetics . [ 7 ] As an example for the case of pharmaceuticals, the non-steroidal anti-inflammatory drug diclofenac, which is commercialised as sodium diclofenac, undergoes this process in groundwater environments. [ 6 ]
Acid dissociation is a homogeneous reaction that yields dissociation of a dissolved acid (in the water phase) into cationic and anionic forms, while aqueous complexation denotes its reverse process. [ 7 ] The aqueous speciation of a solution is determined on the basis of the P k a {\displaystyle Pk_{a}} coefficient, that typically ranges between 3 and 50 (approximately) for organic compounds, such as pharmaceuticals. [ 19 ] [ 24 ] Being the latter ones weak acids and considering that this process is always accomplished upon instantaneous achievement of thermochemical equilibrium conditions, it is then reasonable to assume that the undissociated form of the original contaminant is predominant in the water speciation for most practical cases in the field of hydrogeochemistry.
Pharmaceuticals can undergo biotransformation or transformation processes in groundwater systems. [ 7 ]
Aquifers are indeed rich reserves in terms of minerals and other dissolved chemical species, such as organic matter , dissolved oxygen , nitrates , ferrous and manganese compounds, sulfates , etc., as well as dissolved cations, such as calcium , magnesium and sodium ones. [ 7 ] All of these compounds interact through complex reaction networks embedding reactive processes of different nature, such as carbonates precipitation / dissolution, acid–base reactions , sorption and redox reactions. [ 7 ] With reference to the latter kind of processes, several pathways are typically possible in aquifers because the environment is often rich in both reducing (like organic matter) and oxidizing agents (like dissolved oxygen, nitrates, ferrous and Manganese oxides, sulfates etc.). [ 7 ] Pharmaceuticals can act as substrates as well in this scenario, i.e., they can represent either the reducing, or the oxidizing agent in the context of redox processes. In fact, most chemical reactions involving organic molecules are typically accomplished upon gain or loss of electrons , so that the oxidation state of the molecule changes along the reactive pathway. [ 7 ] In this context, the aquifer acts as a "chemical reactor". [ 25 ]
There are innumerable kinds of chemical reactions that pharmaceuticals can undergo in this environment, which depend on the availability of other reactants , pH and other environmental conditions, but all of these processes typically share common mechanisms . The main ones involve addition , elimination or substitution of functional groups . [ 7 ] The mechanism of reaction is important in the field of hydrogeochemical modeling of aquifer systems because all of these reactions are typically governed by kinetic laws . Therefore, recognizing the correct molecular mechanisms through which a chemical reaction progresses is fundamental to the purpose of modelling the reaction rates correctly (for example, it is often possible to identify a rate limiting step within multistep reactions and relate the rate of reaction progress to that particular step). [ 19 ] Modelling these reactions typically follows the classic kinetic laws, except for the case in which reactions involving the contaminant are accomplished in the context of bacterial metabolism . [ 7 ] While in the former case the ensemble of reactions is addressed as transformation pathway, in the latter one the terms biodegradation or biotransformation are used, depending on the extent to which the chemical reactions effectively degrade the original organic molecule to innocuous compounds in their maximum oxidation state (i.e., carbon dioxide , methane and water ). [ 20 ] In case of biologically mediated pathways of reaction, which are relevant in the study of groundwater contamination by pharmaceuticals, there are appropriate kinetic laws that can be employed to model these processes in hydrogeochemical contexts. For example, the Monod and Michaelis-Menten equations are suitable options in case of biotic transformation processes involving organic compounds (such as pharmaceuticals) as substrates. [ 7 ]
Despite most hydrogeochemical literature addresses these processes through linear biodegradation models, several studies have been carried out since the second decade of the twenty-first century, as the former ones are typically too simplified to ensure reliable predictions of pharmaceuticals fate in groundwater and might bias risk estimates in the context of risk mitigation applications for the environment. [ 7 ] [ 26 ]
Groundwater contamination by pharmaceuticals is a topic of great interest in the field of the environmental and hydraulic engineering, where most research efforts have been fostered towards studies on this kind of contaminants since the beginning of the twenty-first century. [ 6 ] The general goal of those disciplines is that of developing interpretive models capable to predict the behaviour of aquifer systems in relation to the occurrence of various types of contaminants, among which are included also medical drugs. Such goal is motivated by the necessity to provide mathematical tools to predict, for example, how contaminants concentration fields develop across the aquifer along time. This may provide useful information to support decision-making processes in the context of environmental risk assessment procedures. [ 4 ] [ 6 ] To this purpose, several interdisciplinary strategies and tools are typically employed, the most fundamental ones being listed below:
All of these interdisciplinary tools and strategies are contemporarily employed to analyse the fate of pharmaceuticals in groundwater. | https://en.wikipedia.org/wiki/Groundwater_contamination_by_pharmaceuticals |
Groundwater discharge is the volumetric flow rate of groundwater through an aquifer .
Total groundwater discharge, as reported through a specified area, is similarly expressed as:
where
For example, this can be used to determine the flow rate of water flowing along a plane with known geometry.
The discharge potential is a potential in groundwater mechanics which links the physical properties, hydraulic head , with a mathematical formulation for the energy as a function of position. The discharge potential, Φ {\textstyle \Phi } [L 3 ·T −1 ], is defined in such way that its gradient equals the discharge vector. [ 1 ]
Q x = − ∂ Φ ∂ x {\displaystyle Q_{x}=-{\frac {\partial \Phi }{\partial x}}}
Q y = − ∂ Φ ∂ y {\displaystyle Q_{y}=-{\frac {\partial \Phi }{\partial y}}}
Thus the hydraulic head may be calculated in terms of the discharge potential, for confined flow as
Φ = K H ϕ {\displaystyle \Phi =KH\phi }
and for unconfined shallow flow as
Φ = 1 2 K ϕ 2 + C {\displaystyle \Phi ={\frac {1}{2}}K\phi ^{2}+C}
where
As mentioned the discharge potential may also be written in terms of position. The discharge potential is a function of the Laplace's equation
∂ 2 Φ ∂ x 2 + ∂ 2 Φ ∂ y 2 = 0 {\displaystyle {\frac {\partial ^{2}\Phi }{\partial x^{2}}}+{\frac {\partial ^{2}\Phi }{\partial y^{2}}}=0}
which solution is a linear differential equation. Because the solution is a linear differential equation for which superposition principle holds, it may be combined with other solutions for the discharge potential, e.g. uniform flow, multiple wells, analytical elements ( analytic element method ). | https://en.wikipedia.org/wiki/Groundwater_discharge |
The groundwater energy balance is the energy balance of a groundwater body in terms of incoming hydraulic energy associated with groundwater inflow into the body, energy associated with the outflow, energy conversion into heat due to friction of flow, and the resulting change of energy status and groundwater level.
When multiplying the horizontal velocity of groundwater (dimension, for example, m 3 / day {\displaystyle m^{3}/{\text{day}}} per m 2 {\displaystyle m^{2}} cross-sectional area) with the groundwater potential (dimension energy per volume of water, or E / m 3 {\displaystyle E/m^{3}} ) one obtains an energy flow (flux) in E / day {\displaystyle E/{\text{day}}} for the given flow and cross-sectional area. [ 1 ]
Summation or integration of the energy flux in a vertical cross-section of unit width (say 1m) from the lower flow boundary (the impermeable layer or base) up to the water table in an unconfined aquifer gives the energy flow f E {\displaystyle f_{E}} through the cross-section in E / day {\displaystyle E/{\text{day}}} per m width of the aquifer.
While flowing, the groundwater loses energy due to friction of flow, i.e. hydraulic energy is converted into heat. At the same time, energy may be added with the recharge of water coming into the aquifer through the water table. Thus one can make an hydraulic energy balance of a block of soil between two nearby cross-sections. In steady state, i.e. without change in energy status and without accumulation or depletion of water stored in the soil body, the energy flow in the first section plus the energy added by groundwater recharge between the sections minus the energy flow in the second section must equal the energy loss due to friction of flow.
In mathematical terms this balance can be obtained by differentiating the cross-sectional integral of Fe in the direction of flow using the Leibniz rule, taking into account that the level of the water table may change in the direction of flow.
The mathematics is simplified using the Dupuit–Forchheimer assumption .
The hydraulic friction losses can be described in analogy to Joule's law in electricity (see Joule's law#Hydraulic equivalent ), where the friction losses are proportional to the square value of the current (flow) and the electrical resistance of the material through which the current occurs. In groundwater hydraulics ( fluid dynamics , hydrodynamics ) one often works with hydraulic conductivity (i.e. permeability of the soil for water), which is inversely proportional to the hydraulic resistance.
The resulting equation of the energy balance of groundwater flow can be used, for example, to calculate the shape of the water table between drains under specific aquifer conditions. For this a numerical solution can be used, taking small steps along the impermeable base. The drainage equation is to be solved by trial and error ( iterations ), because the hydraulic potential is taken with respect to a reference level taken as the level of the water table at the water divide midway between the drains. When calculating the shape of the water table, its level at the water divide is initially not known. Therefore, this level is to be assumed before the calculations on the shape of the water table can be started. According to the findings of the calculation procedure, the initial assumption is to be adjusted and the calculations are to be restarted until the level of the water table at the divide does not differ significantly from the assumed level.
The trial and error procedure is cumbersome and therefore computer programs may be developed to aid in the calculations.
The energy balance of groundwater flow can be applied to flow of groundwater to subsurface drains. [ 2 ] The computer program EnDrain [ 3 ] compares the outcome of the traditional drain spacing equation , based on Darcy's law together with the continuity equation (i.e. conservation of mass ), with the solution obtained by the energy balance and it can be seen that drain spacings are wider in the latter case. This is owing to the introduction of the energy supplied by the incoming recharge. | https://en.wikipedia.org/wiki/Groundwater_energy_balance |
In hydrogeology , groundwater flow is defined as the "part of streamflow that has infiltrated the ground, entered the phreatic zone , and has been (or is at a particular time) discharged into a stream channel or springs ; and seepage water ." [ 1 ] It is governed by the groundwater flow equation .
Groundwater is water that is found underground in cracks and spaces in the soil, sand and rocks. Where water has filled these spaces is the phreatic (also called) saturated zone. Groundwater is stored in and moves slowly (compared to surface runoff in temperate conditions and watercourses) through layers or zones of soil, sand and rocks: aquifers . The rate of groundwater flow depends on the permeability (the size of the spaces in the soil or rocks and how well the spaces are connected) and the hydraulic head (water pressure).
In polar regions groundwater flow may be obstructed by permafrost . [ 2 ] | https://en.wikipedia.org/wiki/Groundwater_flow |
Used in hydrogeology , the groundwater flow equation is the mathematical relationship which is used to describe the flow of groundwater through an aquifer . The transient flow of groundwater is described by a form of the diffusion equation , similar to that used in heat transfer to describe the flow of heat in a solid ( heat conduction ). The steady-state flow of groundwater is described by a form of the Laplace equation , which is a form of potential flow and has analogs in numerous fields.
The groundwater flow equation is often derived for a small representative elemental volume (REV), where the properties of the medium are assumed to be effectively constant. A mass balance is done on the water flowing in and out of this small volume, the flux terms in the relationship being expressed in terms of head by using the constitutive equation called Darcy's law , which requires that the flow is laminar. Other approaches are based on Agent Based Models to incorporate the effect of complex aquifers such as karstic or fractured rocks (i.e. volcanic) [ 1 ]
A mass balance must be performed, and used along with Darcy's law , to arrive at the transient groundwater flow equation. This balance is analogous to the energy balance used in heat transfer to arrive at the heat equation . It is simply a statement of accounting, that for a given control volume, aside from sources or sinks, mass cannot be created or destroyed. The conservation of mass states that, for a given increment of time ( Δt ), the difference between the mass flowing in across the boundaries, the mass flowing out across the boundaries, and the sources within the volume, is the change in storage.
Mass can be represented as density times volume , and under most conditions, water can be considered incompressible (density does not depend on pressure). The mass fluxes across the boundaries then become volume fluxes (as are found in Darcy's law ). Using Taylor series to represent the in and out flux terms across the boundaries of the control volume, and using the divergence theorem to turn the flux across the boundary into a flux over the entire volume, the final form of the groundwater flow equation (in differential form) is:
This is known in other fields as the diffusion equation or heat equation, it is a parabolic partial differential equation (PDE). This mathematical statement indicates that the change in hydraulic head with time (left hand side) equals the negative divergence of the flux ( q ) and the source terms ( G ). This equation has both head and flux as unknowns, but Darcy's law relates flux to hydraulic heads, so substituting it in for the flux ( q ) leads to
Now if hydraulic conductivity ( K ) is spatially uniform and isotropic (rather than a tensor ), it can be taken out of the spatial derivative, simplifying them to the Laplacian , this makes the equation
Dividing through by the specific storage ( S s ), puts hydraulic diffusivity ( α = K/S s or equivalently, α = T/S ) on the right hand side. The hydraulic diffusivity is proportional to the speed at which a finite pressure pulse will propagate through the system (large values of α lead to fast propagation of signals). The groundwater flow equation then becomes
Where the sink/source term, G , now has the same units but is divided by the appropriate storage term (as defined by the hydraulic diffusivity substitution).
Especially when using rectangular grid finite-difference models ( e.g. MODFLOW , made by the USGS ), we deal with Cartesian coordinates . In these coordinates the general Laplacian operator becomes (for three-dimensional flow) specifically
MODFLOW code discretizes and simulates an orthogonal 3-D form of the governing groundwater flow equation. However, it has an option to run in a "quasi-3D" mode if the user wishes to do so; in this case the model deals with the vertically averaged T and S , rather than k and S s . In the quasi-3D mode, flow is calculated between 2D horizontal layers using the concept of leakage.
Another useful coordinate system is 3D cylindrical coordinates (typically where a pumping well is a line source located at the origin — parallel to the z axis — causing converging radial flow). Under these conditions the above equation becomes ( r being radial distance and θ being angle),
This equation represents flow to a pumping well (a sink of strength G ), located at the origin. Both this equation and the Cartesian version above are the fundamental equation in groundwater flow, but to arrive at this point requires considerable simplification. Some of the main assumptions which went into both these equations are:
Despite these large assumptions, the groundwater flow equation does a good job of representing the distribution of heads in aquifers due to a transient distribution of sources and sinks.
If the aquifer has recharging boundary conditions a steady-state may be reached (or it may be used as an approximation in many cases), and the diffusion equation (above) simplifies to the Laplace equation .
This equation states that hydraulic head is a harmonic function , and has many analogs in other fields. The Laplace equation can be solved using techniques, using similar assumptions stated above, but with the additional requirements of a steady-state flow field.
A common method for solution of this equations in civil engineering and soil mechanics is to use the graphical technique of drawing flownets ; where contour lines of hydraulic head and the stream function make a curvilinear grid , allowing complex geometries to be solved approximately.
Steady-state flow to a pumping well (which never truly occurs, but is sometimes a useful approximation) is commonly called the Thiem solution .
The above groundwater flow equations are valid for three dimensional flow. In unconfined aquifers , the solution to the 3D form of the equation is complicated by the presence of a free surface water table boundary condition: in addition to solving for the spatial distribution of heads, the location of this surface is also an unknown. This is a non-linear problem, even though the governing equation is linear.
An alternative formulation of the groundwater flow equation may be obtained by invoking the Dupuit–Forchheimer assumption , where it is assumed that heads do not vary in the vertical direction (i.e., ∂ h / ∂ z = 0 {\displaystyle \partial h/\partial z=0} ). A horizontal water balance is applied to a long vertical column with area δ x δ y {\displaystyle \delta x\delta y} extending from the aquifer base to the unsaturated surface. This distance is referred to as the saturated thickness , b . In a confined aquifer , the saturated thickness is determined by the height of the aquifer, H , and the pressure head is non-zero everywhere. In an unconfined aquifer , the saturated thickness is defined as the vertical distance between the water table surface and the aquifer base. If ∂ h / ∂ z = 0 {\displaystyle \partial h/\partial z=0} , and the aquifer base is at the zero datum, then the unconfined saturated thickness is equal to the head, i.e., b=h .
Assuming both the hydraulic conductivity and the horizontal components of flow are uniform along the entire saturated thickness of the aquifer (i.e., ∂ q x / ∂ z = 0 {\displaystyle \partial q_{x}/\partial z=0} and ∂ K / ∂ z = 0 {\displaystyle \partial K/\partial z=0} ), we can express Darcy's law in terms of integrated groundwater discharges , Q x and Q y :
Inserting these into our mass balance expression, we obtain the general 2D governing equation for incompressible saturated groundwater flow:
Where n is the aquifer porosity . The source term, N (length per time), represents the addition of water in the vertical direction (e.g., recharge). By incorporating the correct definitions for saturated thickness , specific storage , and specific yield , we can transform this into two unique governing equations for confined and unconfined conditions:
(confined), where S=S s b is the aquifer storativity and
(unconfined), where S y is the specific yield of the aquifer.
Note that the partial differential equation in the unconfined case is non-linear, whereas it is linear in the confined case. For unconfined steady-state flow, this non-linearity may be removed by expressing the PDE in terms of the head squared:
Or, for homogeneous aquifers,
This formulation allows us to apply standard methods for solving linear PDEs in the case of unconfined flow. For heterogeneous aquifers with no recharge, Potential flow methods may be applied for mixed confined/unconfined cases. | https://en.wikipedia.org/wiki/Groundwater_flow_equation |
Groundwater pollution (also called groundwater contamination ) occurs when pollutants are released to the ground and make their way into groundwater . This type of water pollution can also occur naturally due to the presence of a minor and unwanted constituent, contaminant, or impurity in the groundwater, in which case it is more likely referred to as contamination rather than pollution . Groundwater pollution can occur from on-site sanitation systems, landfill leachate, effluent from wastewater treatment plants , leaking sewers, petrol filling stations , hydraulic fracturing (fracking) or from over application of fertilizers in agriculture . Pollution (or contamination) can also occur from naturally occurring contaminants, such as arsenic or fluoride . [ 1 ] Using polluted groundwater causes hazards to public health through poisoning or the spread of disease ( water-borne diseases ).
The pollutant often produces a contaminant plume within an aquifer . Movement of water and dispersion within the aquifer spreads the pollutant over a wider area. Its advancing boundary, often called a plume edge, can intersect with groundwater wells and surface water, such as seeps and springs, making the water supplies unsafe for humans and wildlife. The movement of the plume, called a plume front, may be analyzed through a hydrological transport model or groundwater model . Analysis of groundwater pollution may focus on soil characteristics and site geology , hydrogeology , hydrology , and the nature of the contaminants. Different mechanisms have influence on the transport of pollutants, e.g. diffusion , adsorption , precipitation , decay , in the groundwater.
The interaction of groundwater contamination with surface waters is analyzed by use of hydrology transport models. Interactions between groundwater and surface water are complex. For example, many rivers and lakes are fed by groundwater. This means that damage to groundwater aquifers e.g. by fracking or over abstraction, could therefore affect the rivers and lakes that rely on it. Saltwater intrusion into coastal aquifers is an example of such interactions. [ 2 ] [ 3 ] Prevention methods include: applying the precautionary principle , groundwater quality monitoring, land zoning for groundwater protection, locating on-site sanitation systems correctly and applying legislation. When pollution has occurred, management approaches include point-of-use water treatment , groundwater remediation , or as a last resort, abandonment.
Contaminants found in groundwater cover a broad range of physical, inorganic chemical, organic chemical, bacteriological, and radioactive parameters. Principally, many of the same pollutants that play a role in surface water pollution may also be found in polluted groundwater, although their respective importance may differ.
Arsenic and fluoride have been recognized by the World Health Organization (WHO) as the most serious inorganic contaminants in drinking-water on a worldwide basis. [ 4 ] [ 5 ]
Inorganic arsenic is the most common type of arsenic in soil and water. [ 6 ] The metalloid arsenic can occur naturally in groundwater, as seen most frequently in Asia, including in China , India and Bangladesh . [ 7 ] In the Ganges Plain of northern India and Bangladesh severe contamination of groundwater by naturally occurring arsenic affects 25% of water wells in the shallower of two regional aquifers . Groundwater in these areas is also contaminated by the use of arsenic-based pesticides . [ 8 ]
Arsenic in groundwater can also be present where there are mining operations or mine waste dumps that will leach arsenic.
Natural fluoride in groundwater is of growing concern as deeper groundwater is being used, "with more than 200 million people at risk of drinking water with elevated concentrations." [ 9 ] Fluoride can especially be released from acidic volcanic rocks and dispersed volcanic ash when water hardness is low. High levels of fluoride in groundwater is a serious problem in the Argentinean Pampas , Chile , Mexico , India , Pakistan , the East African Rift , and some volcanic islands ( Tenerife ) [ 10 ]
In areas that have naturally occurring high levels of fluoride in groundwater which is used for drinking water, both dental and skeletal fluorosis can be prevalent and severe. [ 11 ]
The lack of proper sanitation measures, as well as improperly placed wells , can lead to drinking water contaminated with pathogens carried in feces and urine . Such fecal–oral transmitted diseases include typhoid , cholera and diarrhea . [ 12 ] [ 13 ] Of the four pathogen types that are present in feces ( bacteria , viruses , protozoa , and helminths or helminth eggs), the first three can be commonly found in polluted groundwater, whereas the relatively large helminth eggs are usually filtered out by the soil matrix. [ citation needed ]
Deep, confined aquifers are usually considered the safest source of drinking water with respect to pathogens. Pathogens from treated or untreated wastewater can contaminate certain, especially shallow, aquifers. [ 14 ] [ 15 ]
Nitrate is the most common chemical contaminant in the world's groundwater and aquifers. [ 16 ] In some low-income countries, nitrate levels in groundwater are extremely high, causing significant health problems. It is also stable (it does not degrade) under high oxygen conditions. [ 4 ]
Elevated nitrate levels in groundwater can be caused by on-site sanitation, sewage sludge disposal and agricultural activities. [ 17 ] It can therefore have an urban or agricultural origin. [ 10 ]
Nitrate levels above 10 mg/L (10 ppm) in groundwater can cause " blue baby syndrome " (acquired methemoglobinemia ). [ 18 ] Drinking water quality standards in the European Union stipulate less than 50 mg/L for nitrate in drinking water . [ 19 ]
The linkages between nitrates in drinking water and blue baby syndrome have been disputed in other studies. [ 20 ] [ 21 ] The syndrome outbreaks might be due to other factors than elevated nitrate concentrations in drinking water. [ 22 ]
Volatile organic compounds (VOCs) are a dangerous contaminant of groundwater. They are generally introduced to the environment through careless industrial practices. Many of these compounds were not known to be harmful until the late 1960s and it was some time before regular testing of groundwater identified these substances in drinking water sources. [ citation needed ]
Primary VOC pollutants found in groundwater include aromatic hydrocarbons such as BTEX compounds ( benzene , toluene , ethylbenzene and xylenes ), and chlorinated solvents including tetrachloroethylene (PCE), trichloroethylene (TCE), and vinyl chloride (VC). BTEX are important components of gasoline . PCE and TCE are industrial solvents used in dry cleaning processes and as a metal degreaser, respectively. [ citation needed ]
Other organic pollutants present in groundwater and derived from industrial operations are the polycyclic aromatic hydrocarbons (PAHs). Due to its molecular weight, naphthalene is the most soluble and mobile PAH found in groundwater, whereas benzo(a)pyrene is the most toxic one. PAHs are generally produced as byproducts by incomplete combustion of organic matter. [ citation needed ]
Organic pollutants can also be found in groundwater as insecticides and herbicides . As many other synthetic organic compounds, most pesticides have very complex molecular structures. This complexity determines the water solubility, adsorption capacity, and mobility of pesticides in the groundwater system. Thus, some types of pesticides are more mobile than others so they can more easily reach a drinking-water source. [ 9 ]
Several trace metals occur naturally in certain rock formations and can enter in the environment from natural processes such as weathering. However, industrial activities such as mining , metallurgy , solid waste disposal, paint and enamel works, etc. can lead to elevated concentrations of toxic metals including lead , cadmium and chromium . These contaminants have the potential to make their way into groundwater. [ 17 ]
The migration of metals (and metalloids) in groundwater will be affected by several factors, in particular by chemical reactions which determine the partitioning of contaminants among different phases and species. Thus, the mobility of metals primarily depends on the pH and redox state of groundwater. [ 9 ]
Trace amounts of pharmaceuticals from treated wastewater infiltrating into the aquifer are among emerging ground-water contaminants being studied throughout the United States. [ 23 ] Popular pharmaceuticals such as antibiotics, anti-inflammatories, antidepressants, decongestants,
tranquilizers, etc. are normally found in treated wastewater. [ 24 ] This wastewater is discharged from the treatment facility, and often makes its way into the aquifer or source of surface water used for drinking water. [ citation needed ]
Trace amounts of pharmaceuticals in both groundwater and surface water are far below what is considered dangerous or of concern in most areas, but it could be an increasing problem as population grows and more reclaimed wastewater is utilized for municipal water supplies. [ 24 ] [ 25 ]
Other organic pollutants include a range of organohalides and other chemical compounds, petroleum hydrocarbons , various chemical compounds found in personal hygiene and cosmetic products, drug pollution involving pharmaceutical drugs and their metabolites. Inorganic pollutants might include other nutrients such as ammonia and phosphate , and radionuclides such as uranium (U) or radon (Rn) naturally present in some geological formations. Saltwater intrusion is also an example of natural contamination, but is very often intensified by human activities. [ citation needed ]
Groundwater pollution is a worldwide issue. A study of the groundwater quality of the principal aquifers of the United States conducted between 1991 and 2004, showed that 23% of domestic wells had contaminants at levels greater than human-health benchmarks. [ 26 ] Another study suggested that the major groundwater pollution problems in Africa, considering the order of importance are: (1) nitrate pollution, (2) pathogenic agents, (3) organic pollution, (4) salinization, and (5) acid mine drainage. [ 27 ]
Causes of groundwater pollution include (further details below):
"Geogenic" refers to naturally occurring as a result from geological processes. [ citation needed ]
The natural arsenic pollution occurs because aquifer sediments contain organic matter that generates anaerobic conditions in the aquifer. These conditions result in the microbial dissolution of iron oxides in the sediment and, thus, the release of the arsenic , normally strongly bound to iron oxides, into the water. As a consequence, arsenic-rich groundwater is often iron-rich, although secondary processes often obscure the association of dissolved arsenic and dissolved iron. [ citation needed ] . Arsenic is found in groundwater most commonly as the reduced species arsenite and the oxidized species arsenate, the acute toxicity of arsenite being somewhat greater than that of arsenate. [ 28 ] Investigations by WHO indicated that 20% of 25,000 boreholes tested in Bangladesh had arsenic concentrations exceeding 50 μg/L. [ 4 ]
The occurrence of fluoride is close related to the abundance and solubility of fluoride-containing minerals such as fluorite (CaF 2 ) . [ 28 ] Considerably high concentrations of fluoride in groundwater are typically caused by a lack of calcium in the aquifer. [ 4 ] Health problems associated with dental fluorosis may occur when fluoride concentrations in groundwater exceed 1.5 mg/L, which is the WHO guideline value since 1984. [ 4 ]
The Swiss Federal Institute of Aquatic Science and Technology (EAWAG) has recently developed the interactive Groundwater Assessment Platform (GAP), where the geogenic risk of contamination in a given area can be estimated using geological, topographical and other environmental data without having to test samples from every single groundwater resource. This tool also allows the user to produce probability risk mapping for both arsenic and fluoride. [ 29 ]
High concentrations of parameters like salinity, iron, manganese, uranium, radon and chromium, in groundwater, may also be of geogenic origin. This contaminants can be important locally but they are not as widespread as arsenic and fluoride. [ 28 ]
Groundwater pollution with pathogens and nitrate can also occur from the liquids infiltrating into the ground from on-site sanitation systems such as pit latrines and septic tanks , depending on the population density and the hydrogeological conditions. [ 12 ]
Factors controlling the fate and transport of pathogens are quite complex and the interaction among them is not well understood. [ 4 ] If the local hydrogeological conditions (which can vary within a space of a few square kilometers) are ignored, simple on-site sanitation infrastructures such as pit latrines can cause significant public health risks via contaminated groundwater. [ citation needed ]
Liquids leach from the pit and pass the unsaturated soil zone (which is not completely filled with water). Subsequently, these liquids from the pit enter the groundwater where they may lead to groundwater pollution. This is a problem if a nearby water well is used to supply groundwater for drinking water purposes. During the passage in the soil, pathogens can die off or be adsorbed significantly, mostly depending on the travel time between the pit and the well. [ 30 ] Most, but not all pathogens die within 50 days of travel through the subsurface. [ 31 ]
The degree of pathogen removal strongly varies with soil type, aquifer type, distance and other environmental factors. [ 32 ] For example, the unsaturated zone becomes "washed" during extended periods of heavy rain, providing hydraulic pathway for the quick pass of pathogens. [ 4 ] It is difficult to estimate the safe distance between a pit latrine or a septic tank and a water source. In any case, such recommendations about the safe distance are mostly ignored by those building pit latrines. In addition, household plots are of a limited size and therefore pit latrines are often built much closer to groundwater wells than what can be regarded as safe. This results in groundwater pollution and household members falling sick when using this groundwater as a source of drinking water. [ citation needed ]
Groundwater pollution can be caused by untreated waste discharge leading to diseases like skin lesions, bloody diarrhea and dermatitis. This is more common in locations having limited wastewater treatment infrastructure, or where there are systematic failures of the on-site sewage disposal system. [ 32 ] Along with pathogens and nutrients, untreated sewage can also have an important load of heavy metals that may seep into the groundwater system. [ citation needed ]
The treated effluent from sewage treatment plants may also reach the aquifer if the effluent is infiltrated or discharged to local surface water bodies. Therefore, those substances that are not removed in conventional sewage treatment plants may reach the groundwater as well. [ 33 ] For example, detected concentrations of pharmaceutical residues in groundwater were in the order of 50 mg/L in several locations in Germany. [ 34 ] This is because in conventional sewage treatment plants, micropollutants such as hormones , pharmaceutical residues and other micropollutants contained in urine and feces are only partially removed and the remainder is discharged into surface water, from where it may also reach the groundwater.
Groundwater pollution can also occur from leaking sewers which has been observed for example in Germany. [ 35 ] This can also lead to potential cross-contamination of drinking-water supplies. [ 36 ]
Spreading wastewater or sewage sludge in agriculture may also be included as sources of fecal contamination in groundwater. [ 4 ]
Nitrate can also enter the groundwater via excessive use of fertilizers, including manure spreading. This is because only a fraction of the nitrogen-based fertilizers is converted to produce and other plant matter. The remainder accumulates in the soil or lost as run-off. [ 37 ] High application rates of nitrogen-containing fertilizers combined with the high water-solubility of nitrate leads to increased runoff into surface water as well as leaching into groundwater, thereby causing groundwater pollution. [ 38 ] The excessive use of nitrogen-containing fertilizers (be they synthetic or natural) is particularly damaging, as much of the nitrogen that is not taken up by plants is transformed into nitrate which is easily leached. [ 39 ]
The nutrients, especially nitrates, in fertilizers can cause problems for natural habitats and for human health if they are washed off soil into watercourses or leached through soil into groundwater. The heavy use of nitrogenous fertilizers in cropping systems is the largest contributor to anthropogenic nitrogen in groundwater worldwide. [ 40 ]
Feedlots/animal corrals can also lead to the potential leaching of nitrogen and metals to groundwater. [ 36 ] Over application of animal manure may also result in groundwater pollution with pharmaceutical residues derived from veterinary drugs. [ citation needed ]
The US Environmental Protection Agency (EPA) and the European Commission are seriously dealing with the nitrate problem related to agricultural development, as a major water supply problem that requires appropriate management and governance. [ 10 ] [ 41 ]
Runoff of pesticides may leach into groundwater causing human health problems from contaminated water wells. [ 4 ] Pesticide concentrations found in groundwater are typically low, and often the regulatory human health-based limits exceeded are also very low. [ 4 ] The organophosphorus insecticide monocrotophos (MCP) appears to be one of a few hazardous, persistent, soluble and mobile (it does not bind with minerals in soils) pesticides able to reach a drinking-water source. [ 42 ] In general, more pesticide compounds are being detected as groundwater quality monitoring programs have become more extensive; however, much less monitoring has been conducted in developing countries due to the high analysis costs. [ 4 ]
A wide variety of both inorganic and organic pollutants have been found in aquifers underlying commercial and industrial activities. [ citation needed ]
Ore mining and metal processing facilities are the primary responsible of the presence of metals in groundwater of anthropogenic origin, including arsenic. The low pH associated with acid mine drainage (AMD) contributes to the solubility of potential toxic metals that can eventually enter the groundwater system. [ citation needed ]
There is an increasing concern over the groundwater pollution by gasoline leaked from petroleum underground storage tanks (USTs) of gas stations . [ 4 ] BTEX compounds are the most common additives of the gasoline. BTEX compounds, including benzene, have densities lower than water (1 g/mL). Similar to the oil spills on the sea, the non-miscible phase, referred to as Light Non-Aqueous Phase Liquid (LNAPL) , will "float" upon the water table in the aquifer. [ 4 ]
Chlorinated solvents are used in nearly any industrial practice where degreasing removers are required. [ 4 ] PCE is a highly utilized solvent in the dry cleaning industry because of its cleaning effectiveness and relatively low cost. It has also been used for metal-degreasing operations. Because it is highly volatile, it is more frequently found in groundwater than in surface water. [ 43 ] [ unreliable source? ] TCE has historically been used as a metal cleaning. The military facility Anniston Army Depot (ANAD) in the United States was placed on the EPA Superfund National Priorities List (NPL) because of groundwater contamination with as much as 27 million pounds of TCE. [ 44 ] Both PCE and TCE may degrade to vinyl chloride (VC), the most toxic chlorinated hydrocarbon. [ 4 ]
Many types of solvents may have also been disposed illegally, leaking over time to the groundwater system. [ 4 ]
Chlorinated solvents such as PCE and TCE have densities higher than water and the non-miscible phase is referred to as Dense Non-Aqueous Phase Liquids (DNAPL) . [ 4 ] Once they reach the aquifer, they will "sink" and eventually accumulate on the top of low-permeability layers. [ 4 ] [ 45 ] Historically, wood-treating facilities have also release insecticides such as pentachlorophenol (PCP) and creosote into the environment, impacting the groundwater resources. [ 46 ] PCP is a highly soluble and toxic obsolete pesticide recently listed in the Stockholm Convention on Persistent Organic Pollutants . PAHs and other semi-VOCs are the common contaminants associated with creosote.
Although non-miscible, both LNAPLs and DNAPLs still have the potential to slowly dissolve into the aqueous (miscible) phase to create a plume and thus become a long-term source of contamination. DNAPLs (chlorinated solvents, heavy PAHs, creosote, PCBs ) tend to be difficult to manage as they can reside very deep in the groundwater system. [ 4 ]
The recent growth of hydraulic fracturing ("Fracking") wells in the United States has raised concerns regarding its potential risks of contaminating groundwater resources. [ 47 ] EPA, along with many other researchers, has been delegated to study the relationship between hydraulic fracturing and drinking water resources. [ 48 ] While it is possible to perform hydraulic fracturing without having a relevant impact on groundwater resources if stringent controls and quality management measures are in place, there are a number of cases where groundwater pollution due to improper handling or technical failures was observed. [ citation needed ]
While the EPA has not found significant evidence of a widespread, systematic impact on drinking water by hydraulic fracturing , this may be due to insufficient systematic pre- and post- hydraulic fracturing data on drinking water quality, and the presence of other agents of contamination that preclude the link between tight oil and shale gas extraction and its impact. [ 49 ]
Despite the EPA's lack of profound widespread evidence, other researchers have made significant observations of rising groundwater contamination in close proximity to major shale oil/gas drilling sites located in Marcellus [ 50 ] [ 51 ] ( British Columbia , Canada). Within one kilometer of these specific sites, a subset of shallow drinking water consistently showed higher concentration levels of methane , ethane , and propane concentrations than normal. An evaluation of higher Helium and other noble gas concentration along with the rise of hydrocarbon levels supports the distinction between hydraulic fracturing fugitive gas and naturally occurring "background" hydrocarbon content. This contamination is speculated to be the result of leaky, failing, or improperly installed gas well casings. [ 52 ]
Furthermore, it is theorized that contamination could also result from the capillary migration of deep residual hyper-saline water and hydraulic fracturing fluid, slowly flowing through faults and fractures until finally making contact with groundwater resources ; [ 52 ] however, many researchers argue that the permeability of rocks overlying shale formations are too low to allow this to ever happen sufficiently. [ 53 ] To ultimately prove this theory, there would have to be traces of toxic trihalomethanes (THM) since they are often associated with the presence of stray gas contamination, and typically co-occur with high halogen concentrations in hyper-saline waters. [ 53 ] Besides, highly saline waters are a common natural feature in deep groundwater systems. [ citation needed ]
While conclusions regarding groundwater pollution as the result to hydraulic fracturing fluid flow is restricted in both space and time, researchers have hypothesized that the potential for systematic stray gas contamination depends mainly on the integrity of the shale oil/gas well structure, along with its relative geological location to local fracture systems that could potentially provide flow paths for fugitive gas migration. [ 52 ] [ 53 ]
Though widespread, systematic contamination by hydraulic fracturing has been heavily disputed, one major source of contamination that has the most consensus among researchers of being the most problematic is site-specific accidental spillage of hydraulic fracturing fluid and produced water . So far, a significant majority of groundwater contamination events are derived from surface-level anthropogenic routes rather than the subsurface flow from underlying shale formations. [ 54 ] While the damage can be obvious, and much more effort is being done to prevent these accidents from occurring so frequently, the lack of data from fracking oil spills continue to leave researchers in the dark. In many of these events, the data acquired from the leakage or spillage is often very vague, and thus would lead researchers to lacking conclusions. [ 55 ]
Researchers from the Federal Institute for Geosciences and Natural Resources (BGR) conducted a model study for a deep shale-gas formation in the North German Basin. They concluded that the probability is small that the rise of fracking fluids through the geological underground to the surface will impact shallow groundwater. [ 56 ]
Leachate from sanitary landfills can lead to groundwater pollution. Chemicals can reach into ground water through precipitation and runoff. New landfills are required to be lined with clay or another synthetic material, along with leachate to protect surrounding ground water. However, older landfills do not have these measures and are often close to surface waters and in permeable soils. Closed landfills can still pose a threat to ground water if they are not capped by an impermeable material before closure to prevent leaking of contaminants. [ 57 ]
Love Canal was one of the most widely known examples of groundwater pollution. In 1978, residents of the Love Canal neighborhood in upstate New York noticed high rates of cancer and an alarming number of birth defects . This was eventually traced to organic solvents and dioxins from an industrial landfill that the neighborhood had been built over and around, which had then infiltrated into the water supply and evaporated in basements to further contaminate the air. Eight hundred families were reimbursed for their homes and moved, after extensive legal battles and media coverage. [ citation needed ]
Satellite data in the Mekong Delta in Vietnam have provided evidence that over-pumping of groundwater leads to land subsidence as well as consequential release of arsenic and possibly other heavy metals. [ 58 ] Arsenic is found in clay strata due to their high surface area to volume ratio relative to sand-sized particles. Most pumped groundwater travels through sands and gravels with low arsenic concentration. However, during over-pumping, a high vertical gradient pulls water from less-permeable clays, thus promoting arsenic release into the water. [ 59 ]
Groundwater pollution can be caused by chemical spills from commercial or industrial operations, chemical spills occurring during transport (e.g. spillage of diesel fuels), illegal waste dumping , infiltration from urban runoff or mining operations, road salts , de-icing chemicals from airports and even atmospheric contaminants since groundwater is part of the hydrologic cycle . [ 60 ]
Herbicide use can contribute to groundwater contamination through arsenic infiltration. Herbicides contribute to arsenic desorption through mobilization and transportation of the contaminant. Chlorinated herbicides exhibit a lower impact on arsenic desorption than phosphate type herbicides. This can help to prevent arsenic contamination through choosing herbicides appropriate for different concentrations of arsenic present in certain soils. [ 61 ]
The burial of corpses and their subsequent degradation may also pose a risk of pollution to groundwater. [ 62 ]
The passage of water through the subsurface can provide a reliable natural barrier to contamination but it only works under favorable conditions. [ 12 ]
The stratigraphy of the area plays an important role in the transport of pollutants. An area can have layers of sandy soil, fractured bedrock, clay, or hardpan. Areas of karst topography on limestone bedrock are sometimes vulnerable to surface pollution from groundwater. Earthquake faults can also be entry routes for downward contaminant entry. Water table conditions are of great importance for drinking water supplies, agricultural irrigation, waste disposal (including nuclear waste), wildlife habitat, and other ecological issues. [ 63 ]
Many chemicals undergo reactive decay or chemical change, especially over long periods of time in groundwater reservoirs. A noteworthy class of such chemicals is the chlorinated hydrocarbons such as trichloroethylene (used in industrial metal degreasing and electronics manufacturing) and tetrachloroethylene used in the dry cleaning industry. Both of these chemicals, which are thought to be carcinogens themselves, undergo partial decomposition reactions, leading to new hazardous chemicals (including dichloroethylene and vinyl chloride ). [ 64 ]
Although interrelated, surface water and groundwater have often been studied and managed as separate resources. [ 65 ] Interactions between groundwater and surface water are complex. Surface water seeps through the soil and becomes groundwater. Conversely, groundwater can also feed surface water sources. For example, many rivers and lakes are fed by groundwater. This means that damage to groundwater aquifers e.g. by fracking or over abstraction, could therefore affect the rivers and lakes that rely on it. Saltwater intrusion into coastal aquifers is an example of such interactions. [ 2 ] [ 3 ]
A spill or ongoing release of chemical or radionuclide contaminants into soil (located away from a surface water body) may not create point or non-point source pollution but can contaminate the aquifer below, creating a toxic plume . The movement of the plume, may be analyzed through a hydrological transport model or groundwater model . [ citation needed ]
The precautionary principle , evolved from Principle 15 of the Rio Declaration on Environment and Development , is important in protecting groundwater resources from pollution. The precautionary principle provides that "where there are threats of irreversible damage, lack of full scientific certainty shall not be used as reason for postponing cost-effective measures to prevent environmental degradation ." [ 66 ]
One of the six basic principles of the European Union (EU) water policy is the application of the precautionary principle. [ 67 ]
Groundwater quality monitoring programs have been implemented regularly in many countries around the world. They are important components to understand the hydrogeological system, and for the development of conceptual models and aquifer vulnerability maps. [ 68 ]
Groundwater quality must be regularly monitored across the aquifer to determine trends. Effective groundwater monitoring should be driven by a specific objective, for example, a specific contaminant of concern. [ 9 ] Contaminant levels can be compared to the World Health Organization (WHO) guidelines for drinking-water quality. [ 69 ] It is not rare that limits of contaminants are reduced as more medical experience is gained. [ 10 ]
Sufficient investment should be given to continue monitoring over the long term. When a problem is found, action should be taken to correct it. [ 9 ] Waterborne outbreaks in the United States decreased with the introduction of more stringent monitoring (and treatment) requirements in the early 90s. [ 4 ]
The community can also help monitor the groundwater quality. [ 68 ]
Scientists have developed methods by which hazard maps could be produced for geogenic toxic substances in groundwater. [ 70 ] [ 71 ] [ 72 ] This provides an efficient way of determining which wells should be tested. [ citation needed ]
The development of land-use zoning maps has been implemented by several water authorities at different scales around the world. There are two types of zoning maps: aquifer vulnerability maps and source protection maps. [ 9 ]
It refers to the intrinsic (or natural) vulnerability of a groundwater system to pollution. [ 9 ] Intrinsically, some aquifers are more vulnerable to pollution than other aquifers. [ 68 ] Shallow unconfined aquifers are more at risk of pollution because there are fewer layers to filter out contaminants. [ 9 ]
The unsaturated zone can play an important role in retarding (and in some cases eliminating) pathogens and so must be considered when assessing aquifer vulnerability. [ 4 ] The biological activity is greatest in the top soil layers where the attenuation of pathogens is generally most effective. [ 4 ]
Preparation of the vulnerability maps typically involves overlaying several thematic maps of physical factors that have been selected to describe the aquifer vulnerability. [ 68 ] The index-based parametric mapping method GOD developed by Foster and Hirata (1988) uses three generally available or readily estimated parameters, the degree of G roundwater hydraulic confinement, geological nature of the O verlying strata and D epth to groundwater. [ 68 ] [ 73 ] [ 74 ] A further approach developed by EPA, a rating system named "DRASTIC", employs seven hydrogeological factors to develop an index of vulnerability: D epth to water table, net R echarge, A quifer media, S oil media, T opography (slope), I mpact on the vadose zone , and hydraulic C onductivity . [ 68 ] [ 75 ]
There is a particular debate among hydrogeologists as to whether aquifer vulnerability should be established in a general (intrinsic) way for all contaminants, or specifically for each pollutant. [ 68 ]
It refers to the capture areas around an individual groundwater source, such as a water well or a spring, to especially protect them from pollution. Thus, potential sources of degradable pollutants, such as pathogens, can be located at distances which travel times along the flowpaths are long enough for the pollutant to be eliminated through filtration or adsorption. [ 9 ]
Analytical methods using equations to define groundwater flow and contaminant transport are the most widely used. [ 76 ] The WHPA is a semi-analytical groundwater flow simulation program developed by the US EPA for delineating capture zones in a wellhead protection area. [ 77 ]
The simplest form of zoning employs fixed-distance methods where activities are excluded within a uniformly applied specified distance around abstraction points. [ 76 ]
As the health effects of most toxic chemicals arise after prolonged exposure, risk to health from chemicals is generally lower than that from pathogens. [ 4 ] Thus, the quality of the source protection measures is an important component in controlling whether pathogens may be present in the final drinking-water. [ 76 ]
On-site sanitation systems can be designed in such a way that groundwater pollution from these sanitation systems is prevented from occurring. [ 12 ] [ 31 ] Detailed guidelines have been developed to estimate safe distances to protect groundwater sources from pollution from on-site sanitation . [ 78 ] [ 79 ] The following criteria have been proposed for safe siting (i.e. deciding on the location) of on-site sanitation systems: [ 12 ]
As a very general guideline it is recommended that the bottom of the pit should be at least 2 m above groundwater level, and a minimum horizontal distance of 30 m between a pit and a water source is normally recommended to limit exposure to microbial contamination. [1] However, no general statement should be made regarding the minimum lateral separation distances required to prevent contamination of a well from a pit latrine. [ 12 ] For example, even 50 m lateral separation distance might not be sufficient in a strongly karstified system with a downgradient supply well or spring, while 10 m lateral separation distance is completely sufficient if there is a well developed clay cover layer and the annular space of the groundwater well is well sealed. [ citation needed ]
Institutional and legal issues are critical in determining the success or failure of groundwater protection policies and strategies. [ 4 ] In the United States the Resource Conservation and Recovery Act protects groundwater by regulating the disposal of solid waste and hazardous waste , and the Comprehensive Environmental Response, Compensation, and Liability Act , also known as "Superfund", requires remediation of abandoned hazardous waste sites. [ 80 ] [ 81 ]
Options for remediation of contaminated groundwater can be grouped into the following categories:
Portable water purification devices or "point-of-use" (POU) water treatment systems and field water disinfection techniques can be used to remove some forms of groundwater pollution prior to drinking, namely any fecal pollution. Many commercial portable water purification systems or chemical additives are available which can remove pathogens, chlorine, bad taste, odors, and heavy metals like lead and mercury. [ 83 ]
Techniques include boiling, filtration, activated charcoal absorption, chemical disinfection, ultraviolet purification, ozone water disinfection, solar water disinfection, solar distillation, homemade water filters. [ citation needed ]
Arsenic removal filters (ARF) are dedicated technologies typically installed to remove arsenic. Many of these technologies require a capital investment and long-term maintenance. Filters in Bangladesh are usually abandoned by the users due to their high cost and complicated maintenance, which is also quite expensive. [ citation needed ]
Groundwater pollution is much more difficult to abate than surface pollution because groundwater can move great distances through unseen aquifers . Non-porous aquifers such as clays partially purify water of bacteria by simple filtration (adsorption and absorption), dilution, and, in some cases, chemical reactions and biological activity; however, in some cases, the pollutants merely transform to soil contaminants . Groundwater that moves through open fractures and caverns is not filtered and can be transported as easily as surface water. In fact, this can be aggravated by the human tendency to use natural sinkholes as dumps in areas of karst topography . [ 84 ]
Pollutants and contaminants can be removed from ground water by applying various techniques thereby making it safe for use. Ground water treatment (or remediation) techniques span biological, chemical, and physical treatment technologies. Most ground water treatment techniques utilize a combination of technologies. Some of the biological treatment techniques include bioaugmentation , bioventing , biosparging , bioslurping , and phytoremediation . Some chemical treatment techniques include ozone and oxygen gas injection, chemical precipitation , membrane separation , ion exchange , carbon absorption, aqueous chemical oxidation, and surfactant-enhanced recovery. Some chemical techniques may be implemented using nanomaterials . Physical treatment techniques include, but are not limited to, pump and treat, air sparging , and dual phase extraction.
If treatment or remediation of the polluted groundwater is deemed to be too difficult or expensive, then abandoning the use of this aquifer's groundwater and finding an alternative source of water is the only other option.
The peri-urban areas of Lusaka, the capital of Zambia, have ground conditions which are strongly karstified and for this reason – together with the increasing population density in these peri-urban areas – pollution of water wells from pit latrines is a major public health threat there. [ 85 ]
In Tanzania, many residents rely on groundwater sources, mainly from shallow on-site wells, for drinking and other domestic purposes. The cost of the official water supply has resulted in many households relying on private wells rather than Babati's urban water and sanitation facilities. The consumption of water from temporary water sources of unknown quality (mainly shallow wells) has resulted in large numbers of people suffering from water-borne diseases. In Tanzania, 23,900 children under the age of 5 are reported to die each year from dysentery and diarrhoea associated with drinking unsafe water. [ 86 ]
The Ganga River Basin (GRB) which is a sacred body of water for the Hindus is facing severe arsenic contamination. India covers 79% of the GRB, and thus numerous states have been affected. Affected states include Uttarakhand , Uttar Pradesh , Delhi , Madhya Pradesh , Bihar , Jharkhand , Rajasthan , Chhattisgarh , Punjab , Haryana , and West Bengal . The arsenic levels are up to 4730 μg/L in the groundwater, ~1000 μg/L in irrigation water, and up to 3947 μg/kg in food materials all of which all exceed the United Nations Food and Agricultural Organization 's standard for irrigation water and the World Health Organization 's standards for drinking water. As a result, individuals who are exposed suffer from diseases that affect their dermal, neurological, reproductive and cognitive functioning, and can even result in cancer. [ 87 ]
In India the government has proceeded to promote sanitation development in order to combat the rise in ground water contamination in several regions of the country. The effort has proved to show results and has decreased the groundwater pollution and has decreased the chance of sickness for mothers and children who were mainly affected by this issue. This was something greatly needed as according to the study, over 117,000 children under five die every year due to consuming polluted water. The countries effort has seen success in the more economically developed sections of the country. [ 88 ]
The town of Hinkley, California (U.S.), had its groundwater contaminated with hexavalent chromium starting in 1952, resulting in a legal case against Pacific Gas & Electric (PG&E) and a multimillion-dollar settlement in 1996. The legal case was dramatized in the film Erin Brockovich , released in 2000.
San Joaquin, U.S.
Intensive pumping in San Joaquin county, California, has resulted in arsenic pollution. San Joaquin county has faced serious intensive pumping which has caused the ground below San Joaquin to sink and in turn damaged infrastructure. This intensive pumping into groundwater has allowed arsenic to move into groundwater aquifers which supply drinking water to at least a million residents and used in irrigation for crops in some of the richest farmland in the US. Aquifers are made up of sand and gravel that are separated by thin layers of clay which acts as a sponge that holds onto water and arsenic. When water is pumped intensively, the aquifer compresses and ground sinks which leads to the clay releasing arsenic. Study shows that aquifers contaminated as a result from over pumping, they can recover if withdrawals stop. [ 89 ]
The town of Norco, California was affected by Trichloroethylene and Hydrazine groundwater contamination as a result of improper and negligent hazardous material handling and disposal practices from the Wyle Laboratories facility, which was located next to the Norco High School . Trichlorethylene levels were detected as high as 128 times higher than the states safe limit for drinking water and Hydrazine was found in 2 nearby wells. [ 90 ]
The site is no longer in operation, and is an active hazmat cleanup site.
In the year 2000, groundwater pollution occurred in the small town of Walkerton, Canada leading to seven deaths in what is known as the Walkerton E. Coli outbreak . The water supply which was drawn from groundwater became contaminated with the highly dangerous O157:H7 strain of E. coli bacteria. [ 91 ] This contamination was due to farm runoff into an adjacent water well that was vulnerable to groundwater pollution. | https://en.wikipedia.org/wiki/Groundwater_pollution |
Groundwater recharge or deep drainage or deep percolation is a hydrologic process, where water moves downward from surface water to groundwater . Recharge is the primary method through which water enters an aquifer . This process usually occurs in the vadose zone below plant roots and is often expressed as a flux to the water table surface. Groundwater recharge also encompasses water moving away from the water table farther into the saturated zone. [ 1 ] Recharge occurs both naturally (through the water cycle ) and through anthropogenic processes (i.e., "artificial groundwater recharge"), where rainwater and/or reclaimed water is routed to the subsurface.
The most common methods to estimate recharge rates are: chloride mass balance (CMB); soil physics methods; environmental and isotopic tracers; groundwater-level fluctuation methods; water balance (WB) methods (including groundwater models (GMs)); and the estimation of baseflow (BF) to rivers. [ 2 ]
Groundwater recharge can occur through diffuse or focused mechanisms. Diffuse recharge occurs when precipitation infiltrates through the soil to the water table, and is by definition distributed over large areas. Focused recharge occurs where water leaks from surface water sources (rivers, lakes, wadis, wetlands) or land surface depressions, and generally becomes more dominant with aridity. [ 2 ]
Water is recharged naturally by rain and snow melt and to a smaller extent by surface water (rivers and lakes). Recharge may be impeded somewhat by human activities including paving, development, or logging . These activities can result in loss of topsoil resulting in reduced water infiltration, enhanced surface runoff and reduction in recharge. Use of groundwater, especially for irrigation , may also lower the water tables. Groundwater recharge is an important process for sustainable groundwater management, since the volume-rate abstracted from an aquifer in the long term should be less than or equal to the volume-rate that is recharged.
Recharge can help move excess salts that accumulate in the root zone to deeper soil layers, or into the groundwater system. Tree roots increase water saturation into groundwater reducing water runoff . [ 3 ] Flooding temporarily increases river bed permeability by moving clay soils downstream, and this increases aquifer recharge. [ 4 ]
Wetlands help maintain the level of the water table and exert control on the hydraulic head. [ 5 ] [ 6 ] This provides force for groundwater recharge and discharge to other waters as well. The extent of groundwater recharge by a wetland is dependent upon soil , vegetation , site, perimeter to volume ratio, and water table gradient. [ 7 ] [ 8 ] Groundwater recharge occurs through mineral soils found primarily around the edges of wetlands. [ 9 ] The soil under most wetlands is relatively impermeable. A high perimeter to volume ratio, such as in small wetlands, means that the surface area through which water can infiltrate into the groundwater is high. [ 8 ] Groundwater recharge is typical in small wetlands such as prairie potholes , which can contribute significantly to recharge of regional groundwater resources. [ 8 ] Researchers have discovered groundwater recharge of up to 20% of wetland volume per season. [ 8 ]
Managed aquifer recharge (MAR) strategies to augment freshwater availability include streambed channel modification, bank filtration , water spreading and recharge wells. [ 10 ] : 110 A facility in Orange County, California cleans and injects 100 million gallons per day; [ 11 ] or 90 billion gallons per year. [ 12 ]
Artificial groundwater recharge is becoming increasingly important in India, where over-pumping of groundwater by farmers has led to underground resources becoming depleted. In 2007, on the recommendations of the International Water Management Institute , the Indian government allocated ₹ 1,800 crore (equivalent to ₹ 54 billion or US$640 million in 2023) to fund dug-well recharge projects (a dug-well is a wide, shallow well, often lined with concrete) in 100 districts within seven states where water stored in hard-rock aquifers had been over-exploited. Another environmental issue is the disposal of waste through the water flux such as dairy farms, industrial, and urban runoff.
Pollution in stormwater run-off collects in retention basins . Concentrating degradable contaminants can accelerate biodegradation . However, where and when water tables are high this affects appropriate design of detention ponds , retention ponds and rain gardens .
If water falls uniformly over a field such that field capacity of the soil is not exceeded, then negligible water percolates to groundwater . If instead water puddles in low-lying areas, the same water volume concentrated over a smaller area may exceed field capacity resulting in water that percolates down to recharge groundwater. The larger the relative contributing runoff area is, the more focused infiltration is. The recurring process of water that falls relatively uniformly over an area, flowing to groundwater selectively under surface depressions is depression focused recharge . Water tables rise under such depressions.
Depression focused groundwater recharge can be very important in arid regions . More rain events are capable of contributing to groundwater supply.
Depression focused groundwater recharge also profoundly effects contaminant transport into groundwater. This is of great concern in regions with karst geological formations because water can eventually dissolve tunnels all the way to aquifers , or otherwise disconnected streams. This extreme form of preferential flow, accelerates the transport of contaminants and the erosion of such tunnels . In this way depressions intended to trap runoff water—before it flows to vulnerable water resources—can connect underground over time. Cavitation of surfaces above into the tunnels, results in potholes or caves.
Deeper ponding exerts pressure that forces water into the ground faster. Faster flow dislodges contaminants otherwise adsorbed on soil and carries them along. This can carry pollution directly to the raised water table below and into the groundwater supply. Thus, the quality of water collecting in infiltration basins is of special concern.
Rates of groundwater recharge are difficult to quantify. [ 13 ] [ 2 ] This is because other related processes, such as evaporation , transpiration (or evapotranspiration ) and infiltration processes must first be measured or estimated to determine the balance. There are no widely applicable method available that can directly and accurately quantify the volume of rainwater that reaches the water table. [ 2 ]
The most common methods to estimate recharge rates are: chloride mass balance (CMB); soil physics methods; environmental and isotopic tracers; groundwater-level fluctuation methods; water balance (WB) methods (including groundwater models (GMs)); and the estimation of baseflow (BF) to rivers. [ 2 ]
Regional, continental and global estimates of recharge commonly derive from global hydrological models . [ 2 ]
Physical methods use the principles of soil physics to estimate recharge. The direct physical methods are those that attempt to actually measure the volume of water passing below the root zone. Indirect physical methods rely on the measurement or estimation of soil physical parameters, which along with soil physical principles, can be used to estimate the potential or actual recharge. After months without rain the level of the rivers under humid climate is low and represents solely drained groundwater. Thus, the recharge can be calculated from this base flow if the catchment area is already known.
Chemical methods use the presence of relatively inert water-soluble substances, such as an isotopic tracer [ 14 ] [ 15 ] [ 16 ] or chloride , [ 17 ] moving through the soil, as deep drainage occurs.
Recharge can be estimated using numerical methods , using such codes as Hydrologic Evaluation of Landfill Performance , UNSAT-H, SHAW (short form of Simultaneous Heat and Water Transfer model), WEAP , and MIKE SHE . The 1D-program HYDRUS1D is available online. The codes generally use climate and soil data to arrive at a recharge estimate and use the Richards equation in some form to model groundwater flow in the vadose zone .
The impacts of climate change on groundwater may be greatest through its indirect effects on irrigation water demand via increased evapotranspiration . [ 18 ] : 5 There is an observed declined in groundwater storage in many parts of the world. This is due to more groundwater being used for irrigation activities in agriculture, particularly in drylands . [ 19 ] : 1091 Some of this increase in irrigation can be due to water scarcity issues made worse by effects of climate change on the water cycle . Direct redistribution of water by human activities amounting to ~24,000 km 3 per year is about double the global groundwater recharge each year. [ 19 ]
Climate change causes changes to the water cycle which in turn affect groundwater in several ways: There can be a decline in groundwater storage, and reduction in groundwater recharge and water quality deterioration due to extreme weather events. [ 20 ] : 558 In the tropics intense precipitation and flooding events appear to lead to more groundwater recharge. [ 20 ] : 582
However, the exact impacts of climate change on groundwater are still under investigation. [ 20 ] : 579 This is because scientific data derived from groundwater monitoring is still missing, such as changes in space and time, abstraction data and "numerical representations of groundwater recharge processes". [ 20 ] : 579
Further implications of groundwater recharge are a consequence of urbanization . Research shows that the recharge rate can be up to ten times higher [ 22 ] in urban areas compared to rural regions . This is explained through the vast water supply and sewage networks supported in urban regions in which rural areas are not likely to obtain. Recharge in rural areas is heavily supported by precipitation, [ 22 ] and this is the opposite for urban areas. Road networks and infrastructure within cities prevent surface water from percolating into the soil, resulting in most surface runoff entering storm drains for local water supply. As urban development continues to spread across various regions, groundwater recharge rates will increase relative to the existing rates of the previous rural region. A consequence of sudden influxes in groundwater recharge includes flash flooding . [ 23 ] The ecosystem will have to adjust to the elevated groundwater surplus due to groundwater recharge rates. Additionally, road networks are less permeable compared to soil, resulting in higher amounts of surface runoff. Therefore, urbanization increases the rate of groundwater recharge and reduces infiltration, [ 23 ] resulting in flash floods as the local ecosystem accommodates changes to the surrounding environment. | https://en.wikipedia.org/wiki/Groundwater_recharge |
Groundwater remediation is the process that is used to treat polluted groundwater by removing the pollutants or converting them into harmless products. Groundwater is water present below the ground surface that saturates the pore space in the subsurface. Globally, between 25 per cent and 40 per cent of the world's drinking water is drawn from boreholes and dug wells . [ 1 ] Groundwater is also used by farmers to irrigate crops and by industries to produce everyday goods. Most groundwater is clean, but groundwater can become polluted, or contaminated as a result of human activities or as a result of natural conditions.
The many and diverse activities of humans produce innumerable waste materials and by-products. Historically, the disposal of such waste have not been subject to many regulatory controls. Consequently, waste materials have often been disposed of or stored on land surfaces where they percolate into the underlying groundwater. As a result, the contaminated groundwater is unsuitable for use.
Current practices can still impact groundwater, such as the over application of fertilizer or pesticides , spills from industrial operations, infiltration from urban runoff , and leaking from landfills . Using contaminated groundwater causes hazards to public health through poisoning or the spread of disease, and the practice of groundwater remediation has been developed to address these issues. Contaminants found in groundwater cover a broad range of physical, inorganic chemical, organic chemical, bacteriological, and radioactive parameters. Pollutants and contaminants can be removed from groundwater by applying various techniques, thereby bringing the water to a standard that is commensurate with various intended uses.
Ground water remediation techniques span biological, chemical, and physical treatment technologies. Most ground water treatment techniques utilize a combination of technologies. Some of the biological treatment techniques include bioaugmentation , bioventing , biosparging , bioslurping , and phytoremediation . Some chemical treatment techniques include ozone and oxygen gas injection, chemical precipitation , membrane separation , ion exchange , carbon absorption, aqueous chemical oxidation , and surfactant enhanced recovery . Some chemical techniques may be implemented using nanomaterials . Physical treatment techniques include, but are not limited to, pump and treat , air sparging , and dual phase extraction . [ citation needed ]
If a treatability study shows no degradation (or an extended lab period before significant degradation is achieved) in contamination contained in the groundwater, then inoculation with strains known to be capable of degrading the contaminants may be helpful. This process increases the reactive enzyme concentration within the bioremediation system and subsequently may increase contaminant degradation rates over the nonaugmented rates, at least initially after inoculation. [ 2 ]
Bioventing is an on site remediation technology that uses microorganisms to biodegrade organic constituents in the groundwater system. Bioventing enhances the activity of indigenous bacteria and archaea and stimulates the natural in situ biodegradation of hydrocarbons by inducing air or oxygen flow into the unsaturated zone and, if necessary, by adding nutrients. [ 3 ] During bioventing, oxygen may be supplied through direct air injection into residual contamination in soil. Bioventing primarily assists in the degradation of adsorbed fuel residuals, but also assists in the degradation of volatile organic compounds (VOCs) as vapors move slowly through biologically active soil. [ 4 ]
Biosparging is an in situ remediation technology that uses indigenous microorganisms to biodegrade organic constituents in the saturated zone. In biosparging, air (or oxygen) and nutrients (if needed) are injected into the saturated zone to increase the biological activity of the indigenous microorganisms. Biosparging can be used to reduce concentrations of petroleum constituents that are dissolved in groundwater, adsorbed to soil below the water table , and within the capillary fringe . [ citation needed ]
Bioslurping combines elements of bioventing and vacuum-enhanced pumping of free-product that is lighter than water ( light non-aqueous phase liquid or LNAPL) to recover free-product from the groundwater and soil, and to bioremediate soils. The bioslurper system uses a “slurp” tube that extends into the free-product layer. Much like a straw in a glass draws liquid, the pump draws liquid (including free-product) and soil gas up the tube in the same process stream. Pumping lifts LNAPLs, such as oil, off the top of the water table and from the capillary fringe (i.e., an area just above the saturated zone, where water is held in place by capillary forces). The LNAPL is brought to the surface, where it is separated from water and air. The biological processes in the term “bioslurping” refer to aerobic biological degradation of the hydrocarbons when air is introduced into the unsaturated zone contaminated soil. [ 5 ]
In the phytoremediation process certain plants and trees are planted, whose roots absorb contaminants from ground water over time. This process can be carried out in areas where the roots can tap the ground water. Few examples of plants that are used in this process are Chinese Ladder fern Pteris vittata, also known as the brake fern, is a highly efficient accumulator of arsenic . Genetically altered cottonwood trees are good absorbers of mercury and transgenic Indian mustard plants soak up selenium well. [ 6 ]
Certain types of permeable reactive barriers utilize biological organisms in order to remediate groundwater. [ citation needed ]
Chemical precipitation is commonly used in wastewater treatment to remove hardness and heavy metals . In general, the process involves addition of agent to an aqueous waste stream in a stirred reaction vessel, either batchwise or with steady flow. Most metals can be converted to insoluble compounds by chemical reactions between the agent and the dissolved metal ions. The insoluble compounds (precipitates) are removed by settling and/or filtering. [ citation needed ]
Ion exchange for ground water remediation is virtually always carried out by passing the water downward under pressure through a fixed bed of granular medium (either cation exchange media and anion exchange media) or spherical beads. Cations are displaced by certain cations from the solutions and ions are displaced by certain anions from the solution. Ion exchange media most often used for remediation are zeolites (both natural and synthetic) and synthetic resins. [ 2 ]
The most common activated carbon used for remediation is derived from bituminous coal . Activated carbon adsorbs volatile organic compounds from ground water; the compounds attach to the graphite-like surface of the activated carbon. [ citation needed ]
In this process, called In Situ Chemical Oxidation or ISCO, chemical oxidants are delivered in the subsurface to destroy (converted to water and carbon dioxide or to nontoxic substances) the organics molecules. The oxidants are introduced as either liquids or gasses. Oxidants include air or oxygen, ozone , and certain liquid chemicals such as hydrogen peroxide , permanganate and persulfate . Ozone and oxygen gas can be generated on site from air and electricity and directly injected into soil and groundwater contamination. The process has the potential to oxidize and/or enhance naturally occurring aerobic degradation. Chemical oxidation has proven to be an effective technique for dense non-aqueous phase liquid or DNAPL when it is present. [ citation needed ]
Surfactant enhanced recovery increases the mobility and solubility of the contaminants absorbed to the saturated soil matrix or present as dense non-aqueous phase liquid . Surfactant-enhanced recovery injects surfactants (surface-active agents that are primary ingredient in soap and detergent) into contaminated groundwater. A typical system uses an extraction pump to remove groundwater downstream from the injection point. The extracted groundwater is treated aboveground to separate the injected surfactants from the contaminants and groundwater. Once the surfactants have separated from the groundwater they are re-used. The surfactants used are non-toxic, food-grade, and biodegradable. Surfactant enhanced recovery is used most often when the groundwater is contaminated by dense non-aqueous phase liquids (DNAPLs). These dense compounds, such as trichloroethylene (TCE), sink in groundwater because they have a higher density than water. They then act as a continuous source for contaminant plumes that can stretch for miles within an aquifer. These compounds may biodegrade very slowly. They are commonly found in the vicinity of the original spill or leak where capillary forces have trapped them. [ 7 ]
Some permeable reactive barriers utilize chemical processes to achieve groundwater remediation. [ citation needed ]
Pump and treat is one of the most widely used ground water remediation technologies. In this process ground water is pumped to the surface and is coupled with either biological or chemical treatments to remove the impurities. [ 8 ]
Air sparging is the process of blowing air directly into the ground water. As the bubbles rise, the contaminants are removed from the groundwater by physical contact with the air (i.e., stripping) and are carried up into the unsaturated zone (i.e., soil). As the contaminants move into the soil, a soil vapor extraction system is usually used to remove vapors. [ 9 ]
Dual-phase vacuum extraction (DPVE), also known as multi-phase extraction, is a technology that uses a high-vacuum system to remove both contaminated groundwater and soil vapor. In DPVE systems, a high-vacuum extraction well is installed with its screened section in the zone of contaminated soils and groundwater. Fluid/vapor extraction systems depress the water table and water flows faster to the extraction well. DPVE removes contaminants from above and below the water table. As the water table around the well is lowered from pumping, unsaturated soil is exposed. This area, called the capillary fringe , is often highly contaminated, as it holds undissolved chemicals, chemicals that are lighter than water, and vapors that have escaped from the dissolved groundwater below. Contaminants in the newly exposed zone can be removed by vapor extraction. Once above ground, the extracted vapors and liquid-phase organics and groundwater are separated and treated. Use of dual-phase vacuum extraction with these technologies can shorten the cleanup time at a site, because the capillary fringe is often the most contaminated area. [ 10 ]
Monitoring-wells are often drilled for the purpose of collecting ground water samples for analysis. These wells, which are usually six inches or less in diameter, can also be used to remove hydrocarbons from the contaminant plume within a groundwater aquifer by using a belt-style oil skimmer. Belt oil skimmers, which are simple in design, are commonly used to remove oil and other floating hydrocarbon contaminants from industrial water systems. [ citation needed ]
A monitoring-well oil skimmer remediates various oils, ranging from light fuel oils such as petrol, light diesel or kerosene to heavy products such as No. 6 oil, creosote and coal tar. It consists of a continuously moving belt that runs on a pulley system driven by an electric motor. The belt material has a strong affinity for hydrocarbon liquids and for shedding water. The belt, which can have a vertical drop of 100+ feet, is lowered into the monitoring well past the LNAPL/water interface. As the belt moves through this interface, it picks up liquid hydrocarbon contaminant which is removed and collected at ground level as the belt passes through a wiper mechanism. To the extent that DNAPL hydrocarbons settle at the bottom of a monitoring well, and the lower pulley of the belt skimmer reaches them, these contaminants can also be removed by a monitoring-well oil skimmer. [ citation needed ]
Typically, belt skimmers remove very little water with the contaminant, so simple weir-type separators can be used to collect any remaining hydrocarbon liquid, which often makes the water suitable for its return to the aquifer. Because the small electric motor uses little electricity, it can be powered from solar panels or a wind turbine , making the system self-sufficient and eliminating the cost of running electricity to a remote location. [ 11 ] | https://en.wikipedia.org/wiki/Groundwater_remediation |
In optics , group-velocity dispersion (GVD) is a characteristic of a dispersive medium , used most often to determine how the medium affects the duration of an optical pulse traveling through it. Formally, GVD is defined as the derivative of the inverse of group velocity of light in a material with respect to angular frequency , [ 1 ] [ 2 ]
where ω {\displaystyle \omega } and ω 0 {\displaystyle \omega _{0}} are angular frequencies, and the group velocity v g ( ω ) {\displaystyle v_{g}(\omega )} is defined as v g ( ω ) ≡ ∂ ω / ∂ k {\displaystyle v_{g}(\omega )\equiv \partial \omega /\partial k} . The units of group-velocity dispersion are [time] 2 /[distance], often expressed in fs 2 / mm .
Equivalently, group-velocity dispersion can be defined in terms of the medium-dependent wave vector k ( ω ) {\displaystyle k(\omega )} according to
or in terms of the refractive index n ( ω ) {\displaystyle n(\omega )} according to
Group-velocity dispersion is most commonly used to estimate the amount of chirp that will be imposed on a pulse of light after passing through a material of interest:
A simple illustration of how GVD can be used to determine pulse chirp can be seen by looking at the effect of a transform-limited pulse of duration σ {\displaystyle \sigma } passing through a planar medium of thickness d . Before passing through the medium, the phase offsets of all frequencies are aligned in time, and the pulse can be described as a function of time,
or equivalently, as a function of frequency,
(the parameters A and B are normalization constants).
Passing through the medium results in a frequency-dependent phase accumulation Δ ϕ ( ω ) = k ( ω ) d {\displaystyle \Delta \phi (\omega )=k(\omega )d} , such that the post-medium pulse can be described by
In general, the refractive index n ( ω ) {\displaystyle n(\omega )} , and therefore the wave vector k ( ω ) = n ( ω ) ω / c {\displaystyle k(\omega )=n(\omega )\omega /c} , can be an arbitrary function of ω {\displaystyle \omega } , making it difficult to analytically perform the inverse Fourier transform back into the time domain. However, if the bandwidth of the pulse is narrow relative to the curvature of n {\displaystyle n} , then good approximations of the impact of the refractive index can be obtained by replacing k ( ω ) {\displaystyle k(\omega )} with its Taylor expansion centered about ω 0 {\displaystyle \omega _{0}} :
Truncating this expression and inserting it into the post-medium frequency-domain expression results in a post-medium time-domain expression
On balance, the pulse is lengthened to an intensity standard deviation value of
thus validating the initial expression. Note that a transform-limited pulse has σ ω σ t = 1 / 2 {\displaystyle \sigma _{\omega }\sigma _{t}=1/2} , which makes it appropriate to identify 1/(2 σ t ) as the bandwidth.
An alternate derivation of the relationship between pulse chirp and GVD, which more immediately illustrates the reason why GVD can be defined by the derivative of inverse group velocity, can be outlined as follows. Consider two transform-limited pulses of carrier frequencies ω 1 {\displaystyle \omega _{1}} and ω 2 {\displaystyle \omega _{2}} , which are initially overlapping in time. After passing through the medium, these two pulses will exhibit a time delay between their respective pulse-envelope centers, given by
The expression can be approximated as a Taylor expansion , giving
or
From here it is possible to imagine scaling this expression up two pulses to infinitely many. The frequency difference ω 2 − ω 1 {\displaystyle \omega _{2}-\omega _{1}} must be replaced by the bandwidth, and the time delay Δ T {\displaystyle \Delta T} evolves into the induced chirp.
A closely related yet independent quantity is the group-delay dispersion ( GDD ), defined such that group-velocity dispersion is the group-delay dispersion per unit length. GDD is commonly used as a parameter in characterizing layered mirrors, where the group-velocity dispersion is not particularly well-defined, yet the chirp induced after bouncing off the mirror can be well-characterized. The units of group-delay dispersion are [time] 2 , often expressed in fs 2 .
The group-delay dispersion (GDD) of an optical element is the derivative of the group delay with respect to angular frequency , and also the second derivative of the optical phase:
It is a measure of the chromatic dispersion of the element. GDD is related to the total dispersion parameter D tot {\displaystyle D_{\text{tot}}} as | https://en.wikipedia.org/wiki/Group-velocity_dispersion |
In chemistry , a group (also known as a family ) [ 1 ] is a column of elements in the periodic table of the chemical elements . There are 18 numbered groups in the periodic table; the 14 f-block columns, between groups 2 and 3, are not numbered. The elements in a group have similar physical or chemical characteristics of the outermost electron shells of their atoms (i.e., the same core charge ), because most chemical properties are dominated by the orbital location of the outermost electron.
The modern numbering system of "group 1" to "group 18" has been recommended by the International Union of Pure and Applied Chemistry (IUPAC) since 1988. The 1-18 system is based on each atom's s, p and d electrons beyond those in atoms of the preceding noble gas . Two older incompatible naming schemes can assign the same number to different groups depending on the system being used. The older schemes were used by the Chemical Abstract Service (CAS, more popular in the United States), and by IUPAC before 1988 (more popular in Europe). The system of eighteen groups is generally accepted by the chemistry community, but some dissent exists about membership of elements number 1 and 2 ( hydrogen and helium ). Similar variation on the inner transition metals continues to exist in textbooks, although the correct positioning has been known since 1948 and was twice endorsed by IUPAC in 1988 (together with the 1–18 numbering) and 2021.
Groups may also be identified using their topmost element, or have a specific name. For example, group 16 is also described as the "oxygen group" and as the " chalcogens ". An exception is the " iron group ", which usually refers to group 8 , but in chemistry may also mean iron , cobalt , and nickel , or some other set of elements with similar chemical properties. In astrophysics and nuclear physics , it usually refers to iron, cobalt, nickel, chromium , and manganese .
Modern group names are numbers 1–18, with the 14 f-block columns remaining unnumbered (together making the 32 columns in the periodic table). Also, trivial names (like halogens ) are common. In history, several sets of group names have been used, based on Roman numberings I–VIII, and "A" and "B" suffixes. [ 2 ] [ 3 ]
Two earlier group number systems exist: CAS ( Chemical Abstracts Service ) and old IUPAC . Both use numerals ( Arabic or Roman ) and letters A and B . Both systems agree on the numbers. The numbers indicate approximately the highest oxidation number of the elements in that group, and so indicate similar chemistry with other elements with the same numeral. The number proceeds in a linearly increasing fashion for the most part, once on the left of the table, and once on the right (see List of oxidation states of the elements ), with some irregularities in the transition metals. However, the two systems use the letters differently. For example, potassium (K) has one valence electron . Therefore, it is located in group 1. Calcium (Ca) is in group 2, for it contains two valence electrons.
In the old IUPAC system the letters A and B were designated to the left (A) and right (B) part of the table, while in the CAS system the letters A and B are designated to main group elements (A) and transition elements (B). The old IUPAC system was frequently used in Europe, while the CAS is most common in America. The new IUPAC scheme was developed to replace both systems as they confusingly used the same names to mean different things. The new system simply numbers the groups increasingly from left to right on the standard periodic table. The IUPAC proposal was first circulated in 1985 for public comments, [ 2 ] and was later included as part of the 1990 edition of the Nomenclature of Inorganic Chemistry . [ 19 ]
While groups are defined to be columns in the periodic table, as described above, there are also sets of elements named "group" that are not a column:
Similar sets: noble metals , coinage metals , precious metals , refractory metals . | https://en.wikipedia.org/wiki/Group_(periodic_table) |
Heteroatomic multiple bonding between group 13 and group 15 elements are of great interest in synthetic chemistry due to their isoelectronicity with C-C multiple bonds. Nevertheless, the difference of electronegativity between group 13 and 15 leads to different character of bondings comparing to C-C multiple bonds. Because of the ineffective overlap between p𝝅 orbitals and the inherent lewis acidity/basicity of group 13/15 elements, the synthesis of compounds containing such multiple bonds is challenging and subject to oligomerization . [ 1 ] [ 2 ] The most common example of compounds with 13/15 group multiple bonds are those with B=N units. The boron-nitrogen-hydride compounds are candidates for hydrogen storage . [ 3 ] [ 4 ] [ 5 ] In contrast, multiple bonding between aluminium and nitrogen Al=N, Gallium and nitrogen (Ga=N), boron and phosphorus (B=P), or boron and arsenic (B=As) are less common. [ 2 ]
Suitable precursors are crucial for the synthesis of group 13/15 multiple bond-containing species. In most successfully isolated structures, sterically demanding ligands are utilized to stabilize such bondings.
Boraphosphenes , also known as phosphoboranes, was first reported by Cowley and co-workers in the 1980s. [ 7 ] [(tmp)B=P(Ar)] (tmp= 2,2,6,6,-tetramethylpiperidina, Ar= 2,4,6-t-Bu 3 C 6 H 2 ) was characterized by mass spectroscopy ( EI MS), and the corresponding dimer, diphosphadiboretane, was characterized by X-ray crystallography . [ 7 ] The Power and co-workers later reported the structure of [P(R)=BMes 2 Li(Et 2 O) 2 ] (R = phenyl , cyclohexane , and mesitylene ), which is the first B=P double bond observed in solid state. [ 6 ] The synthesis of [P(R)=BMes 2 Li(Et 2 O) 2 ] starts from treating in-situ generated Mes 2 BPHR with 1 equivalent of t-BuLi in Et 2 O, followed by crystallization at low temperature. [ 6 ]
Isomerization of four-member P-B cycles was investigated by Bourissou and Bertrand . It was reported that cycle-[R 2 PB(R')-B(R')-P(Ph) 2 ] (R = phenyl, isopropyl ; R'= tert-butyl , 2,3,5,6-tetramethyl phenyl) isomerize to form cycle-[R 2 P-B(R')=P(Ph)-B(R')(Ph)] upon irradiation. [ 8 ] An example of five-membered ring was reported by Crossley suggesting that a reaction of 1,2-diphosphinobenzene with n-BuLi and Cl 2 BPh yielded a benzodiphosphaborolediide. [ 9 ] Several six-membered ring systems involving P=B double bonds have been reported. One of the example is an analogue of borazine synthesizing from MesBBr 2 and CyP(H)Li. [ 10 ]
A similar strategy to access litigated arsinideneborate was reported by Power and co-workers after the establishment of synthesizing litigated phosphinideneborates. [ 12 ] Crystallizing [As(Ph)=BMes 2 Li(THF) 3 ] with two equivalence of TMEDA yielded [As(Ph)=BMes 2 ][Li(TMEDA) 2 ]. [ 12 ] Ring-systems containing As-B multiple bonds haven't been reported yet.
Synthesis of group 13 imides usually starts with low valent group 13 species stabilized by bulky ligands. A [2+3] cycloaddition of monomeric [ Dip Nacnc]Al or [ Dip Nacnc]Ga ( Dip Nacnc= HC{(CMe)(NDip)} 2 ) compound with sterically bulky azide, Tip TerN3 ( Tip Ter = -C 6 H 3 -2,6-(C 6 H 2 -2,4,6-iPr 3 ) 2 ), gives the iminotrielenes [{ Dip Nacnc}M=N- Tip Ter] (M=Al, Ga). [ 15 ] Additionally, dimers of Ga(I) or In(I) were reported to form the iminotrielens [( Dip Ter)M=N- Mes' Ter] with Mes' TerN 3 (M = Ga, In; Mes' Ter =C 6 H 3 -2,6(Xyl-4-tBu) 2 ). [ 16 ]
Transient Al≡N triple bond species were also investigated by reacting monomeric alanediyl precursor with organic azides . The unstable Al≡N triple bond species [iPr 2 TIP TerAl≡NR] (R = Ad, SiMe 3 ) was not capture but further rearrange to tetrazole and amino-azide alone, respectively. [ 18 ]
The development of Al=P and Al=As species faced the difficulty due to the tendency of oligomerization of the lewis acidic Al and lewis basic P/As. In 2021, Hering-Junghans, Braunchweig, and co-workers reported the synthesis of phosphaalumens and arsaalumens with Al(I) precursors, [Al(I)Cp*] 4 (Cp* = pentamethylcyclopentadiene ). Reacting [Al(I)Cp*] 4 with Dip Ter-AsPMe 3 or Dip Ter-AsPMe 3 at 1:4 ratio yielded the corresponding phosphaalumens/arsaalumens, which are stable and isolable. [ 19 ]
Synthesis and characterization of Ga=Sb species was reported by Schulz and Cutsail III with the reaction of [ Dip Nacnc]Ga ( Dip Nacnc= HC{(CMe)(NDip)} 2 ) with [Cp*SbCl 2 ]. The resulting Sb radical species, [ Dip Nacnc(Cl)Ga] 2 Sb, was then reduced by KC 8 to give [ Dip NacncGa=Sb-Ga(Cl) Dip Nacnc]. [ 21 ] Utilizing the similar reaction pathway, a Ga=As species, [ Dip NacncGa=AsCp*], was successfully synthesized and stabilized. Interestingly, no radical formation was observed comparing to the case of Ga=Sb species. [ 20 ] With the rapid development of gallium pnictogen in the late 2010s, the first phosphagallene species was reported by Goicoechea and co-workers in 2020. The reaction of [(HC) 2 (NDip) 2 PPCO] with [ Dip NacncGa] gave the phosphagallene, [ Dip NacncGa=P-P(NDip) 2 (CH) 2 ]. [ 22 ]
B=P double bond species has been studied for bond activation. For example, C-F activation of tris(pentafluorophenyl)borane by NHC -stabilized phosphaboranes, [(tmp)(L)B=PMes*] (L = IMe 4 ), was reported by Cowley and co-workers. [ 23 ] The C-F bond activation takes place at the para position, leading to the formation of C-P bond. [ 23 ] Reactions of phenyl acetylene with the dimer of [Mes*P=B(tmp)] give an analogue of cycle-butene, [Mes*P=C(Ph)-C(H)=B(tmp)], where C-C triple bond undergoes a [2+2]-cycloaddition to P=B double bond. [ 23 ]
Transient boraphosphene [(tmp)B=PMes*)] (tmp = 2,2,6,6-tetramethylpiperidine , Mes* = 2,4,6-tri-tert-butylphenyl) reacts with aldehyde, ketone, and esters to form phosphaboraoxetanes, which converts to phosphaalkenes [Mes*P=CRR'] and [(tmp)NBO] x heterocycles. [ 25 ] [ 24 ] This method provides direct access of phosphaalkenes from carbonyl compounds. [ 25 ] [ 24 ]
Compounds with group 13-N multiple bonds are capable of small molecule activation. Reactions of PhCCH or PhNH 2 with NHC-stabilized iminoalane result in the addition of proton to N and -CCPh or -NHPh fragment to Al. [ 26 ] The reaction with CO leads to the insertion of CO between the Al=N bond. [ 26 ]
Small molecule activation takes place across the P-P=Ga bonds in phosphanyl-phosphagallenes species, where the Ga=P species behave as frustrated Lewis pairs . For example, the reaction of CO 2 with [ Dip NacncGa=P-P(NDip) 2 (CH 2 ) 2 ] results in the formation of a P=P-C-O-Ga five-membered ring species. In contrast, H 2 addition to the P-P=Ga fragment in a 1,3-activation manner. [ 22 ] E-H bond activation of protic and hydridic reagents was investigated as well. Reactions of [ Dip NacncGa=P-P(NDip) 2 (CH 2 ) 2 ] toward amines, phosphines, alkynes resulted in the formation of [ Dip Nacnc(E)Ga-P-P(H)(NDip) 2 (CH 2 ) 2 ]. [ 2 ] Reversible ammonia activation was observed under 1 bar pressure in the presence of a Lewis acid. [ 2 ]
Natural bond orbital analysis of a borophosphide anion, [(Mes*)P=BClCp*] − , suggested that the B-P double bonds are polarized to the P atom. The B=P 𝝈-bond is mostly non-polar while the 𝝅-bond is polarized to the phosphorus (71%). [ 28 ] DFT calculation at B3LYP/6-31G level revealed that the HOMO of [(Mes*)P=BClCp*] − has great B-P 𝝅-bonding character. [ 28 ] In most reported phosphinideneborates, the phosphorus chemical shifts are much more deshielded than the starting materials, phosphinoboranes. The down-field resonances of phosphorus in 31 P NMR suggest the delocalization of lone pairs into the empty p-orbital of boron. [ 28 ]
Natural bond orbital analysis was reported for Ga=Sb and Ga=Bi containing species, where electron populates more on Sb and Bi (62% and 59%, respectively). The Lewis acidic Ga results in the delocalization of electrons in Sb and Bi. [ 21 ] | https://en.wikipedia.org/wiki/Group_13/15_multiple_bonds |
Group 2 organometallic chemistry refers to the organic derivativess of any group 2 element . It is a subtheme to main group organometallic chemistry . [ 2 ] [ 3 ] By far the most common group 2 organometallic compounds are the magnesium-containing Grignard reagents which are widely used in organic chemistry . Other organometallic group 2 compounds are typically limited to academic interests.
As the group 2 elements (also referred to as the alkaline earth metals) contain two valence electrons , their chemistries have similarities group 12 organometallic compounds. Both readily assume a +2 oxidation states with higher and lower states being rare, and are less electronegative than carbon. However, as the group two elements (with the exception of beryllium) have considerably low electronegativity the resulting C-M bonds are more highly polarized and ionic -like, if not entirely ionic for the heavier barium compounds. The lighter organoberyllium and organomagnesium compounds are often considered covalent , but with some ionic bond characteristics owing to the attached carbon bearing a negative dipole moment . This higher ionic character and bond polarization tends to produce high coordination numbers and many compounds (particularly dialklys) are polymeric in solid or liquid states with highly complex structures in solution, though in the gaseous state they are often monomeric.
Metallocene compounds with group 2 elements are rare, but some do exist. Bis(cyclopentadienyl)beryllium or beryllocene (Cp 2 Be), with a molecular dipole moment of 2.2 D , is so-called slipped 5 η/ 1 η sandwich. While magnesocene (Cp 2 Mg) is a regular metallocene, bis(pentamethylcyclopentadienyl)calcium (Cp * ) 2 Ca is bent with an angle of 147°.
Mixed alkyl/aryl-halide compounds, which contain a single C-M bond and a C-X bond, are typically prepared by oxidative addition. Magnesium-containing compounds of this configuration are known as the Grignard reagents , though some calcium Grignard's are known and more reactive and sensitive to decomposition. Calcium grignard's must be pre-activated prior to synthesis. [ 6 ]
There are three key reaction pathways for dialkyl and diaryl group 2 metal compounds.
Although organomagnesium compounds are widespread in the form of Grignard reagents , the other organo-group 2 compound are almost exclusively of academic interest. Organoberyllium chemistry is limited due to the cost and toxicity of beryllium. Calcium is nontoxic and cheap but organocalcium compounds are difficult to prepare, strontium and barium compounds even more so. One use for these type of compounds is in chemical vapor deposition .
Beryllium derivatives and reagents are often prepared by alkylation of beryllium chloride . [ 7 ] Examples of known organoberyllium compounds are dineopentylberyllium, [ 8 ] beryllocene (Cp 2 Be), [ 9 ] [ 10 ] [ 11 ] [ 12 ] diallylberyllium (by exchange reaction of diethyl beryllium with triallyl boron), [ 13 ] bis(1,3-trimethylsilylallyl)beryllium [ 14 ] and Be(mes)2. [ 7 ] [ 15 ] Ligands can also be aryls [ 16 ] and alkynyls. [ 17 ]
The distinctive feature of the Grignard reagents is their formation from the organic halide and magnesium metal. Most other group II organic compounds are generated by salt metathesis , which limits their accessibility. The formation of the Grignard reagents has received intense scrutiny. It proceeds by a SET process. For less reactive organic halides, activated forms of magnesium have been produced in the form of Rieke magnesium . Examples of Grignard reagents are phenylmagnesium bromide and ethylmagnesium bromide . These simplified formulas are deceptive: Grignard reagents generally exist as dietherates, RMgX(ether)2. As such they obey the octet rule .
Grignard reagents participate in the Schlenk equilibrium. Exploiting this reaction is a way to generate dimethylmagnesium . Beyond Grignard reagents, another organomagnesium compound is magnesium anthracene . This orange solid is used as a source of highly active magnesium. Butadiene -magnesium serves as a source for the butadiene dianion. Ate complexes of magnesium are also well known, e.g LiMgBu 3 . [ 18 ]
Dimethylcalcium is obtained by metathesis reaction of calcium bis(trimethylsilyl)amide and methyllithium in diethyl ether : [ 19 ]
A well known organocalcium compound is ( Cp )calcium(I). [ citation needed ] Bis(allyl)calcium was described in 2009. [ 20 ] It forms in a metathesis reaction of allylpotassium and calcium iodide as a stable non- pyrophoric off-white powder:
The bonding mode is η 3 . This compound is also reported to give access to an η 1 polymeric (CaCH 2 CHCH 2 ) n compound. [ 21 ]
The compound [(thf) 3 Ca{μ-C 6 H 3 -1,3,5-Ph 3 }Ca(thf) 3 ] also described in 2009 [ 22 ] [ 23 ] is an inverse sandwich compound with two calcium atoms at either side of an arene.
Olefins tethered to cyclopentadienyl ligands have been shown to coordinate to calcium(II), strontium(II), and barium(II): [ 24 ]
Organocalcium compounds have been investigated as catalysts. [ 25 ]
Organostrontium compounds have been reported as intermediates in Barbier-type reactions. [ 26 ] [ 27 ] [ 28 ]
Organobarium compounds [ 29 ] of the type (allyl)BaCl can be prepared by reaction of activated barium (Rieke method reduction of barium iodide with lithium biphenylide) with allyl halides. [ 30 ] [ 31 ] These allylbarium compounds react with carbonyl compounds. Such reagents are more alpha-selective and more stereoselective than the related Grignards or organocalcium compounds. The metallocene ( Cp* ) 2 Ba has also been reported. [ 32 ]
The only known organoradium compound is the gas-phase acetylide . [ citation needed ] | https://en.wikipedia.org/wiki/Group_2_organometallic_chemistry |
In mathematics , a group action of a group G {\displaystyle G} on a set S {\displaystyle S} is a group homomorphism from G {\displaystyle G} to some group (under function composition ) of functions from S {\displaystyle S} to itself. It is said that G {\displaystyle G} acts on S {\displaystyle S} .
Many sets of transformations form a group under function composition ; for example, the rotations around a point in the plane. It is often useful to consider the group as an abstract group , and to say that one has a group action of the abstract group that consists of performing the transformations of the group of transformations. The reason for distinguishing the group from the transformations is that, generally, a group of transformations of a structure acts also on various related structures; for example, the above rotation group also acts on triangles by transforming triangles into triangles.
If a group acts on a structure, it will usually also act on objects built from that structure. For example, the group of Euclidean isometries acts on Euclidean space and also on the figures drawn in it; in particular, it acts on the set of all triangles . Similarly, the group of symmetries of a polyhedron acts on the vertices , the edges , and the faces of the polyhedron.
A group action on a vector space is called a representation of the group. In the case of a finite-dimensional vector space, it allows one to identify many groups with subgroups of the general linear group GL ( n , K ) {\displaystyle \operatorname {GL} (n,K)} , the group of the invertible matrices of dimension n {\displaystyle n} over a field K {\displaystyle K} .
The symmetric group S n {\displaystyle S_{n}} acts on any set with n {\displaystyle n} elements by permuting the elements of the set. Although the group of all permutations of a set depends formally on the set, the concept of group action allows one to consider a single group for studying the permutations of all sets with the same cardinality .
If G {\displaystyle G} is a group with identity element e {\displaystyle e} , and X {\displaystyle X} is a set, then a ( left ) group action α {\displaystyle \alpha } of G {\displaystyle G} on X is a function
that satisfies the following two axioms : [ 1 ]
for all g and h in G and all x in X {\displaystyle X} .
The group G {\displaystyle G} is then said to act on X {\displaystyle X} (from the left). A set X {\displaystyle X} together with an action of G {\displaystyle G} is called a ( left ) G {\displaystyle G} - set .
It can be notationally convenient to curry the action α {\displaystyle \alpha } , so that, instead, one has a collection of transformations α g : X → X , with one transformation α g for each group element g ∈ G . The identity and compatibility relations then read
and
The second axiom states that the function composition is compatible with the group multiplication; they form a commutative diagram . This axiom can be shortened even further, and written as α g ∘ α h = α g h {\displaystyle \alpha _{g}\circ \alpha _{h}=\alpha _{gh}} .
With the above understanding, it is very common to avoid writing α {\displaystyle \alpha } entirely, and to replace it with either a dot, or with nothing at all. Thus, α ( g , x ) can be shortened to g ⋅ x or gx , especially when the action is clear from context. The axioms are then
From these two axioms, it follows that for any fixed g in G {\displaystyle G} , the function from X to itself which maps x to g ⋅ x is a bijection , with inverse bijection the corresponding map for g −1 . Therefore, one may equivalently define a group action of G on X as a group homomorphism from G into the symmetric group Sym( X ) of all bijections from X to itself. [ 2 ]
Likewise, a right group action of G {\displaystyle G} on X {\displaystyle X} is a function
that satisfies the analogous axioms: [ 3 ]
(with α ( x , g ) often shortened to xg or x ⋅ g when the action being considered is clear from context)
for all g and h in G and all x in X .
The difference between left and right actions is in the order in which a product gh acts on x . For a left action, h acts first, followed by g second. For a right action, g acts first, followed by h second. Because of the formula ( gh ) −1 = h −1 g −1 , a left action can be constructed from a right action by composing with the inverse operation of the group. Also, a right action of a group G on X can be considered as a left action of its opposite group G op on X .
Thus, for establishing general properties of group actions, it suffices to consider only left actions. However, there are cases where this is not possible. For example, the multiplication of a group induces both a left action and a right action on the group itself—multiplication on the left and on the right, respectively.
Let G be a group acting on a set X . The action is called faithful or effective if g ⋅ x = x for all x ∈ X implies that g = e G . Equivalently, the homomorphism from G to the group of bijections of X corresponding to the action is injective .
The action is called free (or semiregular or fixed-point free ) if the statement that g ⋅ x = x for some x ∈ X already implies that g = e G . In other words, no non-trivial element of G fixes a point of X . This is a much stronger property than faithfulness.
For example, the action of any group on itself by left multiplication is free. This observation implies Cayley's theorem that any group can be embedded in a symmetric group (which is infinite when the group is). A finite group may act faithfully on a set of size much smaller than its cardinality (however such an action cannot be free). For instance the abelian 2-group ( Z / 2 Z ) n (of cardinality 2 n ) acts faithfully on a set of size 2 n . This is not always the case, for example the cyclic group Z / 2 n Z cannot act faithfully on a set of size less than 2 n .
In general the smallest set on which a faithful action can be defined can vary greatly for groups of the same size. For example, three groups of size 120 are the symmetric group S 5 , the icosahedral group A 5 × Z / 2 Z and the cyclic group Z / 120 Z . The smallest sets on which faithful actions can be defined for these groups are of size 5, 7, and 16 respectively.
The action of G on X is called transitive if for any two points x , y ∈ X there exists a g ∈ G so that g ⋅ x = y .
The action is simply transitive (or sharply transitive , or regular ) if it is both transitive and free. This means that given x , y ∈ X there is exactly one g ∈ G such that g ⋅ x = y . If X is acted upon simply transitively by a group G then it is called a principal homogeneous space for G or a G -torsor.
For an integer n ≥ 1 , the action is n -transitive if X has at least n elements, and for any pair of n -tuples ( x 1 , ..., x n ), ( y 1 , ..., y n ) ∈ X n with pairwise distinct entries (that is x i ≠ x j , y i ≠ y j when i ≠ j ) there exists a g ∈ G such that g ⋅ x i = y i for i = 1, ..., n . In other words, the action on the subset of X n of tuples without repeated entries is transitive. For n = 2, 3 this is often called double, respectively triple, transitivity. The class of 2-transitive groups (that is, subgroups of a finite symmetric group whose action is 2-transitive) and more generally multiply transitive groups is well-studied in finite group theory.
An action is sharply n -transitive when the action on tuples without repeated entries in X n is sharply transitive.
The action of the symmetric group of X is transitive, in fact n -transitive for any n up to the cardinality of X . If X has cardinality n , the action of the alternating group is ( n − 2) -transitive but not ( n − 1) -transitive.
The action of the general linear group of a vector space V on the set V ∖ {0} of non-zero vectors is transitive, but not 2-transitive (similarly for the action of the special linear group if the dimension of v is at least 2). The action of the orthogonal group of a Euclidean space is not transitive on nonzero vectors but it is on the unit sphere .
The action of G on X is called primitive if there is no partition of X preserved by all elements of G apart from the trivial partitions (the partition in a single piece and its dual , the partition into singletons ).
Assume that X is a topological space and the action of G is by homeomorphisms .
The action is wandering if every x ∈ X has a neighbourhood U such that there are only finitely many g ∈ G with g ⋅ U ∩ U ≠ ∅ . [ 4 ]
More generally, a point x ∈ X is called a point of discontinuity for the action of G if there is an open subset U ∋ x such that there are only finitely many g ∈ G with g ⋅ U ∩ U ≠ ∅ . The domain of discontinuity of the action is the set of all points of discontinuity. Equivalently it is the largest G -stable open subset Ω ⊂ X such that the action of G on Ω is wandering. [ 5 ] In a dynamical context this is also called a wandering set .
The action is properly discontinuous if for every compact subset K ⊂ X there are only finitely many g ∈ G such that g ⋅ K ∩ K ≠ ∅ . This is strictly stronger than wandering; for instance the action of Z on R 2 ∖ {(0, 0)} given by n ⋅( x , y ) = (2 n x , 2 − n y ) is wandering and free but not properly discontinuous. [ 6 ]
The action by deck transformations of the fundamental group of a locally simply connected space on a universal cover is wandering and free. Such actions can be characterized by the following property: every x ∈ X has a neighbourhood U such that g ⋅ U ∩ U = ∅ for every g ∈ G ∖ { e G } . [ 7 ] Actions with this property are sometimes called freely discontinuous , and the largest subset on which the action is freely discontinuous is then called the free regular set . [ 8 ]
An action of a group G on a locally compact space X is called cocompact if there exists a compact subset A ⊂ X such that X = G ⋅ A . For a properly discontinuous action, cocompactness is equivalent to compactness of the quotient space X / G .
Now assume G is a topological group and X a topological space on which it acts by homeomorphisms. The action is said to be continuous if the map G × X → X is continuous for the product topology .
The action is said to be proper if the map G × X → X × X defined by ( g , x ) ↦ ( x , g ⋅ x ) is proper . [ 9 ] This means that given compact sets K , K ′ the set of g ∈ G such that g ⋅ K ∩ K ′ ≠ ∅ is compact. In particular, this is equivalent to proper discontinuity if G is a discrete group .
It is said to be locally free if there exists a neighbourhood U of e G such that g ⋅ x ≠ x for all x ∈ X and g ∈ U ∖ { e G } .
The action is said to be strongly continuous if the orbital map g ↦ g ⋅ x is continuous for every x ∈ X . Contrary to what the name suggests, this is a weaker property than continuity of the action. [ citation needed ]
If G is a Lie group and X a differentiable manifold , then the subspace of smooth points for the action is the set of points x ∈ X such that the map g ↦ g ⋅ x is smooth . There is a well-developed theory of Lie group actions , i.e. action which are smooth on the whole space.
If g acts by linear transformations on a module over a commutative ring , the action is said to be irreducible if there are no proper nonzero g -invariant submodules. It is said to be semisimple if it decomposes as a direct sum of irreducible actions.
Consider a group G acting on a set X . The orbit of an element x in X is the set of elements in X to which x can be moved by the elements of G . The orbit of x is denoted by G ⋅ x : G ⋅ x = { g ⋅ x : g ∈ G } . {\displaystyle G{\cdot }x=\{g{\cdot }x:g\in G\}.}
The defining properties of a group guarantee that the set of orbits of (points x in) X under the action of G form a partition of X . The associated equivalence relation is defined by saying x ~ y if and only if there exists a g in G with g ⋅ x = y . The orbits are then the equivalence classes under this relation; two elements x and y are equivalent if and only if their orbits are the same, that is, G ⋅ x = G ⋅ y .
The group action is transitive if and only if it has exactly one orbit, that is, if there exists x in X with G ⋅ x = X . This is the case if and only if G ⋅ x = X for all x in X (given that X is non-empty).
The set of all orbits of X under the action of G is written as X / G (or, less frequently, as G \ X ), and is called the quotient of the action. In geometric situations it may be called the orbit space , while in algebraic situations it may be called the space of coinvariants , and written X G , by contrast with the invariants (fixed points), denoted X G : the coinvariants are a quotient while the invariants are a subset . The coinvariant terminology and notation are used particularly in group cohomology and group homology , which use the same superscript/subscript convention.
If Y is a subset of X , then G ⋅ Y denotes the set { g ⋅ y : g ∈ G and y ∈ Y } . The subset Y is said to be invariant under G if G ⋅ Y = Y (which is equivalent G ⋅ Y ⊆ Y ). In that case, G also operates on Y by restricting the action to Y . The subset Y is called fixed under G if g ⋅ y = y for all g in G and all y in Y . Every subset that is fixed under G is also invariant under G , but not conversely.
Every orbit is an invariant subset of X on which G acts transitively . Conversely, any invariant subset of X is a union of orbits. The action of G on X is transitive if and only if all elements are equivalent, meaning that there is only one orbit.
A G -invariant element of X is x ∈ X such that g ⋅ x = x for all g ∈ G . The set of all such x is denoted X G and called the G -invariants of X . When X is a G -module , X G is the zeroth cohomology group of G with coefficients in X , and the higher cohomology groups are the derived functors of the functor of G -invariants.
Given g in G and x in X with g ⋅ x = x , it is said that " x is a fixed point of g " or that " g fixes x ". For every x in X , the stabilizer subgroup of G with respect to x (also called the isotropy group or little group [ 10 ] ) is the set of all elements in G that fix x : G x = { g ∈ G : g ⋅ x = x } . {\displaystyle G_{x}=\{g\in G:g{\cdot }x=x\}.} This is a subgroup of G , though typically not a normal one. The action of G on X is free if and only if all stabilizers are trivial. The kernel N of the homomorphism with the symmetric group, G → Sym( X ) , is given by the intersection of the stabilizers G x for all x in X . If N is trivial, the action is said to be faithful (or effective).
Let x and y be two elements in X , and let g be a group element such that y = g ⋅ x . Then the two stabilizer groups G x and G y are related by G y = gG x g −1 . Proof: by definition, h ∈ G y if and only if h ⋅( g ⋅ x ) = g ⋅ x . Applying g −1 to both sides of this equality yields ( g −1 hg )⋅ x = x ; that is, g −1 hg ∈ G x . An opposite inclusion follows similarly by taking h ∈ G x and x = g −1 ⋅ y .
The above says that the stabilizers of elements in the same orbit are conjugate to each other. Thus, to each orbit, we can associate a conjugacy class of a subgroup of G (that is, the set of all conjugates of the subgroup). Let ( H ) denote the conjugacy class of H . Then the orbit O has type ( H ) if the stabilizer G x of some/any x in O belongs to ( H ) . A maximal orbit type is often called a principal orbit type .
Orbits and stabilizers are closely related. For a fixed x in X , consider the map f : G → X given by g ↦ g ⋅ x . By definition the image f ( G ) of this map is the orbit G ⋅ x . The condition for two elements to have the same image is f ( g ) = f ( h ) ⟺ g ⋅ x = h ⋅ x ⟺ g − 1 h ⋅ x = x ⟺ g − 1 h ∈ G x ⟺ h ∈ g G x . {\displaystyle f(g)=f(h)\iff g{\cdot }x=h{\cdot }x\iff g^{-1}h{\cdot }x=x\iff g^{-1}h\in G_{x}\iff h\in gG_{x}.} In other words, f ( g ) = f ( h ) if and only if g and h lie in the same coset for the stabilizer subgroup G x . Thus, the fiber f −1 ({ y }) of f over any y in G ⋅ x is contained in such a coset, and every such coset also occurs as a fiber. Therefore f induces a bijection between the set G / G x of cosets for the stabilizer subgroup and the orbit G ⋅ x , which sends gG x ↦ g ⋅ x . [ 11 ] This result is known as the orbit-stabilizer theorem .
If G is finite then the orbit-stabilizer theorem, together with Lagrange's theorem , gives | G ⋅ x | = [ G : G x ] = | G | / | G x | , {\displaystyle |G\cdot x|=[G\,:\,G_{x}]=|G|/|G_{x}|,} in other words the length of the orbit of x times the order of its stabilizer is the order of the group . In particular that implies that the orbit length is a divisor of the group order.
This result is especially useful since it can be employed for counting arguments (typically in situations where X is finite as well).
A result closely related to the orbit-stabilizer theorem is Burnside's lemma : | X / G | = 1 | G | ∑ g ∈ G | X g | , {\displaystyle |X/G|={\frac {1}{|G|}}\sum _{g\in G}|X^{g}|,} where X g is the set of points fixed by g . This result is mainly of use when G and X are finite, when it can be interpreted as follows: the number of orbits is equal to the average number of points fixed per group element.
Fixing a group G , the set of formal differences of finite G -sets forms a ring called the Burnside ring of G , where addition corresponds to disjoint union , and multiplication to Cartesian product .
The notion of group action can be encoded by the action groupoid G ′ = G ⋉ X associated to the group action. The stabilizers of the action are the vertex groups of the groupoid and the orbits of the action are its components.
If X and Y are two G -sets, a morphism from X to Y is a function f : X → Y such that f ( g ⋅ x ) = g ⋅ f ( x ) for all g in G and all x in X . Morphisms of G -sets are also called equivariant maps or G - maps .
The composition of two morphisms is again a morphism. If a morphism f is bijective, then its inverse is also a morphism. In this case f is called an isomorphism , and the two G -sets X and Y are called isomorphic ; for all practical purposes, isomorphic G -sets are indistinguishable.
Some example isomorphisms:
With this notion of morphism, the collection of all G -sets forms a category ; this category is a Grothendieck topos (in fact, assuming a classical metalogic , this topos will even be Boolean).
We can also consider actions of monoids on sets, by using the same two axioms as above. This does not define bijective maps and equivalence relations however. See semigroup action .
Instead of actions on sets, we can define actions of groups and monoids on objects of an arbitrary category: start with an object X of some category, and then define an action on X as a monoid homomorphism into the monoid of endomorphisms of X . If X has an underlying set, then all definitions and facts stated above can be carried over. For example, if we take the category of vector spaces, we obtain group representations in this fashion.
We can view a group G as a category with a single object in which every morphism is invertible . [ 15 ] A (left) group action is then nothing but a (covariant) functor from G to the category of sets , and a group representation is a functor from G to the category of vector spaces . [ 16 ] A morphism between G -sets is then a natural transformation between the group action functors. [ 17 ] In analogy, an action of a groupoid is a functor from the groupoid to the category of sets or to some other category.
In addition to continuous actions of topological groups on topological spaces, one also often considers smooth actions of Lie groups on smooth manifolds , regular actions of algebraic groups on algebraic varieties , and actions of group schemes on schemes . All of these are examples of group objects acting on objects of their respective category. | https://en.wikipedia.org/wiki/Group_action |
Group actions are central to Riemannian geometry and defining orbits (control theory) .
The orbits of computational anatomy consist of anatomical shapes and medical images ; the anatomical shapes are submanifolds of differential geometry consisting of points, curves, surfaces and subvolumes,.
This generalized the ideas of the more familiar orbits of linear algebra which are linear vector spaces . Medical images are scalar and tensor images from medical imaging . The group actions are used to define models of human shape which accommodate variation. These orbits are deformable templates as originally formulated more abstractly in pattern theory .
The central model of human anatomy in computational anatomy is a Groups and group action , a classic formulation from differential geometry . The orbit is called the space of shapes and forms . [ 1 ] The space of shapes are denoted m ∈ M {\displaystyle m\in {\mathcal {M}}} , with the group ( G , ∘ ) {\displaystyle ({\mathcal {G}},\circ )} with law of composition ∘ {\displaystyle \circ } ; the action of the group on shapes is denoted g ⋅ m {\displaystyle g\cdot m} , where the action of the group g ⋅ m ∈ M , m ∈ M {\displaystyle g\cdot m\in {\mathcal {M}},m\in {\mathcal {M}}} is defined to satisfy
The orbit M {\displaystyle {\mathcal {M}}} of the template becomes the space of all shapes, M ≐ { m = g ⋅ m t e m p , g ∈ G } {\displaystyle {\mathcal {M}}\doteq \{m=g\cdot m_{\mathrm {temp} },g\in {\mathcal {G}}\}} .
The central group in CA defined on volumes in R 3 {\displaystyle {\mathbb {R} }^{3}} are the diffeomorphism group G ≐ D i f f {\displaystyle {\mathcal {G}}\doteq \mathrm {Diff} } which are mappings with 3-components ϕ ( ⋅ ) = ( ϕ 1 ( ⋅ ) , ϕ 2 ( ⋅ ) , ϕ 3 ( ⋅ ) ) {\displaystyle \phi (\cdot )=(\phi _{1}(\cdot ),\phi _{2}(\cdot ),\phi _{3}(\cdot ))} , law of composition of functions ϕ ∘ ϕ ′ ( ⋅ ) ≐ ϕ ( ϕ ′ ( ⋅ ) ) {\displaystyle \phi \circ \phi ^{\prime }(\cdot )\doteq \phi (\phi ^{\prime }(\cdot ))} , with inverse ϕ ∘ ϕ − 1 ( ⋅ ) = ϕ ( ϕ − 1 ( ⋅ ) ) = id {\displaystyle \phi \circ \phi ^{-1}(\cdot )=\phi (\phi ^{-1}(\cdot ))=\operatorname {id} } .
For sub- manifolds X ⊂ R 3 ∈ M {\displaystyle X\subset {\mathbb {R} }^{3}\in {\mathcal {M}}} , parametrized by a chart or immersion m ( u ) , u ∈ U {\displaystyle m(u),u\in U} , the diffeomorphic action the flow of the position
Most popular are scalar images, I ( x ) , x ∈ R 3 {\displaystyle I(x),x\in {\mathbb {R} }^{3}} , with action on the right via the inverse.
Many different imaging modalities are being used with various actions. For images such that I ( x ) {\displaystyle I(x)} is a three-dimensional vector then
Cao et al. [ 2 ] examined actions for mapping MRI images measured via diffusion tensor imaging and represented via there principle eigenvector.
For tensor fields a positively oriented orthonormal basis I ( x ) = ( I 1 ( x ) , I 2 ( x ) , I 3 ( x ) ) {\displaystyle I(x)=(I_{1}(x),I_{2}(x),I_{3}(x))} of R 3 {\displaystyle {\mathbb {R} }^{3}} , termed frames, vector cross product denoted I 1 × I 2 {\displaystyle I_{1}\times I_{2}} then
The Frénet frame of three orthonormal vectors, I 1 {\displaystyle I_{1}} deforms as a tangent, I 3 {\displaystyle I_{3}} deforms like
a normal to the plane generated by I 1 × I 2 {\displaystyle I_{1}\times I_{2}} , and I 3 {\displaystyle I_{3}} . H is uniquely constrained by the
basis being positive and orthonormal.
For 3 × 3 {\displaystyle 3\times 3} non-negative symmetric matrices, an action would become φ ⋅ I = ( D φ I D φ T ) ∘ φ − 1 {\displaystyle \varphi \cdot I=(D\varphi \,ID\varphi ^{T})\circ \varphi ^{-1}} .
For mapping MRI DTI images [ 3 ] [ 4 ] (tensors), then eigenvalues are preserved with the diffeomorphism rotating eigenvectors and preserves the eigenvalues.
Given eigenelements { λ i , e i , i = 1 , 2 , 3 } {\displaystyle \{\lambda _{i},e_{i},i=1,2,3\}} , then the action becomes
Orientation distribution function (ODF) characterizes the angular profile of the diffusion probability density function of water molecules and can be reconstructed from High Angular Resolution Diffusion Imaging (HARDI). The ODF is a probability density function defined on a unit sphere, S 2 {\displaystyle {\mathbb {S} }^{2}} . In the field of information geometry , [ 5 ] the space of ODF forms a Riemannian manifold with the Fisher-Rao metric. For the purpose of LDDMM ODF mapping, the square-root representation is chosen because it is one of the most efficient representations found to date as the various Riemannian operations, such as geodesics, exponential maps, and logarithm maps, are available in closed form. In the following, denote square-root ODF ( ODF {\displaystyle {\sqrt {\text{ODF}}}} ) as ψ ( s ) {\displaystyle \psi ({\bf {s}})} , where ψ ( s ) {\displaystyle \psi ({\bf {s}})} is non-negative to ensure uniqueness and ∫ s ∈ S 2 ψ 2 ( s ) d s = 1 {\displaystyle \int _{{\bf {s}}\in {\mathbb {S} }^{2}}\psi ^{2}({\bf {s}})d{\bf {s}}=1} .
Denote diffeomorphic transformation as ϕ {\displaystyle \phi } . Group action of diffeomorphism on ψ ( s ) {\displaystyle \psi ({\bf {s}})} , ϕ ⋅ ψ {\displaystyle \phi \cdot \psi } , needs to guarantee the non-negativity and ∫ s ∈ S 2 ϕ ⋅ ψ 2 ( s ) d s = 1 {\displaystyle \int _{{\bf {s}}\in {\mathbb {S} }^{2}}\phi \cdot \psi ^{2}({\bf {s}})d{\bf {s}}=1} . Based on the derivation in, [ 6 ] this group action is defined as
where ( D ϕ ) {\displaystyle (D\phi )} is the Jacobian of ϕ {\displaystyle \phi } . | https://en.wikipedia.org/wiki/Group_actions_in_computational_anatomy |
Group analysis of differential equations is a branch of mathematics that studies the symmetry properties of differential equations with respect to various transformations of independent and dependent variables. It includes methods and applied aspects of differential geometry , Lie groups and algebras theory, calculus of variations and is, in turn, a powerful research tool in theories of ODEs , PDEs , mathematical and theoretical physics . [ 1 ] [ 2 ] [ 3 ]
This geometry-related article is a stub . You can help Wikipedia by expanding it .
This group theory -related article is a stub . You can help Wikipedia by expanding it . | https://en.wikipedia.org/wiki/Group_analysis_of_differential_equations |
In coding theory , group codes are a type of code . Group codes consist of n {\displaystyle n} linear block codes which are subgroups of G n {\displaystyle G^{n}} , where G {\displaystyle G} is a finite Abelian group .
A systematic group code C {\displaystyle C} is a code over G n {\displaystyle G^{n}} of order | G | k {\displaystyle \left|G\right|^{k}} defined by n − k {\displaystyle n-k} homomorphisms which determine the parity check bits. The remaining k {\displaystyle k} bits are the information bits themselves.
Group codes can be constructed by special generator matrices which resemble generator matrices of linear block codes except that the elements of those matrices are endomorphisms of the group instead of symbols from the code's alphabet. For example, considering the generator matrix
the elements of this matrix are 2 × 2 {\displaystyle 2\times 2} matrices which are endomorphisms. In this scenario, each codeword can be represented as g 1 m 1 g 2 m 2 . . . g r m r {\displaystyle g_{1}^{m_{1}}g_{2}^{m_{2}}...g_{r}^{m_{r}}} where g 1 , . . . g r {\displaystyle g_{1},...g_{r}} are the generators of G {\displaystyle G} . | https://en.wikipedia.org/wiki/Group_code |
In computer science , group coded recording or group code recording ( GCR ) refers to several distinct but related encoding methods for representing data on magnetic media . The first, used in 6250 bpi magnetic tape since 1973, [ 1 ] [ 2 ] is an error-correcting code combined with a run-length limited (RLL) encoding scheme, belonging into the group of modulation codes . [ 3 ] The others are similar encoding methods used in mainframe hard disks or microcomputer floppy disks until the late 1980s. GCR is a modified form of a NRZI code, but necessarily with a higher transition density. [ 3 ]
Group coded recording was first used for magnetic-tape data storage on 9-track reel-to-reel tape . [ 3 ] The term was coined during the development of the IBM 3420 Model 4/6/8 Magnetic Tape Unit [ 1 ] and the corresponding 3803 Model 2 Tape Control Unit, [ 4 ] [ 1 ] both introduced in 1973. [ 1 ] [ 5 ] IBM referred to the error correcting code itself as "group coded recording". However, GCR has come to refer to the recording format of 6250 bpi (250 bits/mm [ 3 ] ) tape as a whole, and later to formats which use similar RLL codes without the error correction code.
In order to reliably read and write to magnetic tape , several constraints on the signal to be written must be followed. The first is that two adjacent flux reversals must be separated by a certain distance on the media, defined by the magnetic properties of the media itself. The second is that there must be a reversal often enough to keep the reader's clock in phase with the written signal; that is, the signal must be self-clocking and most importantly to keep the playback output high enough as this is proportional to the density of flux transitions. Prior to 6250 bpi tapes, 1600 bpi tapes satisfied these constraints using a technique called phase encoding (PE), which was only 50% efficient. For 6250 bpi GCR tapes, a (0, 2) RLL code is used, or more specifically a 4 / 5 (0, 2) block code [ 3 ] sometimes also referred to as GCR (4B-5B) encoding. [ 6 ] This code requires five bits to be written for every four bits of data. [ 3 ] The code is structured so that no more than two zero bits (which are represented by lack of a flux reversal) can occur in a row, [ 3 ] either within a code or between codes, no matter what the data was. This RLL code is applied independently to the data going to each of the nine tracks.
Of the 32 five-bit patterns, eight begin with two consecutive zero bits, six others end with two consecutive zero bits, and one more (10001) contains three consecutive zero bits. Removing the all-ones pattern (11111) from the remainder leaves 16 suitable code words.
The 6250 bpi GCR RLL code: [ 7 ] [ 8 ] [ 9 ] [ 6 ]
11 of the nibbles (other than xx00 and 0001) have their code formed by prepending the complement of the most significant bit ; i.e. abcd is encoded as a abcd. The other five values are assigned codes beginning with 11. Nibbles of the form ab00 have codes 11ba a , i.e. the bit reverse of the code for ab11. The code 0001 is assigned the remaining value 11011.
Because the all-ones code is not used in normal data, at most 8 one-bits can appear in a row. Sequences of 9 or more one-bits (in practice 14 all-ones codes, or 70 one-bits, were used) are used as a synchronization pattern .
Because of the extremely high density (for the time) of 6250 bpi tape, the RLL code is not sufficient to ensure reliable data storage. On top of the RLL code, an error-correcting code called the Optimal Rectangular Code (ORC) is applied. [ 10 ] This code is a combination of a parity track and polynomial code similar to a CRC , but structured for error correction rather than error detection. For every seven bytes written to the tape (before RLL encoding), an eighth check byte is calculated and written to the tape. When reading, the parity is calculated on each byte and exclusive-ORed with the contents of the parity track, and the polynomial check code calculated and exclusive-ORed with the received check code, resulting in two 8-bit syndrome words. If these are both zero, the data is error free. Otherwise, error-correction logic in the tape controller corrects the data before it is forwarded to the host. The error correcting code is able to correct any number of errors in any single track, or in any two tracks if the erroneous tracks can be identified by other means.
In newer IBM half-inch 18-track tape drives recording at 24 000 bpi , 4 / 5 (0, 2) GCR was replaced by a more efficient 8 / 9 (0, 3) modulation code, mapping eight bits to nine bits. [ 3 ]
In the mid-1970s, Sperry Univac , ISS Division was working on large hard drives for the mainframe business using group coding. [ 11 ]
Like magnetic tape drives, floppy disk drives have physical limits on the spacing of flux reversals (also called transitions, represented by one-bits).
Offering GCR-compatible diskette drives and floppy disk controllers (like the 100163-51-8 and 100163-52-6 [ 12 ] ), Micropolis endorsed data encoding with group coded recording [ 13 ] on 5¼-inch 100 tpi 77-track diskette drives to store twelve 512-byte sectors per track since 1977 or 1978. [ 14 ] [ 15 ] [ 16 ] [ 17 ]
Micro Peripherals , Inc. (MPI) marketed double-density 5¼-inch disk drives (like the single-sided B51 and double-sided B52 drives) and a controller solution implementing GCR since early 1978. [ 18 ] [ 19 ]
The Durango Systems F-85 (introduced in September 1978 [ 20 ] [ 21 ] ) used single-sided 5¼-inch 100 tpi diskette drives providing 480 KB utilizing a proprietary high-density 4/5 group coded encoding. The machine was using a Western Digital FD1781 floppy disk controller, designed by a former Sperry ISS engineer, [ 17 ] with 77-track Micropolis drives. [ 22 ] In later models such as the Durango 800 [ 23 ] series this was expanded to a double-sided option for 960 KB (946 KB formatted [ 23 ] [ nb 1 ] ) per diskette. [ 21 ] [ 24 ] [ 22 ] [ 14 ]
For the Apple II floppy drive, Steve Wozniak invented a floppy controller which (along with the Disk II drive itself) imposed two constraints:
The simplest scheme to ensure compliance with these limits is to record an extra "clock" transition before each data bit according to differential Manchester encoding or (digital) FM (frequency modulation). Known as 4-and-4 encoding , the resulting Apple implementation allowed only ten 256-byte sectors per track to be recorded on a single-density 5¼-inch floppy. It uses two bytes for each byte.
Close to a month prior to the shipment of the disk drive in spring 1978, [ 26 ] Wozniak realized that a more complex encoding scheme would allow each eight-bit byte on disk to hold five bits of useful data rather than four bits. This is because there are 34 bytes which have the top bit set and no two zero bits in a row. This encoding scheme became known as 5-and-3 encoding , and allowed 13 sectors per track; it was used for Apple DOS 3.1, 3.2 , and 3.2.1 , as well as for the earliest version of Apple CP/M [ de ] : [ 27 ]
Reserved GCR-codes: 0xAA and 0xD5. [ 27 ]
Wozniak called the system "my most incredible experience at Apple and the finest job I did". [ 26 ]
Later, the design of the floppy drive controller was modified to allow a byte on disk to contain up to one pair of zero bits in a row. This allowed each eight-bit byte to hold six bits of useful data, and allowed 16 sectors per track. This scheme is known as 6-and-2 encoding , [ 27 ] and was used on Apple Pascal , Apple DOS 3.3 [ 27 ] and ProDOS , [ 29 ] and later with Apple FileWare drives in the Apple Lisa and the 400K and 800K 3½-inch disks on the Macintosh and Apple II. [ 30 ] [ 31 ] Apple did not originally call this scheme "GCR", but the term was later applied to it [ 31 ] to distinguish it from IBM PC floppies which used the MFM encoding scheme.
Reserved GCR-codes: 0xAA and 0xD5. [ 27 ] [ 29 ]
Independently, Commodore Business Machines (CBM) created a group coded recording scheme for their Commodore 2040 floppy disk drive (launched in the spring of 1979). The relevant constraints on the 2040 drive were that no more than two zero bits could occur in a row; the drive imposed no special constraint on the first bit in a byte. This allowed the use of a scheme similar to that used in 6250 bpi tape drives. Every four bits of data are translated into five bits on disk, using the same 5-bit codes as IBM to ensure there are never more than two zero bits in a row, but in a different order:
Like the IBM code, at most eight one bits in a row are possible, so Commodore used sequences of ten or more one bits in a row as a synchronization sequence .
This more efficient GCR scheme, combined with an approach at constant bit-density recording by gradually increasing the clock rate ( zone constant angular velocity , ZCAV) and storing more physical sectors on the outer tracks than on the inner ones ( zone bit recording , ZBR), enabled Commodore to fit 170 KiB on a standard single-sided single-density 5.25-inch floppy, where Apple fit 140 KiB (with 6-and-2 encoding) or 114 KiB (with 5-and-3 encoding) and an FM-encoded floppy held only 88 KiB .
Similar, the 5.25-inch floppy drives of the Victor 9000 aka Sirius 1 , designed by Chuck Peddle in 1981/1982, used a combination of GCR and zone bit recording by gradually decreasing a drive's rotational speed for the outer tracks in nine zones while increasing the number of sectors per track [ 33 ] to achieve formatted capacities of 606 KiB (single sided) / 1188 KiB (double-sided) on 96 tpi media. [ 34 ] [ 35 ] [ 36 ] [ 37 ] The GCR code is identical to the Commodore one. [ 38 ]
Starting around 1985, Brother introduced a family of dedicated word processor typewriters with integrated 3.5-inch 38-track [ nb 2 ] diskette drive. Early models of the WP and LW series [ de ] used a Brother-specific group-coded recording scheme with twelve 256-byte sectors to store up to 120 KB [ nb 3 ] on single-sided and up to 240 KB [ nb 3 ] on double-sided double-density (DD) diskettes. [ 17 ] [ 39 ] [ 40 ] [ 41 ] Reportedly, prototypes were already shown at the Internationale Funkausstellung 1979 (IFA) in Berlin.
In 1986, Sharp introduced a turnable 2.5-inch pocket disk drive solution (drives: CE-1600F , CE-140F ; internally based on the FDU-250 chassis; media: CE-1650F ) for their series of pocket computers with a formatted capacity of 62 464 bytes per side (2× 64 kB nominal, 16 tracks, 8 sectors/track, 512 bytes per sector, 48 tpi , 250 kbit/s, 270 rpm) with GCR (4/5) recording. [ 42 ] [ 43 ]
GCR was also evaluated for a possible use in bar code encoding schemes (packing efficiency, timing tolerances, amount of storage bytes for timing information, and DC output level). [ 44 ] | https://en.wikipedia.org/wiki/Group_coded_recording |
In signal processing , group delay and phase delay are functions that describe in different ways the delay times experienced by a signal’s various sinusoidal frequency components as they pass through a linear time-invariant (LTI) system (such as a microphone , coaxial cable , amplifier , loudspeaker , communications system , ethernet cable , digital filter , or analog filter ).
Unfortunately, these delays are sometimes frequency dependent, [ 1 ] which means that different sinusoid frequency components experience different time delays. As a result, the signal's waveform experiences distortion as it passes through the system. This distortion can cause problems such as poor fidelity in analog video and analog audio , or a high bit-error rate in a digital bit stream.
Fourier analysis reveals how signals in time can alternatively be expressed as the sum of sinusoidal frequency components , each based on the trigonometric function sin ( x ) {\displaystyle \sin(x)} with a fixed amplitude and phase and no beginning and no end.
Linear time-invariant systems process each sinusoidal component independently; the property of linearity means they satisfy the superposition principle .
The group delay and phase delay properties of a linear time-invariant (LTI) system are functions of frequency, giving the time from when a frequency component of a time varying physical quantity—for example a voltage signal—appears at the LTI system input, to the time when a copy of that same frequency component—perhaps of a different physical phenomenon—appears at the LTI system output.
A varying phase response as a function of frequency, from which group delay and phase delay can be calculated, typically occurs in devices such as microphones, amplifiers, loudspeakers, magnetic recorders, headphones, coaxial cables, and antialiasing filters. [ 2 ] All frequency components of a signal are delayed when passed through such devices, or when propagating through space or a medium, such as air or water.
While a phase response describes phase shift in angular units (such as degrees or radians ), the phase delay is in units of time and equals the negative of the phase shift at each frequency divided by the value of that frequency. Group delay is the negative derivative of phase shift with respect to frequency.
A linear time-invariant system or device has a phase response property and a phase delay property, where one can be calculated exactly from the other. Phase delay directly measures the device or system time delay of individual sinusoidal frequency components. If the phase delay function at any given frequency—within a frequency range of interest—has the same constant of proportionality between the phase at a selected frequency and the selected frequency itself, the system/device will have the ideal of a flat phase delay property, a.k.a. linear phase . [ 1 ] Since phase delay is a function of frequency giving time delay, a departure from the flatness of its function graph can reveal time delay differences among the signal’s various sinusoidal frequency components , in which case those differences will contribute to signal distortion, which is manifested as the output signal waveform shape being different from that of the input signal.
The phase delay property in general does not give useful information if the device input is a modulated signal. For that, group delay must be used.
The group delay is a convenient measure of the linearity of the phase with respect to frequency in a modulation system. [ 3 ] [ 4 ] For a modulation signal (passband signal), the information carried by the signal is carried exclusively in the wave envelope . Group delay therefore operates only with the frequency components derived from the envelope.
A device's group delay can be exactly calculated from the device's phase response, but not the other way around. The simplest use case for group delay is illustrated in Figure 1 which shows a conceptual modulation system, which is itself an LTI system with a baseband output that is ideally an accurate copy of the baseband signal input. This system as a whole is referred to here as the outer LTI system/device, which contains an inner (red block) LTI system/device. As is often the case for a radio system, the inner red LTI system in Fig 1 can represent two LTI systems in cascade, for example an amplifier driving a transmitting antenna at the sending end and the other an antenna and amplifier at the receiving end.
Amplitude modulation creates the passband signal by shifting the baseband frequency components to a much higher frequency range. Although the frequencies are different, the passband signal carries the same information as the baseband signal. The demodulator does the inverse, shifting the passband frequencies back down to the original baseband frequency range. Ideally, the output (baseband) signal is a time delayed version of the input (baseband) signal where the waveform shape of the output is identical to that of the input.
In Figure 1, the outer system phase delay is the meaningful performance metric. For amplitude modulation, the inner red LTI device group delay becomes the outer LTI device phase delay . If the inner red device group delay is completely flat in the frequency range of interest, the outer device will have the ideal of a phase delay that is also completely flat, where the contribution of distortion due to the outer LTI device's phase response—determined entirely by the inner device's possibly different phase response—is eliminated. In that case, the group delay of the inner red device and the phase delay of the outer device give the same time delay figure for the signal as a whole, from the baseband input to the baseband output. It is significant to note that it is possible for the inner (red) device to have a very non-flat phase delay (but flat group delay), while the outer device has the ideal of a perfectly flat phase delay. This is fortunate because in LTI device design, a flat group delay is easier to achieve than a flat phase delay.
In an angle-modulation system—such as with frequency modulation (FM) or phase modulation (PM)—the (FM or PM) passband signal applied to an LTI system input can be analyzed as two separate passband signals, an in-phase (I) amplitude modulation AM passband signal and a quadrature-phase (Q) amplitude modulation AM passband signal, where their sum exactly reconstructs the original angle-modulation (FM or PM) passband signal. While the (FM/PM) passband signal is not amplitude modulation, and therefore has no apparent outer envelope, the I and Q passband signals do indeed have separate amplitude modulation envelopes. (However, unlike with regular amplitude modulation, the I and Q envelopes do not resemble the wave shape of the baseband signals, even though 100 percent of the baseband signal is represented in a complex manner by their envelopes.) So, for each of the I and Q passband signals, a flat group delay ensures that neither the I pass band envelope nor the Q passband envelope will have wave shape distortion, so when the I passband signal and the Q passband signal are added back together, the sum is the original FM/PM passband signal, which will also be unaltered.
According to LTI system theory (used in control theory and digital or analog signal processing ), the output signal y ( t ) {\displaystyle \displaystyle y(t)} of an LTI system can be determined by convolving the time-domain impulse response h ( t ) {\displaystyle \displaystyle h(t)} of the LTI system with the input signal x ( t ) {\displaystyle \displaystyle x(t)} . Linear time-invariant system § Fourier and Laplace transforms expresses this relationship as:
where ∗ {\displaystyle *} denotes the convolution operation, X ( s ) {\displaystyle \displaystyle X(s)} and H ( s ) {\displaystyle \displaystyle H(s)} are the Laplace transforms of the input x ( t ) {\displaystyle \displaystyle x(t)} and impulse response h ( t ) {\displaystyle \displaystyle h(t)} , respectively, s is the complex frequency , and L − 1 {\displaystyle {\mathcal {L}}^{-1}} is the inverse Laplace transform. H ( s ) {\displaystyle \displaystyle H(s)} is called the transfer function of the LTI system and, like the impulse response h ( t ) {\displaystyle \displaystyle h(t)} , fully defines the input-output characteristics of the LTI system. This convolution can be evaluated by using the integral expression in the time domain , or (according to the rightmost expression) by using multiplication in the Laplace domain and then applying the inverse transform to return to time domain.
Suppose that such a system is driven by a wave packet formed by a sinusoid multiplied by an amplitude envelope A env ( t ) > 0 {\displaystyle \displaystyle A_{\text{env}}(t)>0} , so the input x ( t ) {\displaystyle \displaystyle x(t)} can be expressed in the following form:
Also suppose that the envelope A env ( t ) {\displaystyle \displaystyle A_{\text{env}}(t)} is slowly changing relative to the sinusoid's frequency ω {\displaystyle \displaystyle \omega } . This condition can be expressed mathematically as:
Applying the earlier convolution equation would reveal that the output of such an LTI system is very well approximated [ clarification needed ] as:
Here τ g {\displaystyle \displaystyle \tau _{g}} is the group delay and τ ϕ {\displaystyle \displaystyle \tau _{\phi }} is the phase delay, and they are given by the expressions below (and potentially are functions of the angular frequency ω {\displaystyle \displaystyle \omega } ). The phase of the sinusoid, as indicated by the positions of the zero crossings, is delayed in time by an amount equal to the phase delay, τ ϕ {\displaystyle \displaystyle \tau _{\phi }} . The envelope of the sinusoid is delayed in time by the group delay, τ g {\displaystyle \displaystyle \tau _{g}} .
The group delay , τ g {\displaystyle \displaystyle \tau _{g}} , and phase delay , τ ϕ {\displaystyle \displaystyle \tau _{\phi }} , are (potentially) frequency-dependent [ 5 ] and can be computed from the unwrapped phase shift ϕ ( ω ) {\displaystyle \displaystyle \phi (\omega )} . The phase delay at each frequency equals the negative of the phase shift at that frequency divided by the value of that frequency:
The group delay at each frequency equals the negative of the slope (i.e. the derivative with respect to frequency) of the phase at that frequency: [ 6 ]
In a linear phase system (with non-inverting gain), both τ g {\displaystyle \displaystyle \tau _{g}} and τ ϕ {\displaystyle \displaystyle \tau _{\phi }} are constant (i.e., independent of ω {\displaystyle \displaystyle \omega } ) and equal, and their common value equals the overall delay of the system; and the unwrapped phase shift of the system (namely − ω τ ϕ {\displaystyle \displaystyle -\omega \tau _{\phi }} ) is negative, with magnitude increasing linearly with frequency ω {\displaystyle \displaystyle \omega } .
More generally, it can be shown that for an LTI system with transfer function H ( s ) {\displaystyle \displaystyle H(s)} driven by a complex sinusoid of unit amplitude,
the output is
where the phase shift ϕ {\displaystyle \displaystyle \phi } is
The phase of a 1st-order low-pass filter formed by a RC circuit with cutoff frequency ω o = 1 R C {\displaystyle \omega _{o}{=}{\frac {1}{RC}}} is: [ 7 ]
ϕ ( ω ) = − arctan ( ω ω o ) . {\displaystyle \phi (\omega )=-\arctan({\frac {\omega }{\omega _{o}}})\,.}
Similarly, the phase for a 1st-order RC high-pass filter is:
ϕ ( ω ) = π 2 − arctan ( ω ω o ) . {\displaystyle \phi (\omega )={\frac {\pi }{2}}-\arctan({\frac {\omega }{\omega _{o}}})\,.}
Taking the negative derivative with respect to ω {\displaystyle \omega } for either this low-pass or high-pass filter yields the same group delay of: [ 8 ]
τ g ( ω ) = ω o ω 2 + ω o 2 . {\displaystyle {\begin{aligned}\tau _{g}(\omega )&={\frac {\omega _{o}}{\omega ^{2}+\omega _{o}^{2}}}\,.\\\end{aligned}}}
For frequencies significantly lower than the cutoff frequency, the phase response is approximately linear (arctan for small inputs can be approximated as a line), so the group delay simplifies to a constant value of:
τ g ( ω ≪ ω o ) ≈ 1 ω o = R C . {\displaystyle {\begin{aligned}\tau _{g}(\omega \ll \omega _{o})&\approx {\frac {1}{\omega _{o}}}=RC\,.\\\end{aligned}}}
Similarly, right at the cutoff frequency, τ g ( ω = ω o ) = 1 2 ω o = R C 2 . {\displaystyle \tau _{g}(\omega {=}\omega _{o})={\frac {1}{2\omega _{o}}}={\frac {RC}{2}}\,.}
As frequencies get even larger, the group delay decreases with the inverse square of the frequency and approaches zero as frequency approaches infinity.
Filters will have negative group delay over frequency ranges where its phase response is positively-sloped. If a signal is band-limited within some maximum frequency B, then it is predictable to a small degree (within time periods smaller than 1 ⁄ B ). A filter whose group delay is negative over that signal's entire frequency range is able to use the signal's predictability to provide an illusion of a non-causal time advance. However, if the signal contains an unpredictable event (such as an abrupt change which makes the signal's spectrum exceed its band-limit), then the illusion breaks down. [ 9 ] Circuits with negative group delay (e.g., Figure 2) are possible, though causality is not violated. [ 10 ]
Negative group delay filters can be made in both digital and analog domains. Applications include compensating for the inherent delay of low-pass filters, to create zero phase filters, which can be used to quickly detect changes in the trends of sensor data or stock prices. [ 11 ]
Group delay has some importance in the audio field and especially in the sound reproduction field. [ 12 ] [ 13 ] Many components of an audio reproduction chain, notably loudspeakers and multiway loudspeaker crossover networks , introduce group delay in the audio signal. [ 2 ] [ 13 ] It is therefore important to know the threshold of audibility of group delay with respect to frequency, [ 14 ] [ 15 ] [ 16 ] especially if the audio chain is supposed to provide high fidelity reproduction. The best thresholds of audibility table has been provided by Blauert and Laws. [ 17 ]
Flanagan, Moore and Stone conclude that at 1, 2 and 4 kHz, a group delay of about 1.6 ms is audible with headphones in a non-reverberant condition. [ 18 ] Other experimental results suggest that when the group delay in the frequency range from 300 Hz to 1 kHz is below 1.0 ms, it is inaudible. [ 15 ]
The waveform of any signal can be reproduced exactly by a system that has a flat frequency response and group delay over the bandwidth of the signal. Leach [ 19 ] introduced the concept of differential time-delay distortion, defined as the difference between the phase delay and the group delay:
An ideal system should exhibit zero or negligible differential time-delay distortion. [ 19 ]
It is possible to use digital signal processing techniques to correct the group delay distortion that arises due to the use of crossover networks in multi-way loudspeaker systems. [ 20 ] This involves considerable computational modeling of loudspeaker systems in order to successfully apply delay equalization, [ 21 ] using the Parks-McClellan FIR equiripple filter design algorithm . [ 1 ] [ 4 ] [ 22 ] [ 23 ]
Group delay is important in physics , and in particular in optics .
In an optical fiber , group delay is the transit time required for optical power , traveling at a given mode 's group velocity , to travel a given distance. For optical fiber dispersion measurement purposes, the quantity of interest is group delay per unit length, which is the reciprocal of the group velocity of a particular mode. The measured group delay of a signal through an optical fiber exhibits a wavelength dependence due to the various dispersion mechanisms present in the fiber.
It is often desirable for the group delay to be constant across all frequencies; otherwise there is temporal smearing of the signal. Because group delay is τ g ( ω ) = − d ϕ d ω {\textstyle \tau _{g}(\omega )=-{\frac {d\phi }{d\omega }}} , it therefore follows that a constant group delay can be achieved if the transfer function of the device or medium has a linear phase response (i.e., ϕ ( ω ) = ϕ ( 0 ) − τ g ω {\displaystyle \phi (\omega )=\phi (0)-\tau _{g}\omega } where the group delay τ g {\displaystyle \tau _{g}} is a constant). The degree of nonlinearity of the phase indicates the deviation of the group delay from a constant value.
The differential group delay is the difference in propagation time between the two eigenmodes X and Y polarizations . Consider two eigenmodes that are the 0° and 90° linear polarization states. If the state of polarization of the input signal is the linear state at 45° between the two eigenmodes, the input signal is divided equally into the two eigenmodes. The power of the transmitted signal E T ,total is the combination of the transmitted signals of both x and y modes.
The differential group delay D t is defined as the difference in propagation time between the eigenmodes: D t = | t t , x − t t , y |.
A transmitting apparatus is said to have true time delay (TTD) if the time delay is independent of the frequency of the electrical signal. [ 24 ] [ 25 ] TTD allows for a wide instantaneous signal bandwidth with virtually no signal distortion such as pulse broadening during pulsed operation.
TTD is an important characteristic of lossless and low-loss, dispersion free, transmission lines . Telegrapher's equations § Lossless transmission reveals that signals propagate through them at a speed of 1 / L C {\displaystyle 1/{\sqrt {LC}}} for a distributed inductance L and capacitance C . Hence, any signal's propagation delay through the line simply equals the length of the line divided by this speed.
If a transfer function or Sij of a scattering parameter , is in a polynomial Laplace transform form, then the mathematical definition for group delay above may be solved analytically in closed form. A polynomial transfer function P ( S ) {\displaystyle P(S)} may be taken along the j ω {\displaystyle j\omega } axis and defined as P ( j ω ) {\displaystyle P(j\omega )} . ϕ ( ω ) {\displaystyle \phi (\omega )} may be determined from P ( j ω ) {\displaystyle P(j\omega )} , and then the group delay may be determined by solving for − d ϕ ( ω ) | / d ω {\displaystyle -d\phi (\omega )|/d\omega } .
to determine ϕ ( ω ) {\displaystyle \phi (\omega )} from P ( j ω ) {\displaystyle P(j\omega )} , use the definition of ϕ ( ω ) = t a n − 1 ( P ( j ω ) i m a g / P ( j ω ) r e a l ) {\displaystyle \phi (\omega )=tan^{-1}(P(j\omega )_{imag}/P(j\omega )_{real})} . Given that j 2 N {\displaystyle j^{2N}} is always real, and j 2 N + 1 {\displaystyle j^{2N+1}} is always imaginary, ϕ ( ω ) {\displaystyle \phi (\omega )} may be redefined as ϕ ( ω ) = t a n − 1 ( − j P ( j ω ) o d d / P ( j ω ) e v e n ) {\displaystyle \phi (\omega )=tan^{-1}(-jP(j\omega )_{odd}/P(j\omega )_{even})} where even and odd refer to the polynomials that contain only the even or odd order coefficients respectively. The − j {\displaystyle -j} in the numerator merely converts the imaginary P ( j ω ) o d d {\displaystyle P(j\omega )_{odd}} numerator to a real value, since P ( j ω ) o d d {\displaystyle P(j\omega )_{odd}} by itself is purely imaginary.
d t a n − 1 ( f ( x ) ) d x = d f ( x ) / d x 1 + f ( x ) 2 f ( x ) = − j P ( j ω ) o d d P ( j ω ) e v e n d f ( x ) d x = P ( j ω ) e v e n d ( P ( j ω ) o d d ) d x − − j P ( j ω ) o d d d ( − j P ( j ω ) e v e n ) d x P ( j ω ) e v e n 2 {\displaystyle {\begin{aligned}&{\frac {dtan^{-1}(f(x))}{dx}}={\frac {df(x)/dx}{1+f(x)^{2}}}\\&f(x)={\frac {-jP(j\omega )_{odd}}{P(j\omega )_{even}}}\\&{\frac {df(x)}{dx}}={\frac {P(j\omega )_{even}{\frac {d(P(j\omega )_{odd})}{dx}}--jP(j\omega )_{odd}{\frac {d(-jP(j\omega )_{even})}{dx}}}{P(j\omega )_{even}^{2}}}\end{aligned}}}
The above expressions contain four terms to calculate:
S e = P ( j ω ) e v e n = ∑ k = 0 N / 2 P 2 k ( j ω ) 2 k = ∑ k = 0 N / 2 P 2 k ( − 1 ) k ( ω ) 2 k S o = P ( j ω ) o d d = − j ∑ k = 1 ( N + 1 ) / 2 P 2 k − 1 ( j ω ) 2 k − 1 = ∑ k = 1 ( N + 1 ) / 2 P 2 k − 1 ( − 1 ) k − 1 ( ω ) 2 k − 1 D e = d ( P ( j ω ) e v e n ) d x = − j ∑ k = 1 N / 2 2 k P 2 k ( j ω ) 2 k − 1 = ∑ k = 1 N / 2 2 k P 2 k ( − 1 ) k − 1 ( ω ) 2 k − 1 D o = d ( P ( j ω ) o d d ) d x = ∑ k = 1 ( N + 1 ) / 2 ( 2 k − 1 ) P 2 k − 1 ( j ω ) 2 k − 2 = ∑ k = 1 ( N + 1 ) / 2 ( 2 k − 1 ) P 2 k − 1 ( − 1 ) k − 1 ( ω ) 2 k − 2 d f ( x ) d x = S e D o − S o D e S e 2 {\displaystyle {\begin{array}{lcl}Se=P(j\omega )_{even}&=&\sum _{k=0}^{N/2}P_{2k}(j\omega )^{2k}&=&\sum _{k=0}^{N/2}P_{2k}(-1)^{k}(\omega )^{2k}\\So=P(j\omega )_{odd}&=&-j\sum _{k=1}^{(N+1)/2}P_{2k-1}(j\omega )^{2k-1}&=&\sum _{k=1}^{(N+1)/2}P_{2k-1}(-1)^{k-1}(\omega )^{2k-1}\\De={\frac {d(P(j\omega )_{even})}{dx}}&=&-j\sum _{k=1}^{N/2}2kP_{2k}(j\omega )^{2k-1}&=&\sum _{k=1}^{N/2}2kP_{2k}(-1)^{k-1}(\omega )^{2k-1}\\Do={\frac {d(P(j\omega )_{odd})}{dx}}&=&\sum _{k=1}^{(N+1)/2}{(2k-1)}P_{2k-1}(j\omega )^{2k-2}&=&\sum _{k=1}^{(N+1)/2}{(2k-1)}P_{2k-1}(-1)^{k-1}(\omega )^{2k-2}\\\\{\frac {df(x)}{dx}}&=&{\frac {SeDo-SoDe}{Se^{2}}}\end{array}}}
The equations above may be used to determine the group delay of polynomial P ( S ) {\displaystyle P(S)} in closed form, shown below after the equations have been reduced to a simplified form.
Group Delay = g d ( P ( j ω ) ) = − d ϕ ( ω ) d ω = − ( S o ∗ D e + S e ∗ D o ) ( S e 2 + S o 2 ) sec {\displaystyle {\text{Group Delay}}=gd(P(j\omega ))=-{\frac {d\phi (\omega )}{d\omega }}=-{\frac {(So*De+Se*Do)}{(Se^{2}+So^{2})}}{\text{ sec}}}
A polynomial ratio of the form P 2 ( S ) = P n u m ( S ) / P d e n ( S ) {\displaystyle P2(S)=P_{num}(S)/P_{den}(S)} , such as that typically found in the definition of filter designs , may have the group delay determined by taking advantage of the phase relation, ϕ ( P 1 / P 2 ) = ϕ ( P 1 ) − ϕ ( P 2 ) {\displaystyle \phi (P1/P2)=\phi (P1)-\phi (P2)} .
Group Delay = g d ( P 2 ) = g d ( P 2 n u m ) − g d ( P 2 d e n ) s e c {\displaystyle {\text{Group Delay}}=gd(P2)=gd(P2_{num})-gd(P2_{den})sec}
A four pole Legendre filter transfer function used in the Legendre filter example is shown below.
T 4 ( j ω ) = 1 2.4494897 ( j ω ) 4 + 3.8282201 ( j ω ) 3 + 4.6244874 ( j ω ) 2 + 3.0412127 ( j ω ) + 1 {\displaystyle T_{4}(j\omega )={\frac {1}{2.4494897(j\omega )^{4}+3.8282201(j\omega )^{3}+4.6244874(j\omega )^{2}+3.0412127(j\omega )+1}}}
The numerator group delay by inspection is zero, so only the denominator group delay need be determined.
P e d e n = 2.4494897 ω 4 − 4.6244874 ω 2 + 1 P o d e n = − 3.8282201 ω 3 + 3.0412127 ω D e d e n = 4 ( 2.4494897 ) ω 3 − 2 ( 4.6244874 ) ω D o d e n = 3 ( − 3.8282201 ) ω 2 + 3.0412127 {\displaystyle {\begin{aligned}&Pe_{den}=2.4494897\omega ^{4}-4.6244874\omega ^{2}+1\\&Po_{den}=-3.8282201\omega ^{3}+3.0412127\omega \\&De_{den}=4(2.4494897)\omega ^{3}-2(4.6244874)\omega \\&Do_{den}=3(-3.8282201)\omega ^{2}+3.0412127\end{aligned}}}
Evaluating at ω {\displaystyle \omega } = 1 rad/sec:
P e d e n = − 1.1749977 P o d e n = − 0.7870074 D e d e n = − 0.548984 D o d e n = − 8.4434476 {\displaystyle {\begin{aligned}&Pe_{den}=-1.1749977\\&Po_{den}=-0.7870074\\&De_{den}=-0.548984\\&Do_{den}=-8.4434476\end{aligned}}}
Group Delay = g d ( T 4 ( j ω ) ) = − d ϕ ( ω ) d ω = [ 0 − − ( ( − 0.7870074 ∗ − 0.548984 ) + ( − 1.1749977 ∗ − 8.4434476 ) ) ( ( − 1.1749977 ) 2 + ( − 0.7870074 ) 2 ) ] = 5.1765430 sec at ω = 1 rad/sec {\displaystyle {\begin{aligned}&{\text{Group Delay}}=gd(T_{4}(j\omega ))=-{\frac {d\phi (\omega )}{d\omega }}\\&={\bigg [}0--{\frac {((-0.7870074*-0.548984)+(-1.1749977*-8.4434476))}{((-1.1749977)^{2}+(-0.7870074)^{2})}}{\bigg ]}\\&=5.1765430{\text{ sec}}\\&{\text{at }}\omega =1{\text{ rad/sec}}\end{aligned}}}
The group delay calculation procedure and results may be confirmed to be correct by comparing them to the results derived from the digital derivative of the phase angle, ϕ ( ω ) {\displaystyle \phi (\omega )} , using a small delta Δ ω {\displaystyle \Delta \omega } of +/-1.e-04 rad/sec.
Group Delay = g d ( T 4 ( j ω ) ) = − d ϕ ( ω ) d ω = − ( ϕ ( 1 + 1 e − 04 ) − ϕ ( 1. − 1 e 04 ) ) / 2 e − 04 = 5.1765432 sec at ω = 1 rad/sec {\displaystyle {\begin{aligned}&{\text{Group Delay}}=gd(T_{4}(j\omega ))=-{\frac {d\phi (\omega )}{d\omega }}\\&=-(\phi (1+1e-04)-\phi (1.-1e04))/2e-04\\&=5.1765432{\text{ sec}}\\&{\text{at }}\omega =1{\text{ rad/sec}}\end{aligned}}}
Since the group delay calculated by the digital derivative using a small delta is within 7 digits of accuracy when compared to the precise analytical calculation, the group delay calculation procedure and results are confirmed to be correct.
Deviation from Linear Phase , ϕ D L P ( ω ) {\displaystyle \phi _{DLP}(\omega )} , sometimes referred to as just, "phase deviation", is the difference between the phase response, ϕ ( ω ) {\displaystyle \phi (\omega )} , and the linear portion of the phase response ϕ L ( ω ) {\displaystyle \phi _{L}(\omega )} , [ 26 ] and is a useful measurement to determine the linearity of ϕ ( ω ) {\displaystyle \phi (\omega )} .
A convenient means to measure ϕ D L P ( ω ) {\displaystyle \phi _{DLP}(\omega )} is to take the simple linear regression of ϕ ( ω ) {\displaystyle \phi (\omega )} sampled over a frequency range of interest, and subtract it from the actual ϕ ( ω ) {\displaystyle \phi (\omega )} . The ϕ D L P ( ω ) {\displaystyle \phi _{DLP}(\omega )} of an ideal linear phase response would be expected to have a value of 0 across the frequency range of interest (such as the pass band of a filter), while the ϕ D L P ( ω ) {\displaystyle \phi _{DLP}(\omega )} of a real-world approximately linear phase response may deviate from 0 by a small finite amount across the frequency range of interest.
An advantage of measuring or calculating ϕ D L P ( ω ) {\displaystyle \phi _{DLP}(\omega )} over measuring or calculating group delay, g d ( ω ) {\displaystyle gd(\omega )} , is ϕ D L P ( ω ) {\displaystyle \phi _{DLP}(\omega )} always converges to 0 as the phase becomes linear, whereas g d ( ω ) {\displaystyle gd(\omega )} converges on a finite quantity that may not be known ahead of time. Given this, a linear phase optimizing function may more easily be executed with a ϕ D L P ( ω ) = 0 {\displaystyle \phi _{DLP}(\omega ){=}0} goal than with a g d ( ω ) = c o n s t a n t {\displaystyle gd(\omega ){=}constant} goal when the value for c o n s t a n t {\displaystyle constant} is not necessarily already known.
This article incorporates public domain material from Federal Standard 1037C . General Services Administration . Archived from the original on 2022-01-22. | https://en.wikipedia.org/wiki/Group_delay_and_phase_delay |
In ethology and evolutionary biology , group living is defined as individuals of the same species (conspecifics), maintaining spatial proximity with one another over time with mechanisms of social attraction. [ 1 ] Solitary life in animals is considered to be the ancestral state of living; and group living has thus evolved independently in many species of animals. [ 2 ] Therefore, species that form groups through social interaction will result in a group of individuals that gain an evolutionary advantage, such as increased protection against predators, access to potential mates, increased foraging efficiency and the access to social information.
Important aspects of group living include the frequency and type of social interactions ( egoistic , cooperative , altruistic , revengeful ) between individuals of a group ( social life ), the group size , and the organization of group members in the group . [ citation needed ]
Terminology of animal groups also varies among different taxonomic groups. Groups of sheep are termed herds , whilst groups of birds are referred to as colonies , or flocks .
Most studies on group living focus strictly on groups comprising a single species. However, many mixed-species groups commonly occur in nature. Examples of mixed-species groups include wildebeests forming groups with zebras , [ 3 ] and different species of birds that form large foraging flocks. [ 4 ]
Group living may sometimes be confused with collective animal behavior . Collective animal behavior is the study of how the interactions between individuals of a group give rise to group level patterns and how these patterns have evolved. [ 5 ] Examples include the marching of locusts and flocks of migrating birds. Group living however focuses on the long-term social interactions between individuals of a group and how animals have evolved from solitary living.
It is extremely difficult to distinguish between solitary living and group living. Distinctions between the two are relatively artificial. [ 6 ] This is because many species of animals who spend a majority of their life alone, at some point in their life, will join a group or engage in social behavior . [ 7 ] Some examples of this happens during mating , parental care of their offspring, or even aggregations of conspecifics to an area to exploit resources of food or shelter. [ 2 ] Therefore, multiple definitions of group living have been proposed. Differences in group living definitions vary dependent on the frequency and type of social interactions that members of a group display and the level of coordination and cohesion of group members. [ 6 ] For example, Wilson (2000) [ 8 ] defines a group as “any set of organisms, belonging to the same species, that remain together for a period of time while interacting with one another to a distinctly greater degree than with other conspecific organisms." This definition cannot be applied to situations such as moths drawn to a lamp, or when animals aggregate around a watering hole, [ 1 ] as they are not exampling of a social aggregation. Most definitions however agree that a fundamental characteristic of group living is that individuals need to show spatial proximity over time to be considered a group. [ 2 ] Therefore, the working definition of group living is where two or more individuals display a degree of spatial proximity over time, emphasizing the importance of mechanisms of social attraction to maintain these groups. [ 9 ] [ 1 ]
There have been multiple different hypotheses proposed to explain how group living evolved in animals. Research shows that grouping habits may differ between individuals, and this tendency to group can be inherited. Research also shows that grouping tendency depends heavily on the interaction of many genes, as well as experiences gained by an individual and the environmental conditions surrounding the individual. [ 1 ] Other studies argue that the main driving force of the evolution of social grouping is phylogenetic inertia alongside ecological pressure. [ 8 ] However, it is still unclear how exactly animals have evolved from the ancestral state of solitary life.
A key advantage to group living is the ability for individuals in a group to access information gained by other group members. [ 1 ] This ability to share information can benefit many aspects of a group’s success, such as increased foraging efficiency and increased defenses against predators .
An advantage of information access from group living is increased foraging efficiency. When individuals form a group, they can more effectively locate high quality resources in their environment. Foraging efficiency can be increased by the sheer area of space individuals occupy as well as a greater number of individuals searching for food. [ 10 ] [ 11 ] Once a high-quality resource is found, the individuals may produce signals or cues that guides other members of the group to the location of the resource. [ 1 ] The cues and signals produced thus helps individuals of a group discriminate between low- and high-quality resources. An example of this information transfer to benefit foraging efficiency can be seen in honey-bee ( Apis mellifera ) colonies, in which waggle dances performed by honey-bees share information on where these dancing bees foraged nectar. The waggle dance thus guides other bees to the location of highly productive flowers. [ 12 ] [ 13 ] In some species, for instance the forest tent caterpillar ( Malacosoma disstria ), foraging behaviors change depending on food source. On less favorable food sources, caterpillar groups tend to splinter, thereby potentially increasing the risk for predation, but increasing the potential of finding a more favorable food source. [ 14 ]
Another advantage of living in a group is seen in many prey species in their ability to increase defenses against predatory animals.
A way that a group may increase its defenses against predators is through the ‘many-eyes effect’. This effect states that larger groups of animals are better at detecting predators compared to smaller groups. [ 15 ] This allows the individuals within a group to more effectively identify predators , allowing these individuals to flee or adopt postures to alert the predators that their presence is known. An example can be seen in a study conducted by Siegfried and Underhill (1975) [ 16 ] on laughing doves , in which large groups react to a mock-predator much more rapidly than smaller groups.
Another way in which a group may have decreased risk of predation is through the dilution effect. [ 5 ] The dilution effect shows the idea that an individual in a large group will have a reduced risk of predation compared to an individual in a small group or a solitary individual. Hence the risk is ‘diluted’ among the other members in a group. It is important to note however; this effect only occurs where predators are unable to capture all individuals in a group. For example, a flock of birds preying on a large group of caterpillars will not have any dilution effect, as these birds can rapidly consume all caterpillars at once. [ 1 ] All individuals in a large group however, may not benefit from the dilution effect, and thus the selfish herd theory was developed. The selfish herd theory states that individuals in the periphery of a group is more likely to be preyed upon than those in the center of the group [ 17 ]
It is hypothesized that reproductive success of a female is determined by the number of eggs she can produce, while reproductive success of a male is determined by the number of females he mates with. Furthermore, according to Bateman's principle , [ 18 ] it is expected that females of a population will select mates that result in the best quality offspring, while males compete among each other to mate with a female.
Group living provides the presence of social information within the group, allowing both male and female members to find and select potential mating partners. Alongside this, living in a group allows for higher reproductive success as individuals have access to a greater number of potential mates, and the possibility to choose between them. [ 1 ] Therefore, individuals living in groups have a higher chance of finding a mate and successfully reproducing, on the basis that larger groups present a greater number of accessible mates nearby.
Despite the many benefits of living in groups, individuals of the group may also incur costs when forming groups.
When individuals of the same species aggregate to form groups, there is an increased risk of diseases and parasites spreading throughout the group. Because individuals of a group live together in close proximity, when one individual is infected with a disease or parasite, they bring this disease or parasite into a habitat full of susceptible individuals. [ 19 ] Also, larger groups of animals will produce larger amounts of waste material, allowing for a favorable environment for pathogens, that may spread to individuals. Thus, transmissions of diseases and parasites are more likely to occur and more rapidly than if an individual lived alone. A great example was shown in colonies of cliff swallows by Brown and Brown (1986). [ 20 ] Cliff swallows are commonly parasitized by swallow bugs and this study showed that the number of swallow bugs per nest increased significantly with an increase of the number of cliff swallows per colony, which thus reduced the survivability of the nests’ offspring by up to 50%. Another study shows that bank swallows have an increased likelihood of flea infestations per burrow with the increase of colony size, which also increased mortality rates of offspring in infected burrows. [ 21 ]
A consequence that may arise from forming large groups is the increased intraspecific competition between group members. If resources in a group’s environment becomes limited, group members will then have to compete with one another for the available resources. [ 22 ] The increased competition then results in reduced nutritional intake in some individuals compared to others. An example of this can be seen in a study conducted on leaf monkeys . [ 23 ] This study showed that females in a larger group of leaf monkeys had a reduced energetic intake than females in groups of smaller sizes. The reduction in energy gain seen in females of the larger group also then negatively affected the development rates of any infant offspring. Therefore, despite the benefits of animals forming groups that increases foraging efficiency due to the presence of social information, [ 1 ] large groups of animals may also incur a cost of having to compete for the resources available in the environment.
Another cost to group living is the effect that a larger group size has on the reproductive success of individuals.
While forming groups may benefit the reproductive success of individuals as there are more potential mates, consequentially individuals may also have increased competition between one another to successfully find a mate and reproduce. [ 24 ] This means some individuals will have a reduction in their reproductive success as it now competes with other group members. An example can be seen in a study conducted on the Eurasian badger ( Meles meles ). [ 22 ] This study showed that females belonging to a large group of badgers had a higher failure rate of reproduction in comparison to badgers living in solitary. Therefore, some individuals may actually show reduced reproductive success while living in a group despite the increased presence of potential mates.
It is clear that animals that form groups need to maintain a group size around an optimal level. [ 25 ] Individual group members in group sizes much larger or smaller than the optimum may have increased stress levels. Individuals in groups much larger than their optimum group size may have increased stress levels due to competition for food resources or mates. In contrast, individuals in groups smaller than their optimum have increased stress levels arising from inadequate defense from predators. [ 26 ] An example of this can be seen in a study conducted on a species of ring-tailed lemurs ( Lemur catta ). [ 27 ] This study predicted that the optimum group size of ring-tailed lemurs is 10-20 individuals. The study then showed that groups within the optimum group size produced the lowest level of cortisol (an indicator of stress), while groups larger or smaller than the optimum group size had a significant increase in cortisol production. Therefore, group sizes that are not maintained within their optimum size may incur a cost of increased stress levels of individuals within those groups.
Another proposed cost of group living is the increased risk of inbreeding . [ 28 ] As members of the group in close proximity to one another over long periods of time, this increases the chances that offspring of the group may mate with related individuals. [ 29 ] Offspring resulting from inbreeding have an increased chance to be affected by recessive or deleterious traits, thus reducing its survivability and ability to reproduce. [ 30 ] The risk of inbreeding however, is only prevalent in smaller, isolated groups, as larger group sizes dilutes the chance of an individual mating with its relatives. [ 29 ] | https://en.wikipedia.org/wiki/Group_living |
In mathematics , a group scheme is a type of object from algebraic geometry equipped with a composition law. Group schemes arise naturally as symmetries of schemes , and they generalize algebraic groups , in the sense that all algebraic groups have group scheme structure, but group schemes are not necessarily connected , smooth , or defined over a field. This extra generality allows one to study richer infinitesimal structures, and this can help one to understand and answer questions of arithmetic significance. The category of group schemes is somewhat better behaved than that of group varieties , since all homomorphisms have kernels , and there is a well-behaved deformation theory . Group schemes that are not algebraic groups play a significant role in arithmetic geometry and algebraic topology , since they come up in contexts of Galois representations and moduli problems . The initial development of the theory of group schemes was due to Alexander Grothendieck , Michel Raynaud and Michel Demazure in the early 1960s.
A group scheme is a group object in a category of schemes that has fiber products and some final object S . That is, it is an S -scheme G equipped with one of the equivalent sets of data
A homomorphism of group schemes is a map of schemes that respects multiplication. This can be precisely phrased either by saying that a map f satisfies the equation f μ = μ( f × f ), or by saying that f is a natural transformation of functors from schemes to groups (rather than just sets).
A left action of a group scheme G on a scheme X is a morphism G × S X → X that induces a left action of the group G ( T ) on the set X ( T ) for any S -scheme T . Right actions are defined similarly. Any group scheme admits natural left and right actions on its underlying scheme by multiplication and conjugation . Conjugation is an action by automorphisms, i.e., it commutes with the group structure, and this induces linear actions on naturally derived objects, such as its Lie algebra , and the algebra of left-invariant differential operators.
An S -group scheme G is commutative if the group G ( T ) is an abelian group for all S -schemes T . There are several other equivalent conditions, such as conjugation inducing a trivial action, or inversion map ι being a group scheme automorphism.
Suppose that G is a group scheme of finite type over a field k . Let G 0 be the connected component of the identity, i.e., the maximal connected subgroup scheme. Then G is an extension of a finite étale group scheme by G 0 . G has a unique maximal reduced subscheme G red , and if k is perfect, then G red is a smooth group variety that is a subgroup scheme of G . The quotient scheme is the spectrum of a local ring of finite rank.
Any affine group scheme is the spectrum of a commutative Hopf algebra (over a base S , this is given by the relative spectrum of an O S -algebra). The multiplication, unit, and inverse maps of the group scheme are given by the comultiplication, counit, and antipode structures in the Hopf algebra. The unit and multiplication structures in the Hopf algebra are intrinsic to the underlying scheme. For an arbitrary group scheme G , the ring of global sections also has a commutative Hopf algebra structure, and by taking its spectrum, one obtains the maximal affine quotient group. Affine group varieties are known as linear algebraic groups, since they can be embedded as subgroups of general linear groups.
Complete connected group schemes are in some sense opposite to affine group schemes, since the completeness implies all global sections are exactly those pulled back from the base, and in particular, they have no nontrivial maps to affine schemes. Any complete group variety (variety here meaning reduced and geometrically irreducible separated scheme of finite type over a field) is automatically commutative, by an argument involving the action of conjugation on jet spaces of the identity. Complete group varieties are called abelian varieties . This generalizes to the notion of abelian scheme; a group scheme G over a base S is abelian if the structural morphism from G to S is proper and smooth with geometrically connected fibers. They are automatically projective, and they have many applications, e.g., in geometric class field theory and throughout algebraic geometry. A complete group scheme over a field need not be commutative, however; for example, any finite group scheme is complete.
A group scheme G over a noetherian scheme S is finite and flat if and only if O G is a locally free O S -module of finite rank. The rank is a locally constant function on S , and is called the order of G . The order of a constant group scheme is equal to the order of the corresponding group, and in general, order behaves well with respect to base change and finite flat restriction of scalars .
Among the finite flat group schemes, the constants (cf. example above) form a special class, and over an algebraically closed field of characteristic zero, the category of finite groups is equivalent to the category of constant finite group schemes. Over bases with positive characteristic or more arithmetic structure, additional isomorphism types exist. For example, if 2 is invertible over the base, all group schemes of order 2 are constant, but over the 2-adic integers, μ 2 is non-constant, because the special fiber isn't smooth. There exist sequences of highly ramified 2-adic rings over which the number of isomorphism types of group schemes of order 2 grows arbitrarily large. More detailed analysis of commutative finite flat group schemes over p -adic rings can be found in Raynaud's work on prolongations.
Commutative finite flat group schemes often occur in nature as subgroup schemes of abelian and semi-abelian varieties, and in positive or mixed characteristic, they can capture a lot of information about the ambient variety. For example, the p -torsion of an elliptic curve in characteristic zero is locally isomorphic to the constant elementary abelian group scheme of order p 2 , but over F p , it is a finite flat group scheme of order p 2 that has either p connected components (if the curve is ordinary) or one connected component (if the curve is supersingular ). If we consider a family of elliptic curves, the p -torsion forms a finite flat group scheme over the parametrizing space, and the supersingular locus is where the fibers are connected. This merging of connected components can be studied in fine detail by passing from a modular scheme to a rigid analytic space , where supersingular points are replaced by discs of positive radius.
Cartier duality is a scheme-theoretic analogue of Pontryagin duality taking finite commutative group schemes to finite commutative group schemes.
Finite flat commutative group schemes over a perfect field k of positive characteristic p can be studied by transferring their geometric structure to a (semi-)linear-algebraic setting. The basic object is the Dieudonné ring D = W ( k ){ F , V }/( FV − p ), which is a quotient of the ring of noncommutative polynomials, with coefficients in Witt vectors of k . F and V are the Frobenius and Verschiebung operators, and they may act nontrivially on the Witt vectors. Dieudonne and Cartier constructed an antiequivalence of categories between finite commutative group schemes over k of order a power of "p" and modules over D with finite W ( k )-length. The Dieudonné module functor in one direction is given by homomorphisms into the abelian sheaf CW of Witt co-vectors. This sheaf is more or less dual to the sheaf of Witt vectors (which is in fact representable by a group scheme), since it is constructed by taking a direct limit of finite length Witt vectors under successive Verschiebung maps V : W n → W n+1 , and then completing. Many properties of commutative group schemes can be seen by examining the corresponding Dieudonné modules, e.g., connected p -group schemes correspond to D -modules for which F is nilpotent, and étale group schemes correspond to modules for which F is an isomorphism.
Dieudonné theory exists in a somewhat more general setting than finite flat groups over a field. Oda's 1967 thesis gave a connection between Dieudonné modules and the first de Rham cohomology of abelian varieties, and at about the same time, Grothendieck suggested that there should be a crystalline version of the theory that could be used to analyze p -divisible groups. Galois actions on the group schemes transfer through the equivalences of categories, and the associated deformation theory of Galois representations was used in Wiles 's work on the Shimura–Taniyama conjecture . | https://en.wikipedia.org/wiki/Group_scheme |
The group velocity of a wave is the velocity with which the overall envelope shape of the wave's amplitudes —known as the modulation or envelope of the wave—propagates through space.
For example, if a stone is thrown into the middle of a very still pond, a circular pattern of waves with a quiescent center appears in the water, also known as a capillary wave . The expanding ring of waves is the wave group or wave packet , within which one can discern individual waves that travel faster than the group as a whole. The amplitudes of the individual waves grow as they emerge from the trailing edge of the group and diminish as they approach the leading edge of the group.
The idea of a group velocity distinct from a wave's phase velocity was first proposed by W.R. Hamilton in 1839, and the first full treatment was by Rayleigh in his "Theory of Sound" in 1877. [ 2 ]
The group velocity v g is defined by the equation: [ 3 ] [ 4 ] [ 5 ] [ 6 ]
where ω is the wave's angular frequency (usually expressed in radians per second ), and k is the angular wavenumber (usually expressed in radians per meter). The phase velocity is: v p = ω / k .
The function ω ( k ) , which gives ω as a function of k , is known as the dispersion relation .
One derivation of the formula for group velocity is as follows. [ 8 ] [ 9 ]
Consider a wave packet as a function of position x and time t : α ( x , t ) .
Let A ( k ) be its Fourier transform at time t = 0 ,
By the superposition principle , the wavepacket at any time t is
where ω is implicitly a function of k .
Assume that the wave packet α is almost monochromatic , so that A ( k ) is sharply peaked around a central wavenumber k 0 .
Then, linearization gives
where
(see next section for discussion of this step). Then, after some algebra,
There are two factors in this expression. The first factor, e i ( k 0 x − ω 0 t ) {\displaystyle e^{i\left(k_{0}x-\omega _{0}t\right)}} , describes a perfect monochromatic wave with wavevector k 0 , with peaks and troughs moving at the phase velocity ω 0 / k 0 {\displaystyle \omega _{0}/k_{0}} within the envelope of the wavepacket.
The other factor,
gives the envelope of the wavepacket. This envelope function depends on position and time only through the combination ( x − ω 0 ′ t ) {\displaystyle (x-\omega '_{0}t)} .
Therefore, the envelope of the wavepacket travels at velocity
which explains the group velocity formula.
For light, the refractive index n , vacuum wavelength λ 0 , and wavelength in the medium λ , are related by
with v p = ω / k the phase velocity .
The group velocity, therefore, can be calculated by any of the following formulas,
Part of the previous derivation is the Taylor series approximation that:
If the wavepacket has a relatively large frequency spread, or if the dispersion ω(k) has sharp variations (such as due to a resonance ), or if the packet travels over very long distances, this assumption is not valid, and higher-order terms in the Taylor expansion become important. [ 10 ] : 324–325
As a result, the envelope of the wave packet not only moves, but also distorts, in a manner that can be described by the material's group velocity dispersion . Loosely speaking, different frequency-components of the wavepacket travel at different speeds, with the faster components moving towards the front of the wavepacket and the slower moving towards the back. Eventually, the wave packet gets stretched out. This is an important effect in the propagation of signals through optical fibers and in the design of high-power, short-pulse lasers.
The group velocity of a collection of waves is defined as
When multiple sinusoidal waves are propagating together, the resultant superposition of the waves can result in an "envelope" wave as well as a "carrier" wave that lies inside the envelope. This commonly appears in wireless communication when modulation (a change in amplitude and/or phase) is employed to send data. To gain some intuition for this definition, we consider a superposition of (cosine) waves f(x, t) with their respective angular frequencies and wavevectors.
So, we have a product of two waves: an envelope wave formed by f 1 and a carrier wave formed by f 2 . We call the velocity of the envelope wave the group velocity. We see that the phase velocity of f 1 is
In the context of electromagnetics and optics, the frequency is some function ω ( k ) of the wave number, so in general, the phase velocity and the group velocity depend on specific medium and frequency. The ratio between the speed of light c and the phase velocity v p is known as the refractive index , n = c / v p = ck / ω .
In this way, we can obtain another form for group velocity for electromagnetics. Writing n = n (ω) , a quick way to derive this form is to observe
We can then rearrange the above to obtain
For waves traveling through three dimensions, such as light waves, sound waves, and matter waves, the formulas for phase and group velocity are generalized in a straightforward way: [ 11 ]
where ∇ → k ω {\displaystyle {\vec {\nabla }}_{\mathbf {k} }\,\omega } means the gradient of the angular frequency ω as a function of the wave vector k {\displaystyle \mathbf {k} } , and k ^ {\displaystyle {\hat {\mathbf {k} }}} is the unit vector in direction k .
If the waves are propagating through an anisotropic (i.e., not rotationally symmetric) medium, for example a crystal , then the phase velocity vector and group velocity vector may point in different directions.
The group velocity is often thought of as the velocity at which energy or information is conveyed along a wave. In most cases this is accurate, and the group velocity can be thought of as the signal velocity of the waveform . However, if the wave is travelling through an absorptive or gainful medium, this does not always hold. In these cases the group velocity may not be a well-defined quantity, or may not be a meaningful quantity.
In his text "Wave Propagation in Periodic Structures", [ 12 ] Brillouin argued that in a lossy medium the group velocity ceases to have a clear physical meaning. An example concerning the transmission of electromagnetic waves through an atomic gas is given by Loudon. [ 13 ] Another example is mechanical waves in the solar photosphere : The waves are damped (by radiative heat flow from the peaks to the troughs), and related to that, the energy velocity is often substantially lower than the waves' group velocity. [ 14 ]
Despite this ambiguity, a common way to extend the concept of group velocity to complex media is to consider spatially damped plane wave solutions inside the medium, which are characterized by a complex-valued wavevector. Then, the imaginary part of the wavevector is arbitrarily discarded and the usual formula for group velocity is applied to the real part of wavevector, i.e.,
Or, equivalently, in terms of the real part of complex refractive index , n = n + iκ , one has [ 15 ]
It can be shown that this generalization of group velocity continues to be related to the apparent speed of the peak of a wavepacket. [ 16 ] The above definition is not universal, however: alternatively one may consider the time damping of standing waves (real k , complex ω ), or, allow group velocity to be a complex-valued quantity. [ 17 ] [ 18 ] Different considerations yield distinct velocities, yet all definitions agree for the case of a lossless, gainless medium.
The above generalization of group velocity for complex media can behave strangely, and the example of anomalous dispersion serves as a good illustration.
At the edges of a region of anomalous dispersion, v g {\displaystyle v_{\rm {g}}} becomes infinite (surpassing even the speed of light in vacuum), and v g {\displaystyle v_{\rm {g}}} may easily become negative
(its sign opposes Re k ) inside the band of anomalous dispersion. [ 19 ] [ 20 ] [ 21 ]
Since the 1980s, various experiments have verified that it is possible for the group velocity (as defined above) of laser light pulses sent through lossy materials, or gainful materials, to significantly exceed the speed of light in vacuum c . The peaks of wavepackets were also seen to move faster than c .
In all these cases, however, there is no possibility that signals could be carried faster than the speed of light in vacuum , since the high value of v g does not help to speed up the true motion of the sharp wavefront that would occur at the start of any real signal. Essentially the seemingly superluminal transmission is an artifact of the narrow band approximation used above to define group velocity and happens because of resonance phenomena in the intervening medium. In a wide band analysis it is seen that the apparently paradoxical speed of propagation of the signal envelope is actually the result of local interference of a wider band of frequencies over many cycles, all of which propagate perfectly causally and at phase velocity. The result is akin to the fact that shadows can travel faster than light, even if the light causing them always propagates at light speed; since the phenomenon being measured is only loosely connected with causality, it does not necessarily respect the rules of causal propagation, even if it under normal circumstances does so and leads to a common intuition. [ 15 ] [ 19 ] [ 20 ] [ 22 ] [ 23 ] | https://en.wikipedia.org/wiki/Group_velocity |
In mathematics , the concept of groupoid algebra generalizes the notion of group algebra . [ 1 ]
Given a groupoid ( G , ⋅ ) {\displaystyle (G,\cdot )} (in the sense of a category with all morphisms invertible) and a field K {\displaystyle K} , it is possible to define the groupoid algebra K G {\displaystyle KG} as the algebra over K {\displaystyle K} formed by the vector space having the elements of (the morphisms of) G {\displaystyle G} as generators and having the multiplication of these elements defined by g ∗ h = g ⋅ h {\displaystyle g*h=g\cdot h} , whenever this product is defined, and g ∗ h = 0 {\displaystyle g*h=0} otherwise. The product is then extended by linearity . [ 2 ]
Some examples of groupoid algebras are the following: [ 3 ] | https://en.wikipedia.org/wiki/Groupoid_algebra |
Groups, Geometry, and Dynamics is a quarterly peer-reviewed mathematics journal published quarterly by the European Mathematical Society . It was established in 2007 and covers all aspects of groups , group actions , geometry and dynamical systems . The journal is indexed by Mathematical Reviews and Zentralblatt MATH . Its 2009 MCQ was 0.65, and its 2012 impact factor is 0.867.
This article about a mathematics journal is a stub . You can help Wikipedia by expanding it .
See tips for writing articles about academic journals . Further suggestions might be found on the article's talk page . | https://en.wikipedia.org/wiki/Groups,_Geometry,_and_Dynamics |
The Grove cell was an early electric primary cell named after its inventor, Welsh physical scientist William Robert Grove , and consisted of a zinc anode in dilute sulfuric acid and a platinum cathode in concentrated nitric acid , the two separated by a porous ceramic pot.
The Grove cell voltage is about 1.9 volts and arises from the following reaction:
The Grove cell was the favored power source of the early American telegraph system in the period 1840 – 1860 because it offered a high current output and higher voltage than the earlier Daniell cell (at 1.9 volts and 1.1 volts, respectively).
By the time of the American Civil War , as telegraph traffic increased, the Grove cell's tendency to discharge poisonous nitrogen dioxide (NO 2 ) fumes proved increasingly hazardous to health, and as telegraphs became more complex, the need for constant voltage became critical. The Grove cell was limited in this respect, because as the cell discharged, voltage reduced. Eventually, Grove cells were replaced in use by Daniell cells .
This article about analytical chemistry is a stub . You can help Wikipedia by expanding it . | https://en.wikipedia.org/wiki/Grove_cell |
Growing teeth is a bioengineering technology with the ultimate goal to create new full molars in a person or an animal.
This article about biological engineering is a stub . You can help Wikipedia by expanding it . | https://en.wikipedia.org/wiki/Growing_teeth |
Growth hormone ( GH ) or somatotropin , also known as human growth hormone ( hGH or HGH ) in its human form, is a peptide hormone that stimulates growth, cell reproduction, and cell regeneration in humans and other animals. It is thus important in human development . GH also stimulates production of insulin-like growth factor 1 (IGF-1) and increases the concentration of glucose and free fatty acids . [ 1 ] [ 2 ] It is a type of mitogen which is specific only to the receptors on certain types of cells. GH is a 191- amino acid , single-chain polypeptide that is synthesized, stored and secreted by somatotropic cells within the lateral wings of the anterior pituitary gland.
A recombinant form of HGH called somatropin ( INN ) is used as a prescription drug to treat children's growth disorders and adult growth hormone deficiency . In the United States, it is only available legally from pharmacies by prescription from a licensed health care provider. In recent years in the United States, some health care providers are prescribing growth hormone in the elderly to increase vitality . While legal, the efficacy and safety of this use for HGH has not been tested in a clinical trial. Many of the functions of HGH remain unknown. [ 3 ]
In its role as an anabolic agent, HGH has been used by competitors in sports since at least 1982 and has been banned by the IOC and NCAA . Traditional urine analysis does not detect doping with HGH, so the ban was not enforced until the early 2000s, when blood tests that could distinguish between natural and artificial HGH were starting to be developed. Blood tests conducted by WADA at the 2004 Olympic Games in Athens, Greece , targeted primarily HGH. [ 3 ] Use of the drug for performance enhancement is not currently approved by the FDA .
GH has been studied for use in raising livestock more efficiently in industrial agriculture and several efforts have been made to obtain governmental approval to use GH in livestock production. These uses have been controversial. In the United States, the only FDA-approved use of GH for livestock is the use of a cow-specific form of GH called bovine somatotropin for increasing milk production in dairy cows. Retailers are permitted to label containers of milk as produced with or without bovine somatotropin.
The names somatotropin ( STH ) or somatotropic hormone refer to the growth hormone produced naturally in animals and extracted from carcasses. Hormone extracted from human cadavers is abbreviated hGH . The main growth hormone produced by recombinant DNA technology has the approved generic name ( INN ) somatropin and the brand name Humatrope [ 4 ] and is properly abbreviated rhGH in the scientific literature. Since its introduction in 1992, Humatrope has been a banned sports doping agent [ 5 ] and in this context is referred to as HGH.
The term growth hormone has been incorrectly applied to refer to anabolic sex hormones in the European beef hormone controversy , which initially restricts the use of estradiol , progesterone , testosterone , zeranol , melengestrol acetate and trenbolone acetate . [ 6 ]
Genes for human growth hormone, known as growth hormone 1 (somatotropin; pituitary growth hormone) and growth hormone 2 (placental growth hormone; growth hormone variant), are localized in the q22-24 region of chromosome 17 [ 7 ] [ 8 ] and are closely related to human chorionic somatomammotropin (also known as placental lactogen ) genes. GH, human chorionic somatomammotropin, and prolactin belong to a group of homologous hormones with growth-promoting and lactogenic activity.
The major isoform of the human growth hormone is a protein of 191 amino acids and a molecular weight of 22,124 daltons . The structure includes four helices necessary for functional interaction with the GH receptor. It appears that, in structure, GH is evolutionarily homologous to prolactin and chorionic somatomammotropin. Despite marked structural similarities between growth hormone from different species , only human and Old World monkey growth hormones have significant effects on the human growth hormone receptor . [ 9 ]
Several molecular isoforms of GH exist in the pituitary gland and are released to blood. In particular, a variant of approximately 20 kDa originated by an alternative splicing is present in a rather constant 1:9 ratio, [ 10 ] while recently an additional variant of ~ 23-24 kDa has also been reported in post-exercise states at higher proportions. [ 11 ] This variant has not been identified, but it has been suggested to coincide with a 22 kDa glycosylated variant of 23 kDa identified in the pituitary gland. [ 12 ] Furthermore, these variants circulate partially bound to a protein ( growth hormone-binding protein , GHBP), which is the truncated part of the growth hormone receptor , and an acid-labile subunit (ALS). [ citation needed ]
Secretion of growth hormone (GH) in the pituitary is regulated by the neurosecretory nuclei of the hypothalamus .
These cells release the peptides growth hormone-releasing hormone (GHRH or somatocrinin ) and growth hormone-inhibiting hormone (GHIH or somatostatin ) into the hypophyseal portal venous blood surrounding the pituitary.
GH release in the pituitary is primarily determined by the balance of these two peptides, which in turn is affected by many physiological stimulators (e.g., exercise, nutrition, sleep) and inhibitors (e.g., free fatty acids) of GH secretion. [ 13 ]
Somatotropic cells in the anterior pituitary gland then synthesize and secrete GH in a pulsatile manner, in response to these stimuli by the hypothalamus.
The largest and most predictable of these GH peaks occurs about an hour after onset of sleep with plasma levels of 13 to 72 ng/mL. [ 14 ] Maximal secretion of GH may occur within minutes of the onset of slow-wave (SW) sleep (stage III or IV). [ 15 ] Otherwise there is wide variation between days and individuals. Nearly fifty percent of GH secretion occurs during the third and fourth NREM sleep stages. [ 16 ] Surges of secretion during the day occur at 3- to 5-hour intervals. [ 3 ] The plasma concentration of GH during these peaks may range from 5 to even 45 ng/mL. [ 17 ] Between the peaks, basal GH levels are low, usually less than 5 ng/mL for most of the day and night. [ 14 ] Additional analysis of the pulsatile profile of GH described in all cases less than 1 ng/ml for basal levels while maximum peaks were situated around 10-20 ng/mL. [ 18 ] [ 19 ]
A number of factors are known to affect GH secretion, such as age, sex, diet, exercise, stress, and other hormones. [ 3 ] Young adolescents secrete GH at the rate of about 700 μg/day, while healthy adults secrete GH at the rate of about 400 μg/day. [ 20 ] Sleep deprivation generally suppresses GH release, particularly after early adulthood. [ 21 ]
Stimulators [ quantify ] of growth hormone (GH) secretion include:
Inhibitors [ quantify ] of GH secretion include:
In addition to control by endogenous and stimulus processes, a number of foreign compounds ( xenobiotics such as drugs and endocrine disruptors ) are known to influence GH secretion and function. [ 38 ]
Effects of growth hormone on the tissues of the body can generally be described as anabolic (building up). Like most other peptide hormones, GH acts by interacting with a specific receptor on the surface of cells. [ citation needed ]
Increased height during childhood is the most widely known effect of GH. Height appears to be stimulated by at least two mechanisms:
In addition to increasing height in children and adolescents, growth hormone has many other effects on the body:
GH has a short biological half-life of about 10 to 20 minutes. [ 45 ] [ 46 ]
The most common disease of GH excess is a pituitary tumor composed of somatotroph cells of the anterior pituitary. These somatotroph adenomas are benign and grow slowly, gradually producing more and more GH. For years, the principal clinical problems are those of GH excess. Eventually, the adenoma may become large enough to cause headaches, impair vision by pressure on the optic nerves, or cause deficiency of other pituitary hormones by displacement. [ citation needed ]
Prolonged GH excess thickens the bones of the jaw, fingers and toes, resulting in heaviness of the jaw and increased size of digits, referred to as acromegaly . Accompanying problems can include sweating, pressure on nerves (e.g., carpal tunnel syndrome ), muscle weakness, excess sex hormone-binding globulin (SHBG), insulin resistance or even a rare form of type 2 diabetes , and reduced sexual function. [ citation needed ]
GH-secreting tumors are typically recognized in the fifth decade of life. It is extremely rare for such a tumor to occur in childhood, but, when it does, the excessive GH can cause excessive growth, traditionally referred to as pituitary gigantism . [ citation needed ]
Surgical removal is the usual treatment for GH-producing tumors. In some circumstances, focused radiation or a GH antagonist such as pegvisomant may be employed to shrink the tumor or block function. Other drugs like octreotide (somatostatin agonist) and bromocriptine ( dopamine agonist ) can be used to block GH secretion because both somatostatin and dopamine negatively inhibit GHRH-mediated GH release from the anterior pituitary. [ 47 ]
The effects of growth hormone (GH) deficiency vary depending on the age at which they occur. Alterations in somatomedin can result in growth hormone deficiency with two known mechanisms; failure of tissues to respond to somatomedin , or failure of the liver to produce somatomedin. [ 48 ] Major manifestations of GH deficiency in children are growth failure , the development of a short stature , and delayed sexual maturity. In adults, somatomedin alteration contributes to increased osteoclast activity, resulting in weaker bones that are more prone to pathologic fracture and osteoporosis . [ 48 ] However, deficiency is rare in adults, with the most common cause being a pituitary adenoma . [ 49 ] Other adult causes include a continuation of a childhood problem, other structural lesions or trauma , and very rarely idiopathic GHD. [ 49 ]
Adults with GHD "tend to have a relative increase in fat mass and a relative decrease in muscle mass and, in many instances, decreased energy and quality of life". [ 49 ]
Diagnosis of GH deficiency involves a multiple-step diagnostic process, usually culminating in GH stimulation tests to see if the patient's pituitary gland will release a pulse of GH when provoked by various stimuli. [ citation needed ]
Several studies, primarily involving patients with GH deficiency , have suggested a crucial role of GH in both mental and emotional well-being and maintaining a high energy level. Adults with GH deficiency often have higher rates of depression than those without. [ 50 ] While GH replacement therapy has been proposed to treat depression as a result of GH deficiency, the long-term effects of such therapy are unknown. [ 50 ]
GH has also been studied in the context of cognitive function , including learning and memory. [ 51 ] GH in humans appears to improve cognitive function and may be useful in the treatment of patients with cognitive impairment that is a result of GH deficiency. [ 51 ]
GH is used as replacement therapy in adults with GH deficiency of either childhood-onset or adult-onset (usually as a result of an acquired pituitary tumor). In these patients, benefits have variably included reduced fat mass, increased lean mass, increased bone density, improved lipid profile, reduced cardiovascular risk factors, and improved psychosocial well-being. Long acting growth hormone (LAGH) analogues are now available for treating growth hormone deficiency both in children and adults. These are once weekly injections as compared to conventional growth hormone which has to be taken as daily injections. LAGH injection 4 times a month has been found to be as safe and effective as daily growth hormone injections. [ 52 ]
GH can be used to treat conditions that produce short stature but are not related to deficiencies in GH. However, results are not as dramatic when compared to short stature that is solely attributable to deficiency of GH. Examples of other causes of shortness often treated with GH are Turner syndrome , Growth failure secondary to chronic kidney disease in children, [ 53 ] Prader–Willi syndrome , intrauterine growth restriction , and severe idiopathic short stature . Higher ("pharmacologic") doses are required to produce significant acceleration of growth in these conditions, producing blood levels well above normal ("physiologic"). [ citation needed ]
One version of rHGH has also been FDA approved for maintaining muscle mass in wasting due to AIDS . [ 54 ]
Off-label prescription of HGH is controversial and may be illegal. [ 55 ]
Claims for GH as an anti-aging treatment date back to 1990 when the New England Journal of Medicine published a study wherein GH was used to treat 12 men over 60. [ 56 ] At the conclusion of the study, all the men showed statistically significant increases in lean body mass and bone mineral density, while the control group did not. The authors of the study noted that these improvements were the opposite of the changes that would normally occur over a 10- to 20-year aging period. Despite the fact the authors at no time claimed that GH had reversed the aging process itself, their results were misinterpreted as indicating that GH is an effective anti-aging agent. [ 57 ] [ 58 ] [ 59 ] This has led to organizations such as the controversial American Academy of Anti-Aging Medicine promoting the use of this hormone as an "anti-aging agent". [ 60 ]
A Stanford University School of Medicine meta-analysis of clinical studies on the subject published in early 2007 showed that the application of GH on healthy elderly patients increased muscle by about 2 kg and decreased body fat by the same amount. [ 57 ] However, these were the only positive effects from taking GH. No other critical factors were affected, such as bone density, cholesterol levels, lipid measurements, maximal oxygen consumption, or any other factor that would indicate increased fitness. [ 57 ] Researchers also did not discover any gain in muscle strength, which led them to believe that GH merely let the body store more water in the muscles rather than increase muscle growth. This would explain the increase in lean body mass. [ citation needed ]
GH has also been used experimentally to treat multiple sclerosis , to enhance weight loss in obesity , as well as in fibromyalgia , heart failure , Crohn's disease and ulcerative colitis , and burns. GH has also been used experimentally in patients with short bowel syndrome to lessen the requirement for intravenous total parenteral nutrition . [ citation needed ]
In 1990, the US Congress passed an omnibus crime bill, the Crime Control Act of 1990 , that amended the Federal Food, Drug, and Cosmetic Act , that classified anabolic steroids as controlled substances and added a new section that stated that a person who "knowingly distributes, or possesses with intent to distribute, human growth hormone for any use in humans other than the treatment of a disease or other recognized medical condition, where such use has been authorized by the Secretary of Health and Human Services" has committed a felony . [ 61 ] [ 62 ]
The Drug Enforcement Administration of the US Department of Justice considers off-label prescribing of HGH to be illegal, and to be a key path for illicit distribution of HGH. [ 55 ] This section has also been interpreted by some doctors, most notably [ 63 ] the authors of a commentary article published in the Journal of the American Medical Association in 2005, as meaning that prescribing HGH off-label may be considered illegal. [ 64 ] And some articles in the popular press, such as those criticizing the pharmaceutical industry for marketing drugs for off-label use (with concern of ethics violations) have made strong statements about whether doctors can prescribe HGH off-label: "Unlike other prescription drugs, HGH may be prescribed only for specific uses. U.S. sales are limited by law to treat a rare growth defect in children and a handful of uncommon conditions like short bowel syndrome or Prader-Willi syndrome, a congenital disease that causes reduced muscle tone and a lack of hormones in sex glands." [ 65 ] [ 66 ] At the same time, anti-aging clinics where doctors prescribe, administer, and sell HGH to people are big business. [ 65 ] [ 67 ] In a 2012 article in Vanity Fair , when asked how HGH prescriptions far exceed the number of adult patients estimated to have HGH-deficiency, Dragos Roman, who leads a team at the FDA that reviews drugs in endocrinology, said "The F.D.A. doesn't regulate off-label uses of H.G.H. Sometimes it's used appropriately. Sometimes it's not." [ 67 ]
Injection site reactions are common. More rarely, patients can experience joint swelling, joint pain, carpal tunnel syndrome , and an increased risk of diabetes . [ 57 ] In some cases, the patient can produce an immune response against GH. GH may also be a risk factor for Hodgkin's lymphoma . [ 68 ]
One survey of adults that had been treated with replacement cadaver GH (which has not been used anywhere in the world since 1985) during childhood showed a mildly increased incidence of colon cancer and prostate cancer, but linkage with the GH treatment was not established. [ 69 ]
The first description of the use of GH as a doping agent was Dan Duchaine's "Underground Steroid handbook" which emerged from California in 1982; it is not known where and when GH was first used this way. [ 70 ]
Athletes in many sports have used human growth hormone in order to attempt to enhance their athletic performance. Some recent studies have not been able to support claims that human growth hormone can improve the athletic performance of professional male athletes. [ 71 ] [ 72 ] [ 73 ] Many athletic societies ban the use of GH and will issue sanctions against athletes who are caught using it. However, because GH is a potent endogenous protein, it is very difficult to detect GH doping. In the United States, GH is legally available only by prescription from a medical doctor. [ citation needed ]
To capitalize on the idea that GH might be useful to combat aging, companies selling dietary supplements have websites selling products linked to GH in the advertising text, with medical-sounding names described as "HGH Releasers". Typical ingredients include amino acids, minerals, vitamins, and/or herbal extracts, the combination of which are described as causing the body to make more GH with corresponding beneficial effects. In the United States, because these products are marketed as dietary supplements, it is illegal for them to contain GH, which is a drug. Also, under United States law, products sold as dietary supplements cannot have claims that the supplement treats or prevents any disease or condition, and the advertising material must contain a statement that the health claims are not approved by the FDA. The FTC and the FDA do enforce the law when they become aware of violations. [ 74 ]
In the United States, it is legal to give a bovine GH to dairy cows to increase milk production, and is legal to use GH in raising cows for beef; see article on Bovine somatotropin , cattle feeding , dairy farming and the beef hormone controversy . [ citation needed ]
The use of GH in poultry farming is illegal in the United States. [ 75 ] [ 76 ] Similarly, no chicken meat for sale in Australia is administered hormones. [ 77 ]
Several companies have attempted to have a version of GH for use in pigs (porcine somatotropin) approved by the FDA but all applications have been withdrawn. [ 78 ]
Genentech pioneered the use of recombinant human growth hormone for human therapy, which was approved by the FDA in 1985. [ citation needed ]
Prior to its production by recombinant DNA technology, growth hormone used to treat deficiencies was extracted from the pituitary glands of cadavers . Attempts to create a wholly synthetic HGH failed. Limited supplies of HGH resulted in the restriction of HGH therapy to the treatment of idiopathic short stature. [ 79 ] Very limited clinical studies of growth hormone derived from an Old World monkey, the rhesus macaque , were conducted by John C. Beck and colleagues in Montreal, in the late 1950s. [ 80 ] The study published in 1957, which was conducted on "a 13-year-old male with well-documented hypopituitarism secondary to a crainiophyaryngioma," found that: "Human and monkey growth hormone resulted in a significant enhancement of nitrogen storage ... (and) there was a retention of potassium, phosphorus, calcium, and sodium. ... There was a gain in body weight during both periods. ... There was a significant increase in urinary excretion of aldosterone during both periods of administration of growth hormone. This was most marked with the human growth hormone. ... Impairment of the glucose tolerance curve was evident after 10 days of administration of the human growth hormone. No change in glucose tolerance was demonstrable on the fifth day of administration of monkey growth hormone." [ 80 ] The other study, published in 1958, was conducted on six people: the same subject as the Science paper; an 18-year-old male with statural and sexual retardation and a skeletal age of between 13 and 14 years; a 15-year-old female with well-documented hypopituitarism secondary to a craniopharyngioma; a 53-year-old female with carcinoma of the breast and widespread skeletal metastases; a 68-year-old female with advanced postmenopausal osteoporosis; and a healthy 24-year-old medical student without any clinical or laboratory evidence of systemic disease. [ 81 ]
In 1985, unusual cases of Creutzfeldt–Jakob disease were found in individuals that had received cadaver-derived HGH ten to fifteen years previously. Based on the assumption that infectious prions causing the disease were transferred along with the cadaver-derived HGH, cadaver-derived HGH was removed from the market. [ 20 ]
In 1985, biosynthetic human growth hormone replaced pituitary-derived human growth hormone for therapeutic use in the U.S. and elsewhere. [ citation needed ]
As of 2005, recombinant growth hormones available in the United States (and their manufacturers) included Nutropin ( Genentech ), Humatrope ( Lilly ), Genotropin ( Pfizer ), Norditropin ( Novo ), and Saizen ( Merck Serono ). In 2006, the U.S. Food and Drug Administration (FDA) approved a version of rHGH called Omnitrope (Sandoz). [ 82 ] A sustained-release form of growth hormone, Nutropin Depot (Genentech and Alkermes) was approved by the FDA in 1999, allowing for fewer injections (every 2 or 4 weeks instead of daily); however, the product was discontinued by Genentech/Alkermes in 2004 for financial reasons (Nutropin Depot required significantly more resources to produce than the rest of the Nutropin line [ 83 ] ).
Several growth hormone analogues featuring amino acid substitutions, deletions, and extensions have been developed, resulting in variants with distinct pharmacokinetic and pharmacodynamic profiles. | https://en.wikipedia.org/wiki/Growth_hormone |
Grozny oil field was one of the largest oil-industrial regions in the territory of the Russian Empire and then the USSR .
Oil seeps to the surface of the earth in the North Caucasus were noticed long before the beginning of the industrial development of oil fields on the slopes of the relatively low Tersky and Sunzhensky ridges. Since ancient times, local residents have collected oil here, which was used for household needs, medical and military purposes. They lubricated the axes of the supply with oil, treated people and animals, burned it in lamps, etc. In the 19th century, a whole group of deposits was found on the Grozny Range. Oil was extracted from wells no deeper than two arshins, from which it was simply scooped out with a bucket. Since 1811, oil wells have been farmed out. Such a farmer was originally the Mozdok Regiment, and since 1838, all oil sources have become the property of the Caucasian line troops. This army rented oil wells to merchant farmers, wealthy Cossacks, and other entrepreneurs. From 1833 to 1860, about 140 thousand pounds of oil were mastered in this way. [ 1 ]
Oil production, which at that time was carried out by artisanal methods, peaked in 1885 at 77,000 poods (1 pood = 16.3 kg). Scientists have undertaken a serious study of Grozny oil. Among them was the outstanding Russian chemist, D. I. Mendeleev . The industrial development of the Grozny oil region began. In 1892, 450,000 barrels of oil were produced. Grozny fisheries occupied second place in terms of productivity in the country. [ 2 ]
The Grozny region of oil and gas fields is part of the North Caucasian oil and gas region of Russia. Oil-bearing areas are concentrated in the areas of the Sunzha and Tersky ridges and the Black Mountains. The Grozny region, along with the Baku region, was one of the first oil-producing regions of the USSR. The beginning of industrial oil production was laid back in 1893, when the first fountain of oil gushed from a depth of more than 130 meters in the Starogroznensky district. Over the century-long history of the industry, 420 million tons of oil have been extracted from the bowels of the earth. [ 3 ] [ 4 ]
The largest deposits are: Novogroznenskoye (Oktyabrskoye) and Starogroznenskoye (with the oil-bearing areas of Tashkala and Salt Balka). Oil fields approach anticlinal folds, usually overturned and complicated by ruptures. The main oil deposits belong to the sandstones of the productive strata of the Karagan and Chokrak horizons of the Middle Miocene. Oil is paraffinic, with a high content of light fractions (in particular, gasoline). Geological exploration of the Grozny oil-bearing region began in the second half of the 19th century, and industrial production began in the 1890s. At the end of the 19th century, 7 English companies with a capital of 11 million rubles established themselves in the Grozny oil-industrial region, relegating the French Rothschilds , who had previously occupied a leading position among foreign firms, to the background. In 1913, the Novogroznenskoye field was discovered. The main owners of the Grozny oil fields were the world's largest oil companies and concerns: Nobel , Shell , Oil , Tweedy-Andreis, etc. By 1914, in the oil industry on the territory of the Grozny oil-bearing region, in percentage terms: English 36%, Russian 27%, French 18%, Belgian 10%, Dutch 9%. During the years of Soviet power, the largest fields were put into operation: in 1934, Malgobekneft; in 1937, Goragorskoye; in 1941, Oysungur; and in 1945, Tashkala. Checheno-Ingushetia was the second oil center of the USSR after Azerbaijan (the average oil production by the beginning of the Second World War was from 3 to 4 million tons annually, and its explored reserves amounted to 1.5 billion). [ 5 ] [ 6 ] | https://en.wikipedia.org/wiki/Grozny_oil_field |
Grubbing or clearing is the removal of trees , shrubs , stumps and rubbish from a site. This is often at the site where a transportation or utility corridor , a road or power line , an edifice or a garden is to be constructed. Grubbing is performed following clearance of trees to their stumps, preceding construction . [ 1 ]
In animal behaviour grubbing is a feeding technique , referring to digging and uprooting of roots and rhizomes of plants . It is employed by geese , especially greater and lesser snow geese and Canada geese , [ 2 ] as well as swine . [ 3 ]
This article about a civil engineering topic is a stub . You can help Wikipedia by expanding it . | https://en.wikipedia.org/wiki/Grubbing |
The Gruber Prize in Cosmology , established in 2000, is one of three prestigious international awards worth US$500,000 awarded by the Gruber Foundation , a non-profit organization based at Yale University in New Haven , Connecticut .
Since 2001, the Gruber Prize in Cosmology has been co-sponsored by the International Astronomical Union .
Recipients are selected by a panel from nominations that are received from around the world.
The Gruber Foundation Cosmology Prize honors a leading cosmologist, astronomer, astrophysicist or scientific philosopher for theoretical, analytical or conceptual discoveries leading to fundamental advances in the field. | https://en.wikipedia.org/wiki/Gruber_Prize_in_Cosmology |
Grundlagen der Mathematik (English: Foundations of Mathematics ) is a two-volume work by David Hilbert and Paul Bernays . Originally published in 1934 and 1939, it presents fundamental mathematical ideas and introduced second-order arithmetic .
This article about a mathematical publication is a stub . You can help Wikipedia by expanding it . | https://en.wikipedia.org/wiki/Grundlagen_der_Mathematik |
The Grundmann aldehyde synthesis is a chemical reaction that produces an aldehyde from an acyl halide . [ 1 ]
Because of the Rosenmund reduction and DIBAL-H accomplish similar transformations, this reaction sequence is not practiced much currently.
This chemical reaction article is a stub . You can help Wikipedia by expanding it . | https://en.wikipedia.org/wiki/Grundmann_aldehyde_synthesis |
In physical organic chemistry , the Grunwald–Winstein equation is a linear free energy relationship between relative rate constants and the ionizing power of various solvent systems, describing the effect of solvent as nucleophile on different substrates. The equation, which was developed by Ernest Grunwald and Saul Winstein in 1948, could be written [ 1 ] [ 2 ]
where the k x , sol and k x , 80% EtOH are the solvolysis rate constants for a certain compound in different solvent systems and in the reference solvent, 80% aqueous ethanol , respectively. The parameter m is a parameter measuring the sensitivity of the solvolysis rate with respect to Y , the measure of ionizing power of the solvent. [ 3 ]
The Hammett equation (Equation 1 ) provides the relationship between the substituent on the benzene ring and the ionizing rate constant of the reaction. Hammett used the ionization of benzoic acid as the standard reaction to define a set of substituent parameters σ X , and then to generate the ρ values, which represent ionizing abilities of different substrates. This relationship can be visualized through a Hammett plot.
However, if the solvent of the reaction is changed, but not the structure of the substrate, the rate constant may change too. Following this idea, Grunwald and Winstein plotted the relative rate constant vs. the change of solvent system, and formulated this behavior in the Grunwald–Winstein equation. Since the equation has the same pattern as the Hammett equation but captures the change of the solvent system, it is considered as an extension of the Hammett equation .
The substitution reaction of tert-Butyl chloride was chosen as reference reaction. The first step, ionizing step, is the rate determining step , SO stands for the nucleophilic solvent. The reference solvent is 80% Ethanol and 20% water by volume. Both of them can carry out the nucleophilic attack on the carbocation. [ 4 ] [ 5 ]
The S N 1 reaction is performed through a stable carbocation intermediate, the more nucleophilic solvent can stabilize the carbocation better, thus the rate constant of the reaction could be larger. Since there’s no sharp line between the S N 1 and S N 2 reaction , a reaction that goes through S N 1 mechanism more is preferred to achieve a better linear relationship, hence t -BuCl was chosen.
In equation 2 , k t -BuCl, 80% EtOH stands for the rate constant of t -BuCl reaction in 80% aqueous Ethanol, which is chosen as the reference. The variable k t -BuCl, sol stands for the rate constant of the same reaction in a different solvent system, such as ethanol-water, methanol-water, and acetic acid - formic acid . Thus, Y reflects the ionizing power of different nucleophile solvents.
The equation parameter m , called the sensitivity factor of solvolysis, describes the compound’s ability to form the carbocation intermediate in given solvent system. It is the slope of the plot of log(k sol /k 80%EtOH ) vs Y values. Since the reference reaction has little solvent nucleophilic assistance, the reactions with m equal to 1 or larger than 1 have almost full ionized intermediates. If the compounds are not so sensitive to the ionizing ability of solvent, then the m values are smaller than 1. That is: | https://en.wikipedia.org/wiki/Grunwald–Winstein_equation |
Grupa Azoty ATT Polymers GmbH ( abridged name: Grupa Azoty ATT POLYMERS ) – one of the enterprises producing polyamide 6 (PA6) in Western Europe . Acquired by Zakłady Azotowe w Tarnowie-Mościcach in 2009, currently a member of Grupa Azoty , in production of fertilizers , construction materials and caprolactam . Grupa Azoty ATT POLYMERS is seated in Guben , ( Germany ).
The company has its own laboratory and cooperates with a modern laboratory of the Grupa Azoty located in Tarnów . Its activity is focused on both development of existing products as well as on research on new products.
In 1958, the authorities of the German Democratic Republic decided to increase the volume of national production of synthetic fibres . The main element of that plan was construction of the chemical complex in Guben. The first facilities of Chemiefaserwerk Guben were constructed in 1960 in the place of former arms factories Rheinmetall-Borsig; whereas in 1964, the trial production was commenced. In 1968, polyamide 6 was produced for the first time, which was used to produce carpet yarn. In years 1979–1980 the production possibilities were expanded within the field of PA6.
Upon German reunification , the plants operated under the name of Chemiefaserwerk Guben GmbH. [ 1 ] As a result of multiple ownership changes in 1998, the company Plastomid Polymere GmbH was founded, which was a specialist in polyamide 6 production (20 thousand tons per year). Two years later, the Polymerisation reactors were modernised by their adjustment to two-stage production. Since then, in addition to new types of high-viscose granulate, it was possible to enter the foil market.
In 2005, the company changed its name to Unylon Polymers GmbH and increased the production powers to 47 thousand tons of PA6 per year.
As a result of the financial crisis, it was not possible to maintain the production profitability and bankruptcy proceedings were commenced in 2009. One of its elements was looking for an investor who would take over Unylon Polymers GmbH. In 2010, the plants were taken over in 100% by Zakłady Azotowe in Tarnów-Mościce S.A. [ 2 ] As a result of a merger of two largest Polish chemical firms, the company has operated as Grupa Azoty ATT Polymers GmbH since 2013.
100% of shares is held by the Grupa Azoty, being the controlling shareholder of the major Polish plants of Great Chemical Synthesis ( Kędzierzyn-Koźle , Police, Puławy i Tarnów ).
The company is run by two managing directors: Gabriele Kell [ 3 ] and Jacek Dychtoń, whereas the members of the Supervisory Board are Andrzej Skolmowski, Małgorzata Malec and Witold Szczypiński. [ 4 ]
Grupa Azoty ATT Polymers GmbH mainly specialises in polyamide 6 production. Its high viscosity allows for its application in, e.g. production of bi-oriented foil, which is used in work with foods. A product named Alphalon™ was distinguished at fairs PLASTPOL 2011 in Kielce . [ 5 ]
The production abilities in Grupa Azoty ATT POLYMERS are estimated as 45 thousand tons per year. In 2014, Grupa Azoty gained the possibility to open the next factory of polyamide 6 in Tarnów as a part of Special Economic Zone in Kraków with production rate of 80 thousand tons per year. The synergy of these installations allows the company to produce ca. 170 thousand tons of PA6 per year. [ 6 ] [ 7 ]
In 2011, at XV International Fairs of Plastics and Rubber Processing PLASTPOL in Kielce, the company was distinguished for alphalon E36LN used to produce bi-oriented foil that can have direct contact with foods. [ 8 ] | https://en.wikipedia.org/wiki/Grupa_Azoty_ATT_Polymers_GmbH |
The Grupo de Astronomía y Ciencias del Espacio (Group of Astronomy and Space Sciences, GACE-UV) is an astrophysics research group, part of the Image Processing Laboratory (IPL) in the University of Valencia .
It is located in the Parc Científic in Paterna , Valencia .
This article about an organization or institute connected with astronomy is a stub . You can help Wikipedia by expanding it .
This astrophysics -related article is a stub . You can help Wikipedia by expanding it . | https://en.wikipedia.org/wiki/Grupo_de_Astronomía_y_Ciencias_del_Espacio |
In mathematics , and more specifically in computer algebra , computational algebraic geometry , and computational commutative algebra , a Gröbner basis is a particular kind of generating set of an ideal in a polynomial ring K [ x 1 , … , x n ] {\displaystyle K[x_{1},\ldots ,x_{n}]} over a field K {\displaystyle K} . A Gröbner basis allows many important properties of the ideal and the associated algebraic variety to be deduced easily, such as the dimension and the number of zeros when it is finite. Gröbner basis computation is one of the main practical tools for solving systems of polynomial equations and computing the images of algebraic varieties under projections or rational maps .
Gröbner basis computation can be seen as a multivariate, non-linear generalization of both Euclid's algorithm for computing polynomial greatest common divisors , and Gaussian elimination for linear systems. [ 1 ]
Gröbner bases were introduced by Bruno Buchberger in his 1965 Ph.D. thesis, which also included an algorithm to compute them ( Buchberger's algorithm ). He named them after his advisor Wolfgang Gröbner . In 2007, Buchberger received the Association for Computing Machinery 's Paris Kanellakis Theory and Practice Award for this work.
However, the Russian mathematician Nikolai Günther had introduced a similar notion in 1913, published in various Russian mathematical journals. These papers were largely ignored by the mathematical community until their rediscovery in 1987 by Bodo Renschuch et al. [ 2 ] An analogous concept for multivariate power series was developed independently by Heisuke Hironaka in 1964, who named them standard bases . This term has been used by some authors to also denote Gröbner bases.
The theory of Gröbner bases has been extended by many authors in various directions. It has been generalized to other structures such as polynomials over principal ideal rings or polynomial rings , and also some classes of non-commutative rings and algebras, like Ore algebras .
Gröbner bases are primarily defined for ideals in a polynomial ring R = K [ x 1 , … , x n ] {\displaystyle R=K[x_{1},\ldots ,x_{n}]} over a field K . Although the theory works for any field, most Gröbner basis computations are done either when K is the field of rationals or the integers modulo a prime number.
In the context of Gröbner bases, a nonzero polynomial in R = K [ x 1 , … , x n ] {\displaystyle R=K[x_{1},\ldots ,x_{n}]} is commonly represented as a sum c 1 M 1 + ⋯ + c m M m , {\displaystyle c_{1}M_{1}+\cdots +c_{m}M_{m},} where the c i {\displaystyle c_{i}} are nonzero elements of K , called coefficients , and the M i {\displaystyle M_{i}} are monomials (called power products by Buchberger and some of his followers) of the form x 1 a 1 ⋯ x n a n , {\displaystyle x_{1}^{a_{1}}\cdots x_{n}^{a_{n}},} where the a i {\displaystyle a_{i}} are nonnegative integers. The vector A = [ a 1 , … , a n ] {\displaystyle A=[a_{1},\ldots ,a_{n}]} is called the exponent vector of the monomial. When the list X = [ x 1 , … , x n ] {\displaystyle X=[x_{1},\ldots ,x_{n}]} of the variables is fixed, the notation of monomials is often abbreviated as x 1 a 1 ⋯ x n a n = X A . {\displaystyle x_{1}^{a_{1}}\cdots x_{n}^{a_{n}}=X^{A}.}
Monomials are uniquely defined by their exponent vectors, and, when a monomial ordering (see below) is fixed, a polynomial is uniquely represented by the ordered list of the ordered pairs formed by an exponent vector and the corresponding coefficient. This representation of polynomials is especially efficient for Gröbner basis computation in computers, although it is less convenient for other computations such as polynomial factorization and polynomial greatest common divisor .
If F = { f 1 , … , f k } {\displaystyle F=\{f_{1},\ldots ,f_{k}\}} is a finite set of polynomials in the polynomial ring R , the ideal generated by F is the set of linear combinations of elements of F with coefficients in R ; that is the set of polynomials that can be written ∑ i = 1 k g i f i {\textstyle \sum _{i=1}^{k}g_{i}f_{i}} with g 1 , … , g k ∈ R . {\displaystyle g_{1},\ldots ,g_{k}\in R.}
All operations related to Gröbner bases require the choice of a total order on the monomials, with the following properties of compatibility with multiplication. For all monomials M , N , P ,
A total order satisfying these condition is sometimes called an admissible ordering .
These conditions imply that the order is a well-order , that is, every strictly decreasing sequence of monomials is finite.
Although Gröbner basis theory does not depend on a particular choice of an admissible monomial ordering, three monomial orderings are especially important for the applications:
Gröbner basis theory was initially introduced for the lexicographical ordering. It was soon realised that the Gröbner basis for degrevlex is almost always much easier to compute, and that it is almost always easier to compute a lex Gröbner basis by first computing the degrevlex basis and then using a "change of ordering algorithm". When elimination is needed, degrevlex is not convenient; both lex and lexdeg may be used but, again, many computations are relatively easy with lexdeg and almost impossible with lex.
Once a monomial ordering is fixed, the terms of a polynomial (product of a monomial with its nonzero coefficient) are naturally ordered by decreasing monomials (for this order). This makes the representation of a polynomial as a sorted list of pairs coefficient–exponent vector a canonical representation of the polynomials (that is, two polynomials are equal if and only if they have the same representation).
The first (greatest) term of a polynomial p for this ordering and the corresponding monomial and coefficient are respectively called the leading term , leading monomial and leading coefficient and denoted, in this article, lt( p ), lm( p ) and lc( p ) .
Most polynomial operations related to Gröbner bases involve the leading terms. So, the representation of polynomials as sorted lists make these operations particularly efficient (reading the first element of a list takes a constant time, independently of the length of the list).
The other polynomial operations involved in Gröbner basis computations are also compatible with the monomial ordering; that is, they can be performed without reordering the result:
Let M = x 1 a 1 ⋯ x n a n {\displaystyle M=x_{1}^{a_{1}}\cdots x_{n}^{a_{n}}} and N = x 1 b 1 ⋯ x n b n {\displaystyle N=x_{1}^{b_{1}}\cdots x_{n}^{b_{n}}} be two monomials, with exponent vectors A = [ a 1 , … , a n ] {\displaystyle A=[a_{1},\ldots ,a_{n}]} and B = [ b 1 , … , b n ] . {\displaystyle B=[b_{1},\ldots ,b_{n}].}
One says that M divides N , or that N is a multiple of M , if a i ≤ b i {\displaystyle a_{i}\leq b_{i}} for every i ; that is, if A is componentwise not greater than B . In this case, the quotient N M {\textstyle {\frac {N}{M}}} is defined as N M = x 1 b 1 − a 1 ⋯ x n b n − a n . {\textstyle {\frac {N}{M}}=x_{1}^{b_{1}-a_{1}}\cdots x_{n}^{b_{n}-a_{n}}.} In other words, the exponent vector of N M {\textstyle {\frac {N}{M}}} is the componentwise subtraction of the exponent vectors of N and M .
The greatest common divisor gcd( M , N ) of M and N is the monomial x 1 min ( a 1 , b 1 ) ⋯ x n min ( a n , b n ) {\textstyle x_{1}^{\min(a_{1},b_{1})}\cdots x_{n}^{\min(a_{n},b_{n})}} whose exponent vector is the componentwise minimum of A and B . The least common multiple lcm( M , N ) is defined similarly with max instead of min .
One has
The reduction of a polynomial by other polynomials with respect to a monomial ordering is central to Gröbner basis theory. It is a generalization of both row reduction occurring in Gaussian elimination and division steps of the Euclidean division of univariate polynomials . [ 1 ] When completed as much as possible, it is sometimes called multivariate division although its result is not uniquely defined.
Lead-reduction is a special case of reduction that is easier to compute. It is fundamental for Gröbner basis computation, since general reduction is needed only at the end of a Gröbner basis computation, for getting a reduced Gröbner basis from a non-reduced one.
Let an admissible monomial ordering be fixed, to which refers every monomial comparison that will occur in this section.
A polynomial f is lead-reducible by another polynomial g if the leading monomial lm( f ) is a multiple of lm( g ) . The polynomial f is reducible by g if some monomial of f is a multiple lm( g ) . (So, if f is lead-reducible by g , it is also reducible, but f may be reducible without being lead-reducible.)
Suppose that f is reducible by g , and let cm be a term of f such that the monomial m is a multiple of lm( g ) . A one-step reduction of f by g consists of replacing f by
This operation removes the monomial m from f without changing the terms with a monomial greater than m (for the monomial ordering). In particular, a one step lead-reduction of f produces a polynomial all of whose monomials are smaller than lm( f ) .
Given a finite set G of polynomials, one says that f is reducible or lead-reducible by G if it is reducible or lead-reducible, respectively, by at least one element g of G . In this case, a one-step reduction (resp. one-step lead-reduction) of f by G is any one-step reduction (resp. one-step lead-reduction) of f by an element of G .
The (complete) reduction (resp. lead-reduction) of f by G consists of iterating one-step reductions (respect. one-step lead reductions) until getting a polynomial that is irreducible (resp. lead-irreducible) by G . It is sometimes called a normal form of f by G . In general this form is not uniquely defined because there are, in general, several elements of G that can be used for reducing f ; this non-uniqueness is the starting point of Gröbner basis theory.
The definition of the reduction shows immediately that, if h is a normal form of f by G , one has
where h is irreducible by G and the q g {\displaystyle q_{g}} are polynomials such that lm ( q g g ) ≤ lm ( f ) . {\displaystyle \operatorname {lm} (q_{g}\,g)\leq \operatorname {lm} (f).} In the case of univariate polynomials, if G consists of a single element g , then h is the remainder of the Euclidean division of f by g , and q g is the quotient. Moreover, the division algorithm is exactly the process of lead-reduction. For this reason, some authors use the term multivariate division instead of reduction.
In the example that follows, there are exactly two complete lead-reductions that produce two very different results. The fact that the results are irreducible (not only lead-irreducible) is specific to the example, although this is rather common with such small examples.
In this two variable example, the monomial ordering that is used is the lexicographic order with x > y , {\displaystyle x>y,} and we consider the reduction of f = 2 x 3 − x 2 y + y 3 + 3 y {\displaystyle f=2x^{3}-x^{2}y+y^{3}+3y} , by G = { g 1 , g 2 } , {\displaystyle G=\{g_{1},g_{2}\},} with g 1 = x 2 + y 2 − 1 , g 2 = x y − 2. {\displaystyle {\begin{aligned}g_{1}&=x^{2}+y^{2}-1,\\g_{2}&=xy-2.\end{aligned}}}
For the first reduction step, either the first or the second term of f may be reduced. However, the reduction of a term amounts to removing this term at the cost of adding new lower terms; if it is not the first reducible term that is reduced, it may occur that a further reduction adds a similar term, which must be reduced again. It is therefore always better to reduce first the largest (for the monomial order) reducible term; that is, in particular, to lead-reduce first until getting a lead-irreducible polynomial.
The leading term 2 x 3 {\displaystyle 2x^{3}} of f is reducible by g 1 {\displaystyle g_{1}} and not by g 2 . {\displaystyle g_{2}.} So the first reduction step consists of multiplying g 1 {\displaystyle g_{1}} by −2 x and adding the result to f : f → − 2 x g 1 f 1 = f − 2 x g 1 = − x 2 y − 2 x y 2 + 2 x + y 3 + 3 y . {\displaystyle f\;\xrightarrow {\overset {}{-2xg_{1}}} \;f_{1}=f-2xg_{1}=-x^{2}y-2xy^{2}+2x+y^{3}+3y.}
The leading term − x 2 y {\displaystyle -x^{2}y} of f 1 {\displaystyle f_{1}} is a multiple of the leading monomials of both g 1 {\displaystyle g_{1}} and g 2 , {\displaystyle g_{2},} So, one has two choices for the second reduction step. If one chooses g 2 , {\displaystyle g_{2},} one gets a polynomial that can be reduced again by g 2 : {\displaystyle g_{2}\colon } f → − 2 x g 1 f 1 → x g 2 − 2 x y 2 + y 3 + 3 y → 2 y g 2 f 2 = y 3 − y . {\displaystyle f\;\xrightarrow {\overset {}{-2xg_{1}}} \;f_{1}\;\xrightarrow {xg_{2}} \;-2xy^{2}+y^{3}+3y\;\xrightarrow {2yg_{2}} \;f_{2}=y^{3}-y.} No further reduction is possible, so f 2 {\displaystyle f_{2}} is a complete reduction of f .
One gets a different result with the other choice for the second step: f → − 2 x g 1 f 1 → y g 1 − 2 x y 2 + 2 x + 2 y 3 + 2 y → 2 y g 2 f 3 = 2 x + 2 y 3 − 2 y . {\displaystyle f\;\xrightarrow {\overset {}{-2xg_{1}}} \;f_{1}\;\xrightarrow {yg_{1}} \;-2xy^{2}+2x+2y^{3}+2y\;\xrightarrow {2yg_{2}} \;f_{3}=2x+2y^{3}-2y.} Again, the result f 3 {\displaystyle f_{3}} is irreducible, although only lead reductions were done.
In summary, the complete reduction of f can result in either f 2 = y 3 − y {\displaystyle f_{2}=y^{3}-y} or f 3 = 2 x + 2 y 3 − 2 y . {\displaystyle f_{3}=2x+2y^{3}-2y.}
It is for dealing with the problems set by this non-uniqueness that Buchberger introduced Gröbner bases and S -polynomials. Intuitively, 0 = f − f {\displaystyle 0=f-f} may be reduced to f 2 − f 3 . {\displaystyle f_{2}-f_{3}.} This implies that f 2 − f 3 {\displaystyle f_{2}-f_{3}} belongs to the ideal generated by G . So, this ideal is not changed by adding f 3 − f 2 {\displaystyle f_{3}-f_{2}} to G , and this allows more reductions. In particular, f 3 {\displaystyle f_{3}} can be reduced to f 2 {\displaystyle f_{2}} by f 3 − f 2 {\displaystyle f_{3}-f_{2}} and this restores the uniqueness of the reduced form.
Here Buchberger's algorithm for Gröbner bases would begin by adding to G the polynomial
This polynomial, called S -polynomial by Buchberger, is the difference of the one-step reductions of the least common multiple x 2 y {\displaystyle x^{2}y} of the leading monomials of g 1 {\displaystyle g_{1}} and g 2 {\displaystyle g_{2}} , by g 2 {\displaystyle g_{2}} and g 1 {\displaystyle g_{1}} respectively:
In this example, one has g 3 = f 3 − f 2 . {\displaystyle g_{3}=f_{3}-f_{2}.} This does not complete Buchberger's algorithm, as xy gives different results, when reduced by g 2 {\displaystyle g_{2}} or g 3 . {\displaystyle g_{3}.}
Given monomial ordering, the S-polynomial or critical pair of two polynomials f and g is the polynomial
where lcm denotes the least common multiple of the leading monomials of f and g .
Using the definition of red 1 {\displaystyle \operatorname {red} _{1}} , this translates to:
Using the property that relates the lcm and the gcd , the S -polynomial can also be written as:
where gcd denotes the greatest common divisor of the leading monomials of f and g .
As the monomials that are reducible by both f and g are exactly the multiples of lcm , one can deal with all cases of non-uniqueness of the reduction by considering only the S -polynomials. This is a fundamental fact for Gröbner basis theory and all algorithms for computing them.
For avoiding fractions when dealing with polynomials with integer coefficients, the S polynomial is often defined as
This does not change anything to the theory since the two polynomials are associates .
Let R = F [ x 1 , … , x n ] {\displaystyle R=F[x_{1},\ldots ,x_{n}]} be a polynomial ring over a field F . In this section, we suppose that an admissible monomial ordering has been fixed.
Let G be a finite set of polynomials in R that generates an ideal I . The set G is a Gröbner basis (with respect to the monomial ordering), or, more precisely, a Gröbner basis of I if
or, equivalently,
There are many characterizing properties, which can each be taken as an equivalent definition of Gröbner bases. For conciseness, in the following list, the notation "one-word/another word" means that one can take either "one-word" or "another word" for having two different characterizations of Gröbner bases. All the following assertions are characterizations of Gröbner bases:
Counting the above definition, this provides 12 characterizations of Gröbner bases. The fact that so many characterizations are possible makes Gröbner bases very useful. For example, condition 3 provides an algorithm for testing ideal membership ; condition 4 provides an algorithm for testing whether a set of polynomials is a Gröbner basis and forms the basis of Buchberger's algorithm for computing Gröbner bases; conditions 5 and 6 allow computing in R / I {\displaystyle R/I} in a way that is very similar to modular arithmetic .
For every admissible monomial ordering and every finite set G of polynomials, there is a Gröbner basis that contains G and generates the same ideal. Moreover, such a Gröbner basis may be computed with Buchberger's algorithm .
This algorithm uses condition 4, and proceeds roughly as follows: for any two elements of G , compute the complete reduction by G of their S -polynomial, and add the result to G if it is not zero; repeat this operation with the new elements of G included until, eventually, all reductions produce zero.
The algorithm terminates always because of Dickson's lemma or because polynomial rings are Noetherian ( Hilbert's basis theorem ). Condition 4 ensures that the result is a Gröbner basis, and the definitions of S -polynomials and reduction ensure that the generated ideal is not changed.
The above method is an algorithm for computing Gröbner bases; however, it is very inefficient. Many improvements of the original Buchberger's algorithm, and several other algorithms have been proposed and implemented, which dramatically improve the efficiency. See § Algorithms and implementations , below.
A Gröbner basis is minimal if all leading monomials of its elements are irreducible by the other elements of the basis. Given a Gröbner basis of an ideal I , one gets a minimal Gröbner basis of I by removing the polynomials whose leading monomials are multiple of the leading monomial of another element of the Gröbner basis. However, if two polynomials of the basis have the same leading monomial, only one must be removed. So, every Gröbner basis contains a minimal Gröbner basis as a subset.
All minimal Gröbner bases of a given ideal (for a fixed monomial ordering) have the same number of elements, and the same leading monomials, and the non-minimal Gröbner bases have more elements than the minimal ones.
A Gröbner basis is reduced if every polynomial in it is irreducible by the other elements of the basis, and has 1 as leading coefficient. So, every reduced Gröbner basis is minimal, but a minimal Gröbner basis need not be reduced.
Given a Gröbner basis of an ideal I , one gets a reduced Gröbner basis of I by first removing the polynomials that are lead-reducible by other elements of the basis (for getting a minimal basis); then replacing each element of the basis by the result of the complete reduction by the other elements of the basis; and, finally, by dividing each element of the basis by its leading coefficient.
All reduced Gröbner bases of an ideal (for a fixed monomial ordering) are equal. It follows that two ideals are equal if and only if they have the same reduced Gröbner basis.
Sometimes, reduced Gröbner bases are defined without the condition on the leading coefficients. In this case, the uniqueness of reduced Gröbner bases is true only up to the multiplication of polynomials by a nonzero constant.
When working with polynomials over the field Q {\displaystyle \mathbb {Q} } of the rational numbers , it is useful to work only with polynomials with integer coefficients. In this case, the condition on the leading coefficients in the definition of a reduced basis may be replaced by the condition that all elements of the basis are primitive polynomials with integer coefficients, with positive leading coefficients. This restores the uniqueness of reduced bases.
For every monomial ordering, the empty set of polynomials is the unique Gröbner basis of the zero ideal .
For every monomial ordering, a set of polynomials that contains a nonzero constant is a Gröbner basis of the unit ideal (the whole polynomial ring). Conversely, every Gröbner basis of the unit ideal contains a nonzero constant. The reduced Gröbner basis of the unit is formed by the single polynomial 1 .
In the case of polynomials in a single variable, there is a unique admissible monomial ordering, the ordering by the degree. The minimal Gröbner bases are the singletons consisting of a single polynomial. The reduced Gröbner bases are the monic polynomials .
Let R = Q [ x , y ] {\displaystyle R=\mathbb {Q} [x,y]} be the ring of bivariate polynomials with rational coefficients and consider the ideal I = ⟨ f , g ⟩ {\displaystyle I=\langle f,g\rangle } generated by the polynomials
By reducing g by f , one obtains a new polynomial k such that I = ⟨ f , k ⟩ : {\displaystyle I=\langle f,k\rangle :}
None of f and k is reducible by the other, but xk is reducible by f , which gives another polynomial in I :
Under lexicographic ordering with x > y {\displaystyle x>y} we have
As f , k and h belong to I , and none of them is reducible by the others, none of { f , k } , {\displaystyle \{f,k\},} { f , h } , {\displaystyle \{f,h\},} and { h , k } {\displaystyle \{h,k\}} is a Gröbner basis of I .
On the other hand, { f , k , h } is a Gröbner basis of I , since the S-polynomials
can be reduced to zero by f , k and h .
The method that has been used here for finding h and k , and proving that { f , k , h } is a Gröbner basis is a direct application of Buchberger's algorithm . So, it can be applied mechanically to any similar example, although, in general, there are many polynomials and S-polynomials to consider, and the computation is generally too large for being done without a computer.
Unless explicitly stated, all the results that follow [ 3 ] are true for any monomial ordering (see that article for the definitions of the different orders that are mentioned below).
It is a common misconception that the lexicographical order is needed for some of these results. On the contrary, the lexicographical order is, almost always, the most difficult to compute, and using it makes impractical many computations that are relatively easy with graded reverse lexicographic order (grevlex), or, when elimination is needed, the elimination order (lexdeg) which restricts to grevlex on each block of variables.
Reduced Gröbner bases are unique for any given ideal and any monomial ordering. Thus two ideals are equal if and only if they have the same (reduced) Gröbner basis (usually a Gröbner basis software always produces reduced Gröbner bases).
The reduction of a polynomial f by the Gröbner basis G of an ideal I yields 0 if and only if f is in I . This allows to test the membership of an element in an ideal. Another method consists in verifying that the Gröbner basis of G ∪{ f } is equal to G .
To test if the ideal I generated by f 1 , ..., f k is contained in the ideal J , it suffices to test that every f I is in J . One may also test the equality of the reduced Gröbner bases of J and J ∪ { f 1 , ..., f k } .
Any set of polynomials may be viewed as a system of polynomial equations by equating the polynomials to zero. The set of the solutions of such a system depends only on the generated ideal, and, therefore does not change when the given generating set is replaced by the Gröbner basis, for any ordering, of the generated ideal. Such a solution, with coordinates in an algebraically closed field containing the coefficients of the polynomials, is called a zero of the ideal . In the usual case of rational coefficients, this algebraically closed field is chosen as the complex field .
An ideal does not have any zero (the system of equations is inconsistent ) if and only if 1 belongs to the ideal (this is Hilbert's Nullstellensatz ), or, equivalently, if its Gröbner basis (for any monomial ordering) contains 1, or, also, if the corresponding reduced Gröbner basis is [1].
Given the Gröbner basis G of an ideal I , it has only a finite number of zeros, if and only if, for each variable x , G contains a polynomial with a leading monomial that is a power of x (without any other variable appearing in the leading term). If this is the case, then the number of zeros, counted with multiplicity, is equal to the number of monomials that are not multiples of any leading monomial of G . This number is called the degree of the ideal.
When the number of zeros is finite, the Gröbner basis for a lexicographical monomial ordering provides, theoretically, a solution: the first coordinate of a solution is a root of the greatest common divisor of polynomials of the basis that depend only on the first variable. After substituting this root in the basis, the second coordinate of this solution is a root of the greatest common divisor of the resulting polynomials that depend only on the second variable, and so on. This solving process is only theoretical, because it implies GCD computation and root-finding of polynomials with approximate coefficients, which are not practicable because of numeric instability. Therefore, other methods have been developed to solve polynomial systems through Gröbner bases (see System of polynomial equations for more details).
The dimension of an ideal I in a polynomial ring R is the Krull dimension of the ring R / I and is equal to the dimension of the algebraic set of the zeros of I . It is also equal to number of hyperplanes in general position which are needed to have an intersection with the algebraic set, which is a finite number of points. The degree of the ideal and of its associated algebraic set is the number of points of this finite intersection, counted with multiplicity. In particular, the degree of a hypersurface is equal to the degree of its definition polynomial.
The dimension depend only on the set of the leading monomials of the Gröbner basis of the ideal for any monomial ordering. The same is true for the degree and degree-compatible monomial orderings; a monomial ordering is degree compatible is smaller for the degree implies smaller for the monomial ordering.
The dimension is the maximal size of a subset S of the variables such that there is no leading monomial depending only on the variables in S . Thus, if the ideal has dimension 0, then for each variable x there is a leading monomial in the Gröbner basis that is a power of x .
Both dimension and degree may be deduced from the Hilbert series of the ideal, which is the series ∑ i = 0 ∞ d i t i {\textstyle \sum _{i=0}^{\infty }d_{i}t^{i}} , where d i {\displaystyle d_{i}} is the number of monomials of degree i that are not multiple of any leading monomial in the Gröbner basis. [ 4 ] The Hilbert series may be summed into a rational fraction
where d is the dimension of the ideal and P ( t ) {\displaystyle P(t)} is a polynomial. The number P ( 1 ) {\displaystyle P(1)} is the degree of the algebraic set defined by the ideal, in the case of a homogeneous ideal or a monomial ordering compatible with the degree; that is, to compare two monomials, one compares their total degrees first.
The dimension does not depend on the choice of a monomial ordering, although the Hilbert series and the polynomial P ( t ) {\displaystyle P(t)} may change with changes of the monomial ordering. However, for homogeneous ideals or monomial orderings compatible with the degree, the Hilbert series and the polynomial P ( t ) {\displaystyle P(t)} do not depend on the choice of monomial ordering. [ 5 ]
Most computer algebra systems that provide functions to compute Gröbner bases provide also functions for computing the Hilbert series, and thus also the dimension and the degree.
The computation of Gröbner bases for an elimination monomial ordering allows computational elimination theory . This is based on the following theorem.
Consider a polynomial ring K [ x 1 , … , x n , y 1 , … , y m ] = K [ X , Y ] , {\displaystyle K[x_{1},\ldots ,x_{n},y_{1},\ldots ,y_{m}]=K[X,Y],} in which the variables are split into two subsets X and Y . Let us also choose an elimination monomial ordering "eliminating" X , that is a monomial ordering for which two monomials are compared by comparing first the X -parts, and, in case of equality only, considering the Y -parts. This implies that a monomial containing an X -variable is greater than every monomial independent of X .
If G is a Gröbner basis of an ideal I for this monomial ordering, then G ∩ K [ Y ] {\displaystyle G\cap K[Y]} is a Gröbner basis of I ∩ K [ Y ] {\displaystyle I\cap K[Y]} (this ideal is often called the elimination ideal ). Moreover, G ∩ K [ Y ] {\displaystyle G\cap K[Y]} consists exactly of the polynomials of G whose leading terms belong to K [ Y ] (this makes the computation of G ∩ K [ Y ] {\displaystyle G\cap K[Y]} very easy, as only the leading monomials need to be checked).
This elimination property has many applications, some described in the next sections.
Another application, in algebraic geometry , is that elimination realizes the geometric operation of projection of an affine algebraic set into a subspace of the ambient space: with above notation, the ( Zariski closure of) the projection of the algebraic set defined by the ideal I into the Y -subspace is defined by the ideal I ∩ K [ Y ] . {\displaystyle I\cap K[Y].}
The lexicographical ordering such that x 1 > ⋯ > x n {\displaystyle x_{1}>\cdots >x_{n}} is an elimination ordering for every partition { x 1 , … , x k } , { x k + 1 , … , x n } . {\displaystyle \{x_{1},\ldots ,x_{k}\},\{x_{k+1},\ldots ,x_{n}\}.} Thus a Gröbner basis for this ordering carries much more information than usually necessary. This may explain why Gröbner bases for the lexicographical ordering are usually the most difficult to compute.
If I and J are two ideals generated respectively by { f 1 , ..., f m } and { g 1 , ..., g k }, then a single Gröbner basis computation produces a Gröbner basis of their intersection I ∩ J . For this, one introduces a new indeterminate t , and one uses an elimination ordering such that the first block contains only t and the other block contains all the other variables (this means that a monomial containing t is greater than every monomial that does not contain t ). With this monomial ordering, a Gröbner basis of I ∩ J consists in the polynomials that do not contain t , in the Gröbner basis of the ideal
In other words, I ∩ J is obtained by eliminating t in K .
This may be proven by observing that the ideal K consists of the polynomials ( a − b ) t + b {\displaystyle (a-b)t+b} such that a ∈ I {\displaystyle a\in I} and b ∈ J {\displaystyle b\in J} . Such a polynomial is independent of t if and only if a = b , which means that b ∈ I ∩ J . {\displaystyle b\in I\cap J.}
A rational curve is an algebraic curve that has a set of parametric equations of the form
where f i ( t ) {\displaystyle f_{i}(t)} and g i ( t ) {\displaystyle g_{i}(t)} are univariate polynomials for 1 ≤ i ≤ n . One may (and will) suppose that f i ( t ) {\displaystyle f_{i}(t)} and g i ( t ) {\displaystyle g_{i}(t)} are coprime (they have no non-constant common factors).
Implicitization consists in computing the implicit equations of such a curve. In case of n = 2, that is for plane curves, this may be computed with the resultant . The implicit equation is the following resultant:
Elimination with Gröbner bases allows to implicitize for any value of n , simply by eliminating t in the ideal ⟨ g 1 x 1 − f 1 , … , g n x n − f n ⟩ . {\displaystyle \langle g_{1}x_{1}-f_{1},\ldots ,g_{n}x_{n}-f_{n}\rangle .} If n = 2, the result is the same as with the resultant, if the map t ↦ ( x 1 , x 2 ) {\displaystyle t\mapsto (x_{1},x_{2})} is injective for almost every t . In the other case, the resultant is a power of the result of the elimination.
When modeling a problem by polynomial equations, it is often assumed that some quantities are non-zero, so as to avoid degenerate cases. For example, when dealing with triangles , many properties become false if the triangle degenerates to a line segment, i.e. the length of one side is equal to the sum of the lengths of the other sides. In such situations, one cannot deduce relevant information from the polynomial system unless the degenerate solutions are ignored. More precisely, the system of equations defines an algebraic set which may have several irreducible components , and one must remove the components on which the degeneracy conditions are everywhere zero.
This is done by saturating the equations by the degeneracy conditions, which may be done via the elimination property of Gröbner bases.
The localization of a ring consists in adjoining to it the formal inverses of some elements. This section concerns only the case of a single element, or equivalently a finite number of elements (adjoining the inverses of several elements is equivalent to adjoining the inverse of their product). The localization of a ring R by an element f is the ring R f = R [ t ] / ( 1 − f t ) , {\displaystyle R_{f}=R[t]/(1-ft),} where t is a new indeterminate representing the inverse of f . The localization of an ideal I of R is the ideal I f = R f I {\displaystyle I_{f}=R_{f}I} of R f . {\displaystyle R_{f}.} When R is a polynomial ring, computing in R f {\displaystyle R_{f}} is not efficient because of the need to manage the denominators. Therefore, localization is usually replaced by the operation of saturation .
The saturation with respect to f of an ideal I in R is the inverse image of R f I {\displaystyle R_{f}I} under the canonical map from R to R f . {\displaystyle R_{f}.} It is the ideal I : f ∞ = { g ∈ R ∣ ( ∃ k ∈ N ) f k g ∈ I } {\displaystyle I:f^{\infty }=\{g\in R\mid (\exists k\in \mathbb {N} )f^{k}g\in I\}} consisting in all elements of R whose product with some power of f belongs to I .
If J is the ideal generated by I and 1− ft in R [ t ], then I : f ∞ = J ∩ R . {\displaystyle I:f^{\infty }=J\cap R.} It follows that, if R is a polynomial ring, a Gröbner basis computation eliminating t produces a Gröbner basis of the saturation of an ideal by a polynomial.
The important property of the saturation, which ensures that it removes from the algebraic set defined by the ideal I the irreducible components on which the polynomial f is zero, is the following: The primary decomposition of I : f ∞ {\displaystyle I:f^{\infty }} consists of the components of the primary decomposition of I that do not contain any power of f .
A Gröbner basis of the saturation by f of a polynomial ideal generated by a finite set of polynomials F , may be obtained by eliminating t in F ∪ { 1 − t f } , {\displaystyle F\cup \{1-tf\},} that is by keeping the polynomials independent of t in the Gröbner basis of F ∪ { 1 − t f } {\displaystyle F\cup \{1-tf\}} for an elimination ordering eliminating t .
Instead of using F , one may also start from a Gröbner basis of F . Which method is most efficient depends on the problem. However, if the saturation does not remove any component, that is if the ideal is equal to its saturated ideal, computing first the Gröbner basis of F is usually faster. On the other hand, if the saturation removes some components, the direct computation may be dramatically faster.
If one wants to saturate with respect to several polynomials f 1 , … , f k {\displaystyle f_{1},\ldots ,f_{k}} or with respect to a single polynomial which is a product f = f 1 ⋯ f k , {\displaystyle f=f_{1}\cdots f_{k},} there are three ways to proceed which give the same result but may have very different computation times (it depends on the problem which is the most efficient).
Hilbert's Nullstellensatz has two versions. The first one asserts that a set of polynomials has no common zeros over an algebraic closure of the field of the coefficients, if and only if 1 belongs to the generated ideal. This is easily tested with a Gröbner basis computation, because 1 belongs to an ideal if and only if 1 belongs to the Gröbner basis of the ideal, for any monomial ordering.
The second version asserts that the set of common zeros (in an algebraic closure of the field of the coefficients) of an ideal is contained in the hypersurface of the zeros of a polynomial f , if and only if a power of f belongs to the ideal. This may be tested by saturating the ideal by f ; in fact, a power of f belongs to the ideal if and only if the saturation by f provides a Gröbner basis containing 1.
By definition, an affine rational variety of dimension k may be described by parametric equations of the form
where p 0 , … , p n {\displaystyle p_{0},\ldots ,p_{n}} are n +1 polynomials in the k variables (parameters of the parameterization) t 1 , … , t k . {\displaystyle t_{1},\ldots ,t_{k}.} Thus the parameters t 1 , … , t k {\displaystyle t_{1},\ldots ,t_{k}} and the coordinates x 1 , … , x n {\displaystyle x_{1},\ldots ,x_{n}} of the points of the variety are zeros of the ideal
One could guess that it suffices to eliminate the parameters to obtain the implicit equations of the variety, as it has been done in the case of curves. Unfortunately this is not always the case. If the p i {\displaystyle p_{i}} have a common zero (sometimes called base point ), every irreducible component of the non-empty algebraic set defined by the p i {\displaystyle p_{i}} is an irreducible component of the algebraic set defined by I . It follows that, in this case, the direct elimination of the t i {\displaystyle t_{i}} provides an empty set of polynomials.
Therefore, if k >1, two Gröbner basis computations are needed to implicitize:
Buchberger's algorithm is the oldest algorithm for computing Gröbner bases. It has been devised by Bruno Buchberger together with the Gröbner basis theory. It is straightforward to implement, but it appeared soon that raw implementations can solve only trivial problems. The main issues are the following ones:
For solving 3. many improvements, variants and heuristics have been proposed before the introduction of F4 and F5 algorithms by Jean-Charles Faugère . As these algorithms are designed for integer coefficients or with coefficients in the integers modulo a prime number , Buchberger's algorithm remains useful for more general coefficients.
Roughly speaking, F4 algorithm solves 3. by replacing many S-polynomial reductions by the row reduction of a single large matrix for which advanced methods of linear algebra can be used. This solves partially issue 4., as reductions to zero in Buchberger's algorithm correspond to relations between rows of the matrix to be reduced, and the zero rows of the reduced matrix correspond to a basis of the vector space of these relations.
F5 algorithm improves F4 by introducing a criterion that allows reducing the size of the matrices to be reduced. This criterion is almost optimal, since the matrices to be reduced have full rank in sufficiently regular cases (in particular, when the input polynomials form a regular sequence ). Tuning F5 for a general use is difficult, since its performances depend on an order on the input polynomials and a balance between the incrementation of the working polynomial degree and of the number of the input polynomials that are considered. To date (2022), there is no distributed implementation that is significantly more efficient than F4, but, over modular integers F5 has been used successfully for several cryptographic challenges ; for example, for breaking HFE challenge .
Issue 5. has been solved by the discovery of basis conversion algorithms that start from the Gröbner basis for one monomial ordering for computing a Gröbner basis for another monomial ordering. FGLM algorithm is such a basis conversion algorithm that works only in the zero-dimensional case (where the polynomials have a finite number of complex common zeros) and has a polynomial complexity in the number of common zeros. A basis conversion algorithm that works is the general case is the Gröbner walk algorithm . [ 6 ] In its original form, FGLM may be the critical step for solving systems of polynomial equations because FGML does not take into account the sparsity of involved matrices . This has been fixed by the introduction of sparse FGLM algorithms . [ 7 ]
Most general-purpose computer algebra systems have implementations of one or several algorithms for Gröbner bases, often also embedded in other functions, such as for solving systems of polynomial equations or for simplifying trigonometric functions; this is the case, for example, of CoCoA , GAP , Macaulay 2 , Magma , Maple , Mathematica , SINGULAR , SageMath and SymPy . When F4 is available, it is generally much more efficient than Buchberger's algorithm. The implementation techniques and algorithmic variants are not always documented, although they may have a dramatic effect on efficiency.
Implementations of F4 and (sparse)-FGLM are included in the library Msolve . [ 8 ] Beside Gröbner algorithms, Msolve contains fast algorithms for real-root isolation , and combines all these functions in an algorithm for the real solutions of systems of polynomial equations that outperforms dramatically the other software for this problem (Maple and Magma). [ 8 ] Msolve is available on GitHub , and is interfaced with Julia , Maple and SageMath; this means that Msolve can be used directly from within these software environments.
The complexity of the Gröbner basis computations is commonly evaluated in term of the number n of variables and the maximal degree d of the input polynomials.
In the worst case, the main parameter of the complexity is the maximal degree of the elements of the resulting reduced Gröbner basis. More precisely, if the Gröbner basis contains an element of a large degree D , this element may contain Ω ( D n ) {\displaystyle \Omega (D^{n})} nonzero terms whose computation requires a time of Ω ( D n ) > D Ω ( n ) . {\displaystyle \Omega (D^{n})>D^{\Omega (n)}.} On the other hand, if all polynomials in the reduced Gröbner basis a homogeneous ideal have a degree of at most D , the Gröbner basis can be computed by linear algebra on the vector space of polynomials of degree less than 2 D , which has a dimension O ( D n ) . {\displaystyle O(D^{n}).} [ 1 ] So, the complexity of this computation is O ( D n ) O ( 1 ) = D O ( n ) . {\displaystyle O(D^{n})^{O(1)}=D^{O(n)}.}
The worst-case complexity of a Gröbner basis computation is doubly exponential in n . More precisely, the complexity is upper bounded by a polynomial in d 2 n . {\textstyle d^{2^{n}}.} Using little o notation , it is therefore bounded by d 2 n + o ( n ) . {\textstyle d^{2^{n+o(n)}}.} On the other hand, examples have been given of reduced Gröbner bases containing polynomials of degree d 2 Ω ( n ) , {\textstyle d^{2^{\Omega (n)}},} or containing d 2 Ω ( n ) {\textstyle d^{2^{\Omega (n)}}} elements. As every algorithm for computing a Gröbner basis must write its result, this provides a lower bound of the complexity.
Gröbner basis is EXPSPACE-complete . [ 9 ]
The concept and algorithms of Gröbner bases have been generalized to submodules of free modules over a polynomial ring. In fact, if L is a free module over a ring R , then one may consider the direct sum R ⊕ L {\displaystyle R\oplus L} as a ring by defining the product of two elements of L to be 0 . This ring may be identified with R [ e 1 , … , e l ] / ⟨ { e i e j | 1 ≤ i ≤ j ≤ l } ⟩ {\displaystyle R[e_{1},\ldots ,e_{l}]/\left\langle \{e_{i}e_{j}|1\leq i\leq j\leq l\}\right\rangle } , where e 1 , … , e l {\displaystyle e_{1},\ldots ,e_{l}} is a basis of L . This allows identifying a submodule of L generated by g 1 , … , g k {\displaystyle g_{1},\ldots ,g_{k}} with the ideal of R [ e 1 , … , e l ] {\displaystyle R[e_{1},\ldots ,e_{l}]} generated by g 1 , … , g k {\displaystyle g_{1},\ldots ,g_{k}} and the products e i e j {\displaystyle e_{i}e_{j}} , 1 ≤ i ≤ j ≤ l {\displaystyle 1\leq i\leq j\leq l} . If R is a polynomial ring, this reduces the theory and the algorithms of Gröbner bases of modules to the theory and the algorithms of Gröbner bases of ideals.
The concept and algorithms of Gröbner bases have also been generalized to ideals over various rings, commutative or not, like polynomial rings over a principal ideal ring or Weyl algebras .
Gröbner bases have been applied in the theory of error-correcting codes for algebraic decoding. By using Gröbner basis computation on various forms of error-correcting equations, decoding methods were developed for correcting errors of cyclic codes, [ 10 ] affine variety codes, [ 11 ] algebraic-geometric codes and even general linear block codes. [ 12 ] Applying Gröbner basis in algebraic decoding is still a research area of channel coding theory . | https://en.wikipedia.org/wiki/Gröbner_basis |
In computer algebra , the Gröbner fan of an ideal in the ring of polynomials is a concept in the theory of Gröbner bases . It is defined to be a fan consisting of cones that correspond to different monomial orders on that ideal. The concept was introduced by Mora and Robbiano in 1988. [ 1 ] The result is a weaker version of the result presented in the same issue of the journal by Bayer and Morrison. [ 2 ] Gröbner fan is a base for the nowadays active field of tropical geometry .
One implementation of the Gröbner fan is called Gfan, [ 3 ] based on an article of Fukuda, et al. [ 4 ] which is included in some computer algebra systems such as Singular , [ 5 ] Macaulay2 , [ 6 ] and CoCoA . [ 7 ]
This commutative algebra -related article is a stub . You can help Wikipedia by expanding it . | https://en.wikipedia.org/wiki/Gröbner_fan |
In mathematics , Grönwall's inequality (also called Grönwall's lemma or the Grönwall–Bellman inequality ) allows one to bound a function that is known to satisfy a certain differential or integral inequality by the solution of the corresponding differential or integral equation . There are two forms of the lemma, a differential form and an integral form. For the latter there are several variants.
Grönwall's inequality is an important tool to obtain various estimates in the theory of ordinary and stochastic differential equations . In particular, it provides a comparison theorem that can be used to prove uniqueness of a solution to the initial value problem ; see the Picard–Lindelöf theorem .
It is named for Thomas Hakon Grönwall (1877–1932). Grönwall is the Swedish spelling of his name, but he spelled his name as Gronwall in his scientific publications after emigrating to the United States.
The inequality was first proven by Grönwall in 1919 (the integral form below with α and β being constants). [ 1 ] Richard Bellman proved a slightly more general integral form in 1943. [ 2 ]
A nonlinear generalization of the Grönwall–Bellman inequality is known as Bihari–LaSalle inequality . Other variants and generalizations can be found in Pachpatte, B.G. (1998). [ 3 ]
Let I {\displaystyle I} denote an interval of the real line of the form [ a , ∞ ) {\displaystyle [a,\infty )} or [ a , b ] {\displaystyle [a,b]} or [ a , b ) {\displaystyle [a,b)} with a < b {\displaystyle a<b} . Let β {\displaystyle \beta } and u {\displaystyle u} be real-valued continuous functions defined on I {\displaystyle I} . If u {\displaystyle u} is differentiable in the interior I ∘ {\displaystyle I^{\circ }} of I {\displaystyle I} (the interval I {\displaystyle I} without the end points a {\displaystyle a} and possibly b {\displaystyle b} ) and satisfies the differential inequality
then u {\displaystyle u} is bounded by the solution of the corresponding differential equation v ′ ( t ) = β ( t ) v ( t ) {\displaystyle v'(t)=\beta (t)\,v(t)} :
for all t ∈ I {\displaystyle t\in I} .
Remark: There are no assumptions on the signs of the functions β {\displaystyle \beta } and u {\displaystyle u} .
Define the function
Note that v {\displaystyle v} satisfies
with v ( a ) = 1 {\displaystyle v(a)=1} and v ( t ) > 0 {\displaystyle v(t)>0} for all t ∈ I {\displaystyle t\in I} . By the quotient rule
Thus the derivative of the function u ( t ) / v ( t ) {\displaystyle u(t)/v(t)} is non-positive and the function is bounded above by its value at the initial point a {\displaystyle a} of the interval I {\displaystyle I} :
which is Grönwall's inequality.
Let I denote an interval of the real line of the form [ a , ∞) or [ a , b ] or [ a , b ) with a < b . Let α , β and u be real-valued functions defined on I . Assume that β and u are continuous and that the negative part of α is integrable on every closed and bounded subinterval of I .
Remarks:
(a) Define
Using the product rule , the chain rule , the derivative of the exponential function and the fundamental theorem of calculus , we obtain for the derivative
where we used the assumed integral inequality for the upper estimate. Since β and the exponential are non-negative, this gives an upper estimate for the derivative of v ( s ) {\displaystyle v(s)} . Since v ( a ) = 0 {\displaystyle v(a)=0} , integration of this inequality from a to t gives
Using the definition of v ( t ) {\displaystyle v(t)} from the first step, and then this inequality and the property e a e b = e a + b {\displaystyle e^{a}e^{b}=e^{a+b}} , we obtain
Substituting this result into the assumed integral inequality gives Grönwall's inequality.
(b) If the function α is non-decreasing, then part (a), the fact α ( s ) ≤ α ( t ) , and the fundamental theorem of calculus imply that
Let I denote an interval of the real line of the form [ a , ∞) or [ a , b ] or [ a , b ) with a < b . Let α and u be measurable functions defined on I and let μ be a continuous non-negative measure on the Borel σ-algebra of I satisfying μ ([ a , t ]) < ∞ for all t ∈ I (this is certainly satisfied when μ is a locally finite measure ). Assume that u is integrable with respect to μ in the sense that
and that u satisfies the integral inequality
If, in addition,
then u satisfies Grönwall's inequality
for all t ∈ I , where I s,t denotes to open interval ( s , t ) .
The proof is divided into three steps. The idea is to substitute the assumed integral inequality into itself n times. This is done in Claim 1 using mathematical induction. In Claim 2 we rewrite the measure of a simplex in a convenient form, using the permutation invariance of product measures. In the third step we pass to the limit n to infinity to derive the desired variant of Grönwall's inequality.
For every natural number n including zero,
with remainder
where
is an n -dimensional simplex and
We use mathematical induction . For n = 0 this is just the assumed integral inequality, because the empty sum is defined as zero.
Induction step from n to n + 1 :
Inserting the assumed integral inequality for the function u into the remainder gives
with
Using the Fubini–Tonelli theorem to interchange the two integrals, we obtain
Hence Claim 1 is proved for n + 1 .
For every natural number n including zero and all s < t in I
with equality in case t ↦ μ ([ a , t ]) is continuous for t ∈ I .
For n = 0 , the claim is true by our definitions. Therefore, consider n ≥ 1 in the following.
Let S n denote the set of all permutations of the indices in {1, 2, . . . , n }. For every permutation σ ∈ S n define
These sets are disjoint for different permutations and
Therefore,
Since they all have the same measure with respect to the n -fold product of μ , and since there are n ! permutations in S n , the claimed inequality follows.
Assume now that t ↦ μ ([ a , t ]) is continuous for t ∈ I . Then, for different indices i , j ∈ {1, 2, . . . , n }, the set
is contained in a hyperplane , hence by an application of Fubini's theorem its measure with respect to the n -fold product of μ is zero. Since
the claimed equality follows.
For every natural number n , Claim 2 implies for the remainder of Claim 1 that
By assumption we have μ ( I a , t ) < ∞ . Hence, the integrability assumption on u implies that
Claim 2 and the series representation of the exponential function imply the estimate
for all s < t in I . If the function α is non-negative, then it suffices to insert these results into Claim 1 to derive the above variant of Grönwall's inequality for the function u .
In case t ↦ μ ([ a , t ]) is continuous for t ∈ I , Claim 2 gives
and the integrability of the function α permits to use the dominated convergence theorem to derive Grönwall's inequality.
This article incorporates material from Gronwall's lemma on PlanetMath , which is licensed under the Creative Commons Attribution/Share-Alike License . | https://en.wikipedia.org/wiki/Grönwall's_inequality |
In condensed matter , Grüneisen parameter γ is a dimensionless thermodynamic parameter named after German physicist Eduard Grüneisen , whose original definition was formulated in terms of the phonon nonlinearities. [ 1 ]
Because of the equivalences of many properties and derivatives within thermodynamics (e.g. see Maxwell relations ), there are many formulations of the Grüneisen parameter which are equally valid, leading to numerous interpretations of its meaning. Some formulations for the Grüneisen parameter include: γ = V ( d P d E ) V = α K T C V ρ = α K S C P ρ = α v s 2 C P = − ( ∂ ln T ∂ ln V ) S {\displaystyle \gamma =V\left({\frac {dP}{dE}}\right)_{V}={\frac {\alpha K_{T}}{C_{V}\rho }}={\frac {\alpha K_{S}}{C_{P}\rho }}={\frac {\alpha v_{s}^{2}}{C_{P}}}=-\left({\frac {\partial \ln T}{\partial \ln V}}\right)_{S}} where V is volume, C P {\displaystyle C_{P}} and C V {\displaystyle C_{V}} are the principal (i.e. per-mass) heat capacities at constant pressure and volume, E is energy, S is entropy, α is the volume thermal expansion coefficient , K S {\displaystyle K_{S}} and K T {\displaystyle K_{T}} are the adiabatic and isothermal bulk moduli , v s {\displaystyle v_{s}} is the speed of sound in the medium, and ρ is density. The Grüneisen parameter is dimensionless.
The expression for the Grüneisen constant of a perfect crystal with pair interactions in d {\displaystyle d} -dimensional space has the form: [ 2 ] Γ 0 = − 1 2 d Π ‴ ( a ) a 2 + ( d − 1 ) [ Π ″ ( a ) a − Π ′ ( a ) ] Π ″ ( a ) a + ( d − 1 ) Π ′ ( a ) , {\displaystyle \Gamma _{0}=-{\frac {1}{2d}}{\frac {\Pi '''(a)a^{2}+(d-1)\left[\Pi ''(a)a-\Pi '(a)\right]}{\Pi ''(a)a+(d-1)\Pi '(a)}},} where Π {\displaystyle \Pi } is the interatomic potential , a {\displaystyle a} is the equilibrium distance, d {\displaystyle d} is the space dimensionality. Relations between the Grüneisen constant and parameters of Lennard-Jones , Morse , and Mie [ 3 ] potentials are presented in the table below.
The expression for the Grüneisen constant of a 1D chain with Mie potential exactly coincides with the results of MacDonald and Roy. [ 4 ] Using the relation between the Grüneisen parameter and interatomic potential one can derive the simple necessary and sufficient condition for Negative Thermal Expansion in perfect crystals with pair interactions Π ‴ ( a ) a > − ( d − 1 ) Π ″ ( a ) . {\displaystyle \Pi '''(a)a>-(d-1)\Pi ''(a).} A proper description of the Grüneisen parameter represents a stringent test for any type of interatomic potential.
The physical meaning of the parameter can also be extended by combining thermodynamics with a reasonable microphysics model for the vibrating atoms within a crystal.
When the restoring force acting on an atom displaced from its equilibrium position is linear in the atom's displacement, the frequencies ω i of individual phonons do not depend on the volume of the crystal or on the presence of other phonons, and the thermal expansion (and thus γ) is zero. When the restoring force is non-linear in the displacement, the phonon frequencies ω i change with the volume V {\displaystyle V} . The Grüneisen parameter of an individual vibrational mode i {\displaystyle i} can then be defined as (the negative of) the logarithmic derivative of the corresponding frequency ω i {\displaystyle \omega _{i}} :
γ i = − V ω i ∂ ω i ∂ V . {\displaystyle \gamma _{i}=-{\frac {V}{\omega _{i}}}{\frac {\partial \omega _{i}}{\partial V}}.}
Using the quasi-harmonic approximation for atomic vibrations, the macroscopic Grüneisen parameter ( γ ) can be related to the description of how the vibrational frequencies ( phonons ) within a crystal are altered with changing volume (i.e. γ i 's).
For example, one can show that γ = α K T C V ρ {\displaystyle \gamma ={\frac {\alpha K_{T}}{C_{V}\rho }}} if one defines γ {\displaystyle \gamma } as the weighted average γ = ∑ i γ i c V , i ∑ i c V , i , {\displaystyle \gamma ={\frac {\sum _{i}\gamma _{i}c_{V,i}}{\sum _{i}c_{V,i}}},} where c V , i {\displaystyle c_{V,i}} 's are the partial vibrational mode contributions to the heat capacity, such that C V = 1 ρ V ∑ i c V , i . {\textstyle C_{V}={\frac {1}{\rho V}}\sum _{i}c_{V,i}.}
To prove this relation, it is easiest to introduce the heat capacity per particle C ~ V = ∑ i c V , i {\textstyle {\tilde {C}}_{V}=\sum _{i}c_{V,i}} ; so one can write ∑ i γ i c V , i C ~ V = α K T C V ρ = α V K T C ~ V . {\displaystyle {\frac {\sum _{i}\gamma _{i}c_{V,i}}{{\tilde {C}}_{V}}}={\frac {\alpha K_{T}}{C_{V}\rho }}={\frac {\alpha VK_{T}}{{\tilde {C}}_{V}}}.}
This way, it suffices to prove ∑ i γ i c V , i = α V K T . {\displaystyle \sum _{i}\gamma _{i}c_{V,i}=\alpha VK_{T}.}
Left-hand side (def): ∑ i γ i c V , i = ∑ i [ − V ω i ∂ ω i ∂ V ] [ k B ( ℏ ω i k B T ) 2 exp ( ℏ ω i k B T ) [ exp ( ℏ ω i k B T ) − 1 ] 2 ] {\displaystyle \sum _{i}\gamma _{i}c_{V,i}=\sum _{i}\left[-{\frac {V}{\omega _{i}}}{\frac {\partial \omega _{i}}{\partial V}}\right]\left[k_{\rm {B}}\left({\frac {\hbar \omega _{i}}{k_{\rm {B}}T}}\right)^{2}{\frac {\exp \left({\frac {\hbar \omega _{i}}{k_{\rm {B}}T}}\right)}{\left[\exp \left({\frac {\hbar \omega _{i}}{k_{\rm {B}}T}}\right)-1\right]^{2}}}\right]}
Right-hand side (def): α V K T = [ 1 V ( ∂ V ∂ T ) P ] V [ − V ( ∂ P ∂ V ) T ] = − V ( ∂ V ∂ T ) P ( ∂ P ∂ V ) T {\displaystyle \alpha VK_{T}=\left[{\frac {1}{V}}\left({\frac {\partial V}{\partial T}}\right)_{P}\right]V\left[-V\left({\frac {\partial P}{\partial V}}\right)_{T}\right]=-V\left({\frac {\partial V}{\partial T}}\right)_{P}\left({\frac {\partial P}{\partial V}}\right)_{T}}
Furthermore ( Maxwell relations ): ( ∂ V ∂ T ) P = ∂ ∂ T ( ∂ G ∂ P ) T = ∂ ∂ P ( ∂ G ∂ T ) P = − ( ∂ S ∂ P ) T {\displaystyle \left({\frac {\partial V}{\partial T}}\right)_{P}={\frac {\partial }{\partial T}}\left({\frac {\partial G}{\partial P}}\right)_{T}={\frac {\partial }{\partial P}}\left({\frac {\partial G}{\partial T}}\right)_{P}=-\left({\frac {\partial S}{\partial P}}\right)_{T}}
Thus α V K T = V ( ∂ S ∂ P ) T ( ∂ P ∂ V ) T = V ( ∂ S ∂ V ) T {\displaystyle \alpha VK_{T}=V\left({\frac {\partial S}{\partial P}}\right)_{T}\left({\frac {\partial P}{\partial V}}\right)_{T}=V\left({\frac {\partial S}{\partial V}}\right)_{T}}
This derivative is straightforward to determine in the quasi-harmonic approximation , as only the ω i are V -dependent.
∂ S ∂ V = ∂ ∂ V { − ∑ i k B ln [ 1 − exp ( − ℏ ω i ( V ) k B T ) ] + ∑ i 1 T ℏ ω i ( V ) exp ( ℏ ω i ( V ) k B T ) − 1 } {\displaystyle {\frac {\partial S}{\partial V}}={\frac {\partial }{\partial V}}\left\{-\sum _{i}k_{\rm {B}}\ln \left[1-\exp \left(-{\frac {\hbar \omega _{i}(V)}{k_{\rm {B}}T}}\right)\right]+\sum _{i}{\frac {1}{T}}{\frac {\hbar \omega _{i}(V)}{\exp \left({\frac {\hbar \omega _{i}(V)}{k_{\rm {B}}T}}\right)-1}}\right\}}
V ∂ S ∂ V = − ∑ i V ω i ∂ ω i ∂ V k B ( ℏ ω i k B T ) 2 exp ( ℏ ω i k B T ) [ exp ( ℏ ω i k B T ) − 1 ] 2 = ∑ i γ i c V , i {\displaystyle V{\frac {\partial S}{\partial V}}=-\sum _{i}{\frac {V}{\omega _{i}}}{\frac {\partial \omega _{i}}{\partial V}}\;\;k_{\rm {B}}\left({\frac {\hbar \omega _{i}}{k_{\rm {B}}T}}\right)^{2}{\frac {\exp \left({\frac {\hbar \omega _{i}}{k_{\rm {B}}T}}\right)}{\left[\exp \left({\frac {\hbar \omega _{i}}{k_{\rm {B}}T}}\right)-1\right]^{2}}}=\sum _{i}\gamma _{i}c_{V,i}}
This yields γ = ∑ i γ i c V , i ∑ i c V , i = α V K T C ~ V . {\displaystyle \gamma ={\dfrac {\sum _{i}\gamma _{i}c_{V,i}}{\sum _{i}c_{V,i}}}={\dfrac {\alpha VK_{T}}{{\tilde {C}}_{V}}}.}
Regarding Boltzmann-Gibbs (BG) statistical mechanics, it is reported in the literature that the Grüneisen parameter presents an expressive enhancement close to critical points (CPs) and phase transitions. However, for genuine quantum critical phenomena, i.e., in the complete absence of temperature T , a thermodynamic definition of the Grüneisen parameter is elusive because it embodies dependences with temperature and exactly at T = 0K the Grüneisen parameter is undetermined. Nevertheless, a quantum version Γ 0 K {\displaystyle \Gamma ^{0{\text{K}}}} was recently proposed. [ 5 ] Using the 1D Ising model under a transverse magnetic field (1DIMTF) the authors have shown that, for the quantum CP of such a model, Γ 0 K {\displaystyle \Gamma ^{0{\text{K}}}} shows a divergent-like behavior when the magnetic energy B is comparable to the exchange coupling energy J . Such behavior is associated with the breakdown of the Boltzmann-Gibbs-von Neumann-Shannon entropy extensivity in this regime, which leads to zeros and infinities in physical quantities such as the Grüneisen parameter. However, upon employing the generalized nonadditive entropy S q {\displaystyle S_{q}} , Constantino Tsallis demonstrated that for a unique value of the entropic index q , S q {\displaystyle S_{q}} is extensive right at the CP of the 1DIMTF. Hence, upon making an unprecedented connection of Γ 0 K {\displaystyle \Gamma ^{0{\text{K}}}} in terms of S q {\displaystyle S_{q}} and using the 1DIMTF, researchers from Physics Department - Unesp, Rio Claro, have shown that Γ 0 K {\displaystyle \Gamma ^{0{\text{K}}}} is universally nondivergent when using the appropriate entropy for the critical regime. [ 6 ] Such results suggest that the divergent-like behavior of physical quantities associated with the nonvalidity of BG statistical mechanics should be revisited in terms of S q {\displaystyle S_{q}} . | https://en.wikipedia.org/wiki/Grüneisen_parameter |
In mathematics , the Grünwald–Letnikov derivative is a basic extension of the derivative in fractional calculus that allows one to take the derivative a non-integer number of times. It was introduced by Anton Karl Grünwald (1838–1920) from Prague , in 1867, and by Aleksey Vasilievich Letnikov (1837–1888) in Moscow in 1868.
The formula
for the derivative can be applied recursively to get higher-order derivatives. For example, the second-order derivative would be:
Assuming that the h 's converge synchronously, this simplifies to:
which can be justified rigorously by the mean value theorem . In general, we have (see binomial coefficient ):
Removing the restriction that n be a positive integer, it is reasonable to define:
This defines the Grünwald–Letnikov derivative.
To simplify notation, we set:
Then the Grünwald–Letnikov derivative may be succinctly written as:
In the preceding section, the general first principles equation for integer order derivatives was derived. It can be shown that the equation may also be written as
or removing the restriction that n must be a positive integer:
This equation is called the reverse Grünwald–Letnikov derivative. If the substitution h → − h is made, the resulting equation is called the direct Grünwald–Letnikov derivative: [ 1 ] | https://en.wikipedia.org/wiki/Grünwald–Letnikov_derivative |
The gua operon is responsible for regulating the synthesis of guanosine mono phosphate (GMP), a purine nucleotide, from inosine monophosphate (IMP or inosinate). It consists of two structural genes guaB (encodes for IMP dehydrogenase or and guaA (encodes for GMP synthetase) apart from the promoter and operator region.
The first committed step of purine biosynthesis starts from 5-phosphoribosyl 1 pyrophosphate. This undergoes a series of reactions to form IMP, an important branch point in the pathway. The pathway then branches to form adenylosuccinate and then adenylate (AMP) in one branch and xanthylate (XMP) and then guanylate (GMP) in the other branch. IMP dehydrogenase catalyses the conversion of IMP to XMP and GMP synthetase catalyses the conversion of XMP to GMP. [ 1 ] [ 2 ]
The operon must respond to changes in the metabolic state of the cell. It is subject to growth rate dependent control, stringent control (control during the various stresses the cell is exposed to) and other forms of control. [ 3 ] Hence it stops biosynthesis if guanine can be obtained from the external medium, increases its expression if nucleotides are needed (for example during DNA replication) and balances the production of GMP with respect to AMP and the pyrimidine nucleotides .
The branch point at IMP mentioned above is tightly controlled rigid node [ 4 ] so as to have a balanced production of AMP and GMP. The gua operon is repressed by GMP and is induced by AMP. Similarly AMP synthesis is repressed by AMP itself while it is activated by GMP. This dual control ensures that there is a balance of flux between AMP and GMP and the flux partition remains relatively constant even in the face of perturbations. Some mechanisms by which this control is achieved are discussed below.
Since DNA replication needs a supply of guanine nucleotides, there must be some co-ordination between the DNA replication machinery and the gua operon. One method through which this happens is the DnaA protein. DnaA is a protein which recognises the origin of replication, promotes a local unwinding of an AT rich DNA region and finally guides the helicase DnaB to its entry site. DnaA is the replication initiation factor which causes DNA replication if present in sufficient concentration. [ 5 ]
When replication happens, the origin of replication creates a "sink" for DnaA proteins. So genes which are negatively modulated by DnaA, like those of the gua operon are derepressed. Two potential DnaA binding sites, one on the gua promoter and another 200 bp downstream of the IMP dehydrogenase initiation codon on the guaB gene exist. It is thought that both the former and the latter sequences play a part, the latter part being vital and DNA binding at these sequences negatively affect transcription of the gene. [ 6 ]
While growing on media in which growth rates are low, cAMP binds to a cAMP receptor protein forming a complex which has regulatory properties. This complex binds to a region 100 bp upstream of the guaB transcription start site which then represses the gua operon. It is a matter of debate of how this complex interacts with RNA polymerase from as long as around 100 bit/s away. One view suggests the involvement of an unknown regulatory factor. In any case, the complex confers growth rate dependent control to the promoter region of the operon. [ 7 ]
The repressor purR encoded by purR genes controls the synthesis of enzymes involved in purine biosynthesis . A putative 16 bp pur operator was found in the gua promoter. The purR repressor works with other co-repressors, for example guanine which is a co-repressor in E. coli . [ 3 ]
The gua mRNA leader has the potential for forming a stable stem-loop secondary structure incorporating the first 37 nucleotides of the transcript. As the ribosome binding site is sequestered in the stem loop, this structure may be involved in translational regulation. [ 3 ]
The gua promoter lies back-to-back with the xseA promoter (encoding the mismatch repair enzyme exonuclease VII). [ 3 ] Such close spacing of promoters may have regulatory significance and will lead to steric hindrance as the RNA polymerase molecules try to bind simultaneously. Inactivation of one promoter will naturally lead to greater expression of the other promoter. Other mechanisms include FIS (factor for inversion stimulation) which sterically hinders RNA polymerase binding. But the role of FIS is not yet properly investigated. [ 8 ]
Since the metabolic pathway is conserved across species, the genes are similar too. Extensive studies on the regulation in Saccharomyces cerevisiae have been done. The IMD family of genes and gua1 in yeast correspond to guaB and guaA. Here only IMD genes are guanine sensitive and not gua1 unlike in prokaryotes where the entire operon is sensitive. Mycophenolic acid, a drug which is an inhibitor of IMP dehydrogenase, is an inducer of IMD 2 gene (and hence IMD 2 probably has intrinsic drug activity. [ 9 ] Another aspect in which the eukaryotic regulation is very different is that eukaryotes have differential regulation of the branches leading to AMP and GMP. For example, in yeasts AMP synthesis genes are poorly repressed by guanine whereas GMP synthesis genes are not affected by adenine and in humans IMP dehydrogenase synthesis is repressed in the presence of guanosine but not adenosine. | https://en.wikipedia.org/wiki/Gua_Operon |
In organic chemistry , a guaianolide is a type of sesquiterpene lactone consisting of a gamma- lactone and either a cyclopentane or cyclopentene , both fused to a central cycloheptane or cycloheptene structure. There are two subclasses, structural isomers differing in the location that part of the lactone is bonded to the central ring, known as 6,12-guaianolides and 8,12-guaianolides. [ 1 ]
Because some of the natural products in this class of tricyclic phytochemical have been found to be potentially biologically active , there has been interest in their chemical syntheses. [ 2 ] The full biosynthetic origin of most of the known guaianolides has not been established, but the pathway is generally presumed to begin with the formation of a germacrene lactone derived from farnesyl pyrophosphate . [ 1 ]
This organic chemistry article is a stub . You can help Wikipedia by expanding it . | https://en.wikipedia.org/wiki/Guaianolide |
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