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In thermodynamics , vapor quality is the mass fraction in a saturated mixture that is vapor ; [ 1 ] in other words, saturated vapor has a "quality" of 100%, and saturated liquid has a "quality" of 0%. Vapor quality is an intensive property which can be used in conjunction with other independent intensive properties to specify the thermodynamic state of the working fluid of a thermodynamic system . It has no meaning for substances which are not saturated mixtures (for example, compressed liquids or superheated fluids).
Vapor quality is an important quantity during the adiabatic expansion step in various thermodynamic cycles (like Organic Rankine cycle , Rankine cycle , etc.). Working fluids can be classified by using the appearance of droplets in the vapor during the expansion step.
Quality χ can be calculated by dividing the mass of the vapor by the mass of the total mixture: χ = m vapor m total {\displaystyle \chi ={\frac {m_{\text{vapor}}}{m_{\text{total}}}}} where m indicates mass.
Another definition used in chemical engineering defines quality ( q ) of a fluid as the fraction that is saturated liquid. [ 2 ] By this definition, a saturated liquid has q = 0 . A saturated vapor has q = 1 . [ 3 ]
An alternative definition is the 'equilibrium thermodynamic quality'. It can be used only for single-component mixtures (e.g. water with steam), and can take values < 0 (for sub-cooled fluids) and > 1 (for super-heated vapors):
χ eq = h − h f h f g {\displaystyle \chi _{\text{eq}}={\frac {h-h_{f}}{h_{fg}}}} where h is the mixture specific enthalpy , defined as: h = m f ⋅ h f + m g ⋅ h g m f + m g . {\displaystyle h={\frac {m_{f}\cdot h_{f}+m_{g}\cdot h_{g}}{m_{f}+m_{g}}}.}
Subscripts f and g refer to saturated liquid and saturated gas respectively, and fg refers to vaporization . [ 4 ]
The above expression for vapor quality can be expressed as: χ = y − y f y g − y f {\displaystyle \chi ={\frac {y-y_{f}}{y_{g}-y_{f}}}} where y {\displaystyle y} is equal to either specific enthalpy , specific entropy , specific volume or specific internal energy , y f {\displaystyle y_{f}} is the value of the specific property of saturated liquid state and y g − y f {\displaystyle y_{g}-y_{f}} is the value of the specific property of the substance in dome zone, which we can find both liquid y f {\displaystyle y_{f}} and vapor y g {\displaystyle y_{g}} .
Another expression of the same concept is: χ = m v m l + m v {\displaystyle \chi ={\frac {m_{v}}{m_{l}+m_{v}}}} where m v {\displaystyle m_{v}} is the vapor mass and m l {\displaystyle m_{l}} is the liquid mass.
The origin of the idea of vapor quality was derived from the origins of thermodynamics , where an important application was the steam engine . Low quality steam would contain a high moisture percentage and therefore damage components more easily. [ citation needed ] High quality steam would not corrode the steam engine. Steam engines use water vapor ( steam ) to push pistons or turbines, and that movement creates work . The quantitatively described steam quality (steam dryness) is the proportion of saturated steam in a saturated water/steam mixture. In other words, a steam quality of 0 indicates 100% liquid, while a steam quality of 1 (or 100%) indicates 100% steam.
The quality of steam on which steam whistles are blown is variable and may affect frequency . Steam quality determines the velocity of sound , which declines with decreasing dryness due to the inertia of the liquid phase . Also, the specific volume of steam for a given temperature decreases with decreasing dryness. [ 5 ] [ 6 ]
Steam quality is very useful in determining enthalpy of saturated water/steam mixtures, since the enthalpy of steam (gaseous state) is many orders of magnitude higher than the enthalpy of water (liquid state).
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https://en.wikipedia.org/wiki/Vapor_quality
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Vaporization (or vapo(u)risation) of an element or compound is a phase transition from the liquid phase to vapor . [ 1 ] There are two types of vaporization: evaporation and boiling . Evaporation is a surface phenomenon , whereas boiling is a bulk phenomenon (a phenomenon in which the whole object or substance is involved in the process).
Evaporation is a phase transition from the liquid phase to vapor (a state of substance below critical temperature ) that occurs at temperatures below the boiling temperature at a given pressure. Evaporation occurs on the surface . Evaporation only occurs when the partial pressure of vapor of a substance is less than the equilibrium vapor pressure . For example, due to constantly decreasing pressures, vapor pumped out of a solution will eventually leave behind a cryogenic liquid.
Boiling is also a phase transition from the liquid phase to gas phase, but boiling is the formation of vapor as bubbles of vapor below the surface of the liquid. Boiling occurs when the equilibrium vapor pressure of the substance is greater than or equal to the atmospheric pressure . The temperature at which boiling occurs is the boiling temperature, or boiling point. The boiling point varies with the pressure of the environment.
Sublimation is a direct phase transition from the solid phase to the gas phase, skipping the intermediate liquid phase.
The term vaporization has also been used in a colloquial or hyperbolic way to refer to the physical destruction of an object that is exposed to intense heat or explosive force, where the object is actually blasted into small pieces rather than literally converted to gaseous form. Examples of this usage include the "vaporization" of the uninhabited Marshall Island of Elugelab in the 1952 Ivy Mike thermonuclear test. [ 2 ] Many other examples can be found throughout the various MythBusters episodes that have involved explosives, chief among them being Cement Mix-Up , where they "vaporized" a cement truck with ANFO. [ 3 ]
At the moment of a large enough meteor or comet impact, bolide detonation, a nuclear fission , thermonuclear fusion , or theoretical antimatter weapon detonation, a flux of so many gamma ray , x-ray , ultraviolet , visual light and heat photons strikes matter in a such brief amount of time (a great number of high-energy photons, many overlapping in the same physical space) that all molecules lose their atomic bonds ( atomization ) and "fly apart". All atoms lose their electron shells and become positively charged ions ( ionization ), in turn emitting photons of a slightly lower energy than they had absorbed. All such matter becomes a gas of nuclei and electrons which rise into the air due to the extremely high temperature or bond to each other as they cool. The matter vaporized this way is immediately a plasma in a state of maximum entropy and this state steadily reduces via the factor of passing time due to natural processes in the biosphere and the effects of physics at normal temperatures and pressures .
A similar process occurs during ultrashort pulse laser ablation , where the high flux of incoming electromagnetic radiation strips the target material's surface of electrons, leaving positively charged atoms which undergo a coulomb explosion . [ 4 ]
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In thermodynamics and chemical engineering , the vapor–liquid equilibrium ( VLE ) describes the distribution of a chemical species between the vapor phase and a liquid phase .
The concentration of a vapor in contact with its liquid, especially at equilibrium , is often expressed in terms of vapor pressure , which will be a partial pressure (a part of the total gas pressure) if any other gas(es) are present with the vapor. The equilibrium vapor pressure of a liquid is in general strongly dependent on temperature . At vapor–liquid equilibrium, a liquid with individual components in certain concentrations will have an equilibrium vapor in which the concentrations or partial pressures of the vapor components have certain values depending on all of the liquid component concentrations and the temperature. The converse is also true: if a vapor with components at certain concentrations or partial pressures is in vapor–liquid equilibrium with its liquid, then the component concentrations in the liquid will be determined dependent on the vapor concentrations and on the temperature. The equilibrium concentration of each component in the liquid phase is often different from its concentration (or vapor pressure) in the vapor phase, but there is a relationship. The VLE concentration data can be determined experimentally or approximated with the help of theories such as Raoult's law , Dalton's law , and Henry's law .
Such vapor–liquid equilibrium information is useful in designing columns for distillation , especially fractional distillation , which is a particular specialty of chemical engineers. [ 1 ] [ 2 ] [ 3 ] Distillation is a process used to separate or partially separate components in a mixture by boiling (vaporization) followed by condensation . Distillation takes advantage of differences in concentrations of components in the liquid and vapor phases.
In mixtures containing two or more components, the concentrations of each component are often expressed as mole fractions . The mole fraction of a given component of a mixture in a particular phase (either the vapor or the liquid phase) is the number of moles of that component in that phase divided by the total number of moles of all components in that phase.
Binary mixtures are those having two components. Three-component mixtures are called ternary mixtures. There can be VLE data for mixtures with even more components, but such data is often hard to show graphically. VLE data is a function of the total pressure, such as 1 atm or at the pressure the process is conducted at.
When a temperature is reached such that the sum of the equilibrium vapor pressures of the liquid components becomes equal to the total pressure of the system (it is otherwise smaller), then vapor bubbles generated from the liquid begin to displace the gas that was maintaining the overall pressure, and the mixture is said to boil. This temperature is called the boiling point of the liquid mixture at the given pressure. (It is assumed that the total pressure is held steady by adjusting the total volume of the system to accommodate the specific volume changes that accompany boiling.) The boiling point at an overall pressure of 1 atm is called the normal boiling point of the liquid mixture.
The field of thermodynamics describes when vapor–liquid equilibrium is possible, and its properties. Much of the analysis depends on whether the vapor and liquid consist of a single component, or if they are mixtures.
If the liquid and vapor are pure, in that they consist of only one molecular component and no impurities, then the equilibrium state between the two phases is described by the following equations:
where P liq {\displaystyle P^{\text{liq}}} and P vap {\displaystyle P^{\text{vap}}} are the pressures within the liquid and vapor, T liq {\displaystyle T^{\text{liq}}} and T vap {\displaystyle T^{\text{vap}}} are the temperatures within the liquid and vapor, and G ~ liq {\displaystyle {\tilde {G}}^{\text{liq}}} and G ~ vap {\displaystyle {\tilde {G}}^{\text{vap}}} are the molar Gibbs free energies (units of energy per amount of substance ) within the liquid and vapor, respectively. [ 4 ] : 215 In other words, the temperature, pressure and molar Gibbs free energy are the same between the two phases when they are at equilibrium.
An equivalent, more common way to express the vapor–liquid equilibrium condition in a pure system is by using the concept of fugacity . Under this view, equilibrium is described by the following equation:
where f liq ( T s , P s ) {\displaystyle f^{\text{liq}}(T_{s},P_{s})} and f vap ( T s , P s ) {\displaystyle f^{\text{vap}}(T_{s},P_{s})} are the fugacities of the liquid and vapor, respectively, at the system temperature T s and pressure P s . [ 4 ] : 216, 218 It is often convenient to use the quantity ϕ = f / P {\textstyle \phi =f/P} , the dimensionless fugacity coefficient , which is 1 for an ideal gas .
In a multicomponent system, where the vapor and liquid consist of more than one type of compounds, describing the equilibrium state is more complicated. For all components i in the system, the equilibrium state between the two phases is described by the following equations:
where P and T are the temperature and pressure for each phase, and G ¯ i liq {\displaystyle {\bar {G}}_{i}^{\text{liq}}} and G ¯ i vap {\displaystyle {\bar {G}}_{i}^{\text{vap}}} are the partial molar Gibbs free energy also called chemical potential (units of energy per amount of substance ) within the liquid and vapor, respectively, for each phase. The partial molar Gibbs free energy is defined by:
where G is the ( extensive ) Gibbs free energy, and n i is the amount of substance of component i .
Binary mixture VLE data at a certain overall pressure, such as 1 atm, showing mole fraction vapor and liquid concentrations when boiling at various temperatures can be shown as a two-dimensional graph called a boiling-point diagram . The mole fraction of component 1 in the mixture can be represented by the symbol x 1 . The mole fraction of component 2, represented by x 2 , is related to x 1 in a binary mixture as follows:
In multi-component mixtures in general with n components, this becomes:
The preceding equilibrium equations are typically applied for each phase (liquid or vapor) individually, but the result can be plotted in a single diagram. In a binary boiling-point diagram, temperature ( T ) (or sometimes pressure) is graphed vs. x 1 . At any given temperature (or pressure) where both phases are present, vapor with a certain mole fraction is in equilibrium with liquid with a certain mole fraction. The two mole fractions often differ. These vapor and liquid mole fractions are represented by two points on the same horizontal isotherm (constant T ) line. When an entire range of temperatures vs. vapor and liquid mole fractions is graphed, two (usually curved) lines result. The lower one, representing the mole fraction of the boiling liquid at various temperatures, is called the bubble point curve . The upper one, representing the mole fraction of the vapor at various temperatures, is called the dew point curve . [ 1 ]
These two curves necessarily meet where the mixture becomes purely one component, namely where x 1 = 0 (and x 2 = 1 , pure component 2) or x 1 = 1 (and x 2 = 0 , pure component 1). The temperatures at those two points correspond to the boiling points of each of the two pure components.
For certain pairs of substances, the two curves also coincide at some point strictly between x 1 = 0 and x 1 = 1 . When they meet, they meet tangently; the dew-point temperature always lies above the boiling-point temperature for a given composition when they are not equal. The meeting point is called an azeotrope for that particular pair of substances. It is characterized by an azeotrope temperature and an azeotropic composition, often expressed as a mole fraction. There can be maximum-boiling azeotropes , where the azeotrope temperature is at a maximum in the boiling curves, or minimum-boiling azeotropes , where the azeotrope temperature is at a minimum in the boiling curves.
If one wants to represent a VLE data for a three-component mixture as a boiling point "diagram", a three-dimensional graph can be used. Two of the dimensions would be used to represent the composition mole fractions, and the third dimension would be the temperature. Using two dimensions, the composition can be represented as an equilateral triangle in which each corner represents one of the pure components. The edges of the triangle represent a mixture of the two components at each end of the edge. Any point inside the triangle represents the composition of a mixture of all three components. The mole fraction of each component would correspond to where a point lies along a line starting at that component's corner and perpendicular to the opposite edge. The bubble point and dew point data would become curved surfaces inside a triangular prism, which connect the three boiling points on the vertical temperature "axes". Each face of this triangular prism would represent a two-dimensional boiling-point diagram for the corresponding binary mixture. Due to their three-dimensional complexity, such boiling-point diagrams are rarely seen. Alternatively, the three-dimensional curved surfaces can be represented on a two-dimensional graph by the use of curved isotherm lines at graduated intervals, similar to iso-altitude lines on a map. Two sets of such isotherm lines are needed on such a two-dimensional graph: one set for the bubble point surface and another set for the dew point surface.
The tendency of a given chemical species to partition itself
preferentially between liquid and vapor phases is the Henry's law constant. There can be VLE data for mixtures of four or more components, but such a boiling-point diagram is hard to show in either tabular or graphical form. For such multi-component mixtures, as well as binary
mixtures, the vapor–liquid equilibrium data are represented in terms of K values ( vapor–liquid distribution ratios ) [ 1 ] [ 2 ] defined by
where y i and x i are the mole fractions of component i in the phases y and x respectively.
For Raoult's law
For modified Raoult's law
where γ i {\displaystyle \gamma _{i}} is the activity coefficient , P i is the partial pressure and P is the pressure .
The values of the ratio K i are correlated empirically or theoretically in terms of temperature, pressure and phase compositions in the form of equations, tables or graph such as the DePriester charts. [ 5 ]
For binary mixtures, the ratio of the K values for the two components is called the relative volatility denoted by α
which is a measure of the relative ease or difficulty of separating the two components. Large-scale industrial distillation is rarely undertaken if the relative volatility is less than 1.05 with the volatile component being i and the less volatile component being j . [ 2 ]
K values are widely used in the design calculations of continuous distillation columns for distilling multicomponent mixtures.
For each component in a binary mixture, one could make a vapor–liquid equilibrium diagram. Such a diagram would graph liquid mole fraction on a horizontal axis and vapor mole fraction on a vertical axis. In such VLE diagrams, liquid mole fractions for components 1 and 2 can be represented as x 1 and x 2 respectively, and vapor mole fractions of the corresponding components are commonly represented as y 1 and y 2 . [ 2 ] Similarly for binary mixtures in these VLE diagrams: x 1 + x 2 = 1 y 1 + y 2 = 1 {\displaystyle {\begin{aligned}x_{1}+x_{2}&=1\\y_{1}+y_{2}&=1\end{aligned}}}
Such VLE diagrams are square with a diagonal line running from the ( x 1 = 0, y 1 = 0 ) corner to the ( x 1 = 1, y 1 = 1 ) corner for reference.
These types of VLE diagrams are used in the McCabe–Thiele method to determine the number of equilibrium stages (or theoretical plates ) needed to distill a given composition binary feed mixture into one distillate fraction and one bottoms fraction. Corrections can also be made to take into account the incomplete efficiency of each tray in a distillation column when compared to a theoretical plate.
At boiling and higher temperatures the sum of the individual component partial pressures becomes equal to the overall pressure, which can symbolized as P tot .
Under such conditions, Dalton's law would be in effect as follows: P tot = P 1 + P 2 + ⋯ {\displaystyle P_{\text{tot}}=P_{1}+P_{2}+\cdots }
Then for each component in the vapor phase: y 1 = P 1 P tot , y 2 = P 2 P tot , ⋯ {\displaystyle y_{1}={\frac {P_{1}}{P_{\text{tot}}}},\quad y_{2}={\frac {P_{2}}{P_{\text{tot}}}},\quad \cdots }
where P 1 = partial pressure of component 1, P 2 = partial pressure of component 2, etc.
Raoult's law is approximately valid for mixtures of components between which there is very little interaction other than the effect of dilution by the other components. Examples of such mixtures includes mixtures of alkanes , which are non- polar , relatively inert compounds in many ways, so there is little attraction or repulsion between the molecules. Raoult's law states that for components 1, 2, etc. in a mixture:
P 1 = x 1 P 1 ∘ , P 2 = x 2 P 2 ∘ , ⋯ {\displaystyle P_{1}=x_{1}P_{1}^{\circ },\quad P_{2}=x_{2}P_{2}^{\circ },\quad \cdots }
where P 1 ° , P 2 ° , etc. are the vapor pressures of components 1, 2, etc. when they are pure, and x 1 , x 2 , etc. are mole fractions of the corresponding component in the liquid.
Recall from the first section that vapor pressures of liquids are very dependent on temperature. Thus the P ° pure vapor pressures for each component are a function of temperature ( T ): For example, commonly for a pure liquid component, the Clausius–Clapeyron relation may be used to approximate how the vapor pressure varies as a function of temperature. This makes each of the partial pressures dependent on temperature also regardless of whether Raoult's law applies or not. When Raoult's law is valid these expressions become: P 1 T = x 1 P 1 ∘ T , P 2 T = x 2 P 2 ∘ T , ⋯ {\displaystyle P_{1}T=x_{1}P_{1}^{\circ }T,\quad P_{2}T=x_{2}P_{2}^{\circ }T,\quad \cdots }
At boiling temperatures if Raoult's law applies, the total pressure becomes: P tot = x 1 P 1 ∘ T + x 2 P 2 ∘ T + ⋯ {\displaystyle P_{\text{tot}}=x_{1}P_{1}^{\circ }T+x_{2}P_{2}^{\circ }T+\cdots }
At a given P tot such as 1 atm and a given liquid composition, T can be solved for to give the liquid mixture's boiling point or bubble point, although the solution for T may not be mathematically analytical (i.e., may require a numerical solution or approximation). For a binary mixture at a given P tot , the bubble point T can become a function of x 1 (or x 2 ) and this function can be shown on a two-dimensional graph like a binary boiling point diagram.
At boiling temperatures if Raoult's law applies, a number of the preceding equations in this section can be combined to give the following expressions for vapor mole fractions as a function of liquid mole fractions and temperature: y 1 = x 1 P 1 ∘ T P tot , y 2 = x 2 P 2 ∘ T P tot , ⋯ {\displaystyle y_{1}=x_{1}{\frac {P_{1}^{\circ }T}{P_{\text{tot}}}},\quad y_{2}=x_{2}{\frac {P_{2}^{\circ }T}{P_{\text{tot}}}},\quad \cdots }
Once the bubble point T 's as a function of liquid composition in terms of mole fractions have been determined, these values can be inserted into the above equations to obtain corresponding vapor compositions in terms of mole fractions. When this is finished over a complete range of liquid mole fractions and their corresponding temperatures, one effectively obtains a temperature T function of vapor composition mole fractions. This function effectively acts as the dew point T function of vapor composition.
In the case of a binary mixture, x 2 = 1 − x 1 and the above equations can be expressed as: y 1 = x 1 P 1 ∘ T P tot y 2 = ( 1 − x 1 ) P 2 ∘ T P tot {\displaystyle {\begin{aligned}y_{1}&=x_{1}{\frac {P_{1}^{\circ }T}{P_{\text{tot}}}}\\y_{2}&=(1-x_{1}){\frac {P_{2}^{\circ }T}{P_{\text{tot}}}}\end{aligned}}}
For many kinds of mixtures, particularly where there is interaction between components beyond simply the effects of dilution, Raoult's law does not work well for determining the shapes of the curves in the boiling point or VLE diagrams. Even in such mixtures, there are usually still differences in the vapor and liquid equilibrium concentrations at most points, and distillation is often still useful for separating components at least partially. For such mixtures, empirical data is typically used in determining such boiling point and VLE diagrams. Chemical engineers have done a significant amount of research trying to develop equations for correlating and/or predicting VLE data for various kinds of mixtures which do not obey Raoult's law well.
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https://en.wikipedia.org/wiki/Vapor–liquid_equilibrium
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In chemical engineering , a vapor–liquid separator is a device used to separate a vapor–liquid mixture into its constituent phases . It can be a vertical or horizontal vessel, and can act as a 2-phase or 3-phase separator.
A vapor–liquid separator may also be referred to as a flash drum , breakpot , knock-out drum or knock-out pot , compressor suction drum, suction scrubber or compressor inlet drum, or vent scrubber. When used to remove suspended water droplets from streams of air, it is often called a demister .
In vapor-liquid separators gravity is utilized to cause the denser fluid (liquid) to settle to the bottom of the vessel where it is withdrawn, less dense fluid (vapor) is withdrawn from the top of the vessel. [ 1 ] [ 2 ] [ 3 ] [ 4 ]
In low gravity environments such as a space station , a common liquid separator will not function because gravity is not usable as a separation mechanism. In this case, centrifugal force needs to be utilised in a spinning centrifugal separator to drive liquid towards the outer edge of the chamber for removal. Gaseous components migrate towards the center.
An inlet diffuser reduces the velocity and spreads the incoming mixture across the full cross-section of the vessel. [ 5 ] A mesh pad in the upper part of the vessel aids separation and prevents liquid from being carried over with the vapor. The pad or mist mat traps entrained liquid droplets and allows them to coalesce until they are large enough to fall through the up-flowing vapor to the bottom of the vessel. [ 5 ] Vane packs and cyclonic separators are also used to remove liquid from the outlet vapor. [ 5 ]
The gas outlet may itself be surrounded by a spinning mesh screen or grating, so that any liquid that does approach the outlet strikes the grating, is accelerated, and thrown away from the outlet.
The vapor travels through the gas outlet at a design velocity which minimises the entrainment of any liquid droplets in the vapor as it exits the vessel.
A vortex breaker on the liquid outlet prevents the formation of vortices and of vapor being drawn into the liquid outlet. [ 5 ]
The separator is only effective as long as there is a vapor space inside the chamber. The separator can fail if either the mixed inlet is overwhelmed with supply material, or the liquid drain is unable to handle the volume of liquid being collected. The separator may therefore be combined with some other liquid level sensing mechanism such as a sight glass or float sensor . In this manner, both the supply and drain flow can be regulated to prevent the separator from becoming overloaded.
Vertical separators are generally used when the gas-liquid ratio is high or gas volumes are high. Horizontal separators are used where large volumes of liquid are involved. [ 5 ]
A vapor-liquid separator may operate as a 3-phase separator, with two immiscible liquid phases of different densities. For example natural gas (vapor), water and oil/condensate. The two liquids settle at the bottom of the vessel with oil floating on the water. Separate liquid outlets are provided. [ 5 ]
The feed to a vapor–liquid separator may also be a liquid that is being partially or totally flashed into a vapor and liquid as it enters the separator.
A slug catcher is a type of vapor–liquid separator that is able to receive a large inflow of liquid at random times. It is usually found at the end of gas pipelines where condensate may be present as slugs of liquid. It is usually a horizontal vessel or array of large diameter pipes. [ 5 ]
The liquid capacity of a separator is usually defined by the residence time of the liquid in the vessel. Some typical residence times are as shown. [ 5 ]
Vapor–liquid separators are very widely used in a great many industries and applications, such as:
In refrigeration systems, it is common for the system to contain a mixture of liquid and gas, but for the mechanical gas compressor to be intolerant of liquid.
Some compressor types such as the scroll compressor use a continuously shrinking compression volume. Once liquid completely fills this volume the pump may either stall and overload, or the pump chamber may be warped or otherwise damaged by the fluid that can not fit into a smaller space.
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https://en.wikipedia.org/wiki/Vapor–liquid_separator
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The vapor–liquid–solid method ( VLS ) is a mechanism for the growth of one-dimensional structures, such as nanowires , from chemical vapor deposition . The growth of a crystal through direct adsorption of a gas phase on to a solid surface is generally very slow. The VLS mechanism circumvents this by introducing a catalytic liquid alloy phase which can rapidly adsorb a vapor to supersaturation levels, and from which crystal growth can subsequently occur from nucleated seeds at the liquid–solid interface. The physical characteristics of nanowires grown in this manner depend, in a controllable way, upon the size and physical properties of the liquid alloy.
The VLS mechanism was proposed in 1964 as an explanation for silicon whisker growth from the gas phase in the presence of a liquid gold droplet placed upon a silicon substrate. [ 1 ] The explanation was motivated by the absence of axial screw dislocations in the whiskers (which in themselves are a growth mechanism), the requirement of the gold droplet for growth, and the presence of the droplet at the tip of the whisker during the entire growth process.
The VLS mechanism is typically described in three stages: [ 2 ]
The VLS process takes place as follows:
The requirements for catalysts are: [ 3 ]
The materials system used, as well as the cleanliness of the vacuum system and therefore the amount of contamination and/or the presence of oxide layers at the droplet and wafer surface during the experiment, both greatly influence the absolute magnitude of the forces present at the droplet/surface interface and, in turn, determine the shape of the droplets. The shape of the droplet, i.e. the contact angle (β 0 , see Figure 4) can, be modeled mathematically, however, the actual forces present during growth are extremely difficult to measure experimentally. Nevertheless, the shape of a catalyst particle at the surface of a crystalline substrate is determined by a balance of the forces of surface tension and the liquid–solid interface tension. The radius of the droplet varies with the contact angle as:
R = r o sin ( β o ) , {\displaystyle R={\frac {r_{\mathrm {o} }}{\sin(\beta _{\mathrm {o} })}},}
where r 0 is the radius of the contact area and β 0 is defined by a modified Young’s equation:
σ 1 cos ( β o ) = σ s − σ l s − τ r o {\displaystyle \sigma _{\mathrm {1} }\cos(\beta _{\mathrm {o} })=\sigma _{\mathrm {s} }-\sigma _{\mathrm {ls} }-{\frac {\tau }{r_{\mathrm {o} }}}} ,
It is dependent on the surface (σ s ) and liquid–solid interface (σ ls ) tensions, as well as an additional line tension (τ) which comes into effect when the initial radius of the droplet is small (nanosized). As a nanowire begins to grow, its height increases by an amount dh and the radius of the contact area decreases by an amount dr (see Figure 4). As the growth continues, the inclination angle at the base of the nanowires (α, set as zero before whisker growth) increases, as does β 0 :
σ 1 cos ( β o ) = σ s cos ( α ) − σ l s − τ r o {\displaystyle \sigma _{\mathrm {1} }\cos(\beta _{\mathrm {o} })=\sigma _{\mathrm {s} }\cos(\alpha )-\sigma _{\mathrm {ls} }-{\frac {\tau }{r_{\mathrm {o} }}}} .
The line tension therefore greatly influences the catalyst contact area. The most import result from this conclusion is that different line tensions will result in different growth modes. If the line tensions are too large, nanohillock growth will result and thus stop the growth.
The diameter of the nanowire which is grown depends upon the properties of the alloy droplet. The growth of nano-sized wires requires nano-size droplets to be prepared on the substrate. In an equilibrium situation this is not possible as the minimum radius of a metal droplet is given by [ 4 ]
where V l is the molar volume of the droplet, σ lv the liquid-vapor surface energy , and s is the degree of supersaturation [ 5 ] of the vapor. This equations restricts the minimum diameter of the droplet, and of any crystals which can be grown from it, under typically conditions to well above the nanometer level. Several techniques to generate smaller droplets have been developed, including the use of monodispersed nanoparticles spread in low dilution on the substrate, and the laser ablation of a substrate-catalyst mixture so to form a plasma which allows well-separated nanoclusters of the catalyst to form as the systems cools. [ 6 ]
During VLS whisker growth, the rate at which whiskers grow is dependent on the whisker diameter: the larger the whisker diameter, the faster the nanowire grows axially. This is because the supersaturation of the metal-alloy catalyst ( Δ μ {\displaystyle \Delta \mu } ) is the main driving force for nanowhisker growth and decreases with decreasing whisker diameter (also known as the Gibbs-Thomson effect):
Δ μ = Δ μ o − 4 α Ω d {\displaystyle \Delta \mu =\Delta \mu _{\mathrm {o} }-{\frac {4\alpha \Omega }{d}}} .
Again, Δμ is the main driving force for nanowhisker growth (the supersaturation of the metal droplet). More specifically, Δμ 0 is the difference between the chemical potential of the depositing species (Si in the above example) in the vapor and solid whisker phase. Δμ 0 is the initial difference proceeding whisker growth (when d → ∞ {\displaystyle d\rightarrow \infty } ), while Ω {\displaystyle \Omega } is the atomic volume of Si and α {\displaystyle \alpha } the specific free energy of the wire surface. Examination of the above equation, indeed reveals that small diameters ( < {\displaystyle <} 100 nm) exhibit small driving forces for whisker growth while large wire diameters exhibit large driving forces.
Involves the removal of material from metal-containing solid targets by irradiating the surface with high-powered (~100 mJ/pulse) short (10 Hz) laser pulses, usually with wavelengths in the ultraviolet (UV) region of the light spectrum. When such a laser pulse is adsorbed by a solid target, material from the surface region of the target absorbs the laser energy and either (a) evaporates or sublimates from the surface or is (b) converted into a plasma (see laser ablation ). These particles are easily transferred to the substrate where they can nucleate and grow into nanowires .
The laser-assisted growth technique is particularly useful for growing nanowires with high melting temperatures , multicomponent or doped nanowires, as well as nanowires with extremely high crystalline quality. The high intensity of the laser pulse incident at the target allows the deposition of high melting point materials, without having to try to evaporate the material using extremely high temperature resistive or electron bombardment heating. Furthermore, targets can simply be made from a mixture of materials or even a liquid. Finally, the plasma formed during the laser absorption process allows for the deposition of charged particles as well as a catalytic means to lower the activation barrier of reactions between target constituents.
Some very interesting nanowires microstructures can be obtained by simply thermally evaporating solid materials. This technique can be carried out in a relatively simple setup composed of a dual-zone vacuum furnace. The hot end of the furnace contains the evaporating source material, while the evaporated particles are carrier downstream, (by way of a carrier gas) to the colder end of the furnace where they can absorb, nucleate, and grow on a desired substrate.
Molecular beam epitaxy (MBE) has been used since 2000 to create high-quality semiconductor wires based on the VLS growth mechanism. However, in metal-catalyzed MBE the metal particles do not catalyze a reaction between precursors but rather adsorb vapor phase particles. This is because the chemical potential of the vapor can be drastically lowered by entering the liquid phase.
MBE is carried out under ultra-high vacuum (UHV) conditions where the mean-free-path (distance between collisions) of source atoms or molecules is on the order of meters. Therefore, evaporated source atoms (from, say, an effusion cell) act as a beam of particles directed towards the substrate. The growth rate of the process is very slow, the deposition conditions are very clean, and as a result four superior capabilities arise, when compared to other deposition methods:
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The vapour-phase-mediated antimicrobial activity (VMAA) is the inhibitory or cidal antimicrobial activity of a molecule in a liquid culture , following its initial evaporation and migration via the vapour-phase [ 1 ] Two new in vitro assays i.e. the vapour-phase-mediated patch assay [ 2 ] and the vapour-phase-mediated susceptibility assay [ 1 ] were developed to detect and quantify the VMAA. Both assays belong to the newest class of vaporisation assays i.e. the broth microdilution derived vaporisation assays. In contrast, most other vaporisation assays belong to the class of agar disk diffusion derived vaporisation assays and quantify the antimicrobial activity of the vapour-phase itself. [ 3 ] Both classes of vaporisation assays are useful and measure different aspects of the antimicrobial capacity of molecules.
Possible applications for volatiles like volatile organic compounds with VMAA are: maintaining hygiene in hospitals, treating post-harvest contamination, protecting crops against pathogens and pests , and treating infections of the digestive , vaginal or respiratory tract. [ 4 ]
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The vapor pressure of water is the pressure exerted by molecules of water vapor in gaseous form (whether pure or in a mixture with other gases such as air). The saturation vapor pressure is the pressure at which water vapor is in thermodynamic equilibrium with its condensed state . At pressures higher than saturation vapor pressure, water will condense , while at lower pressures it will evaporate or sublimate . The saturation vapor pressure of water increases with increasing temperature and can be determined with the Clausius–Clapeyron relation . The boiling point of water is the temperature at which the saturated vapor pressure equals the ambient pressure. Water supercooled below its normal freezing point has a higher vapor pressure than that of ice at the same temperature and is, thus, unstable.
Calculations of the (saturation) vapor pressure of water are commonly used in meteorology . The temperature-vapor pressure relation inversely describes the relation between the boiling point of water and the pressure. This is relevant to both pressure cooking and cooking at high altitudes. An understanding of vapor pressure is also relevant in explaining high altitude breathing and cavitation .
There are many published approximations for calculating saturated vapor pressure over water and over ice. Some of these are (in approximate order of increasing accuracy):
Constants are unattributed.
This is of the from that would be derived from the Clausis-Claperyon relation
See also discussion of Clausius-Clapeyron approximations used in meteorology and climatology .
Here is a comparison of the accuracies of these different explicit formulations, showing saturation vapor pressures for liquid water in kPa, calculated at six temperatures with their percentage error from the table values of Lide (2005):
A more detailed discussion of accuracy and considerations of the inaccuracy in temperature measurements is presented in Alduchov and Eskridge (1996). The analysis here shows the simple unattributed formula and the Antoine equation are reasonably accurate at 100 °C, but quite poor for lower temperatures above freezing. Tetens is much more accurate over the range from 0 to 50 °C and very competitive at 75 °C, but Antoine's is superior at 75 °C and above. The unattributed formula must have zero error at around 26 °C, but is of very poor accuracy outside a narrow range. Tetens' equations are generally much more accurate and arguably more straightforward for use at everyday temperatures (e.g., in meteorology). As expected, [ clarification needed ] Buck's equation for T > 0 °C is significantly more accurate than Tetens, and its superiority increases markedly above 50 °C, though it is more complicated to use. The Buck equation is even superior to the more complex Goff-Gratch equation over the range needed for practical meteorology.
For serious computation, Lowe (1977) [ 4 ] developed two pairs of equations for temperatures above and below freezing, with different levels of accuracy. They are all very accurate (compared to Clausius-Clapeyron and the Goff-Gratch ) but use nested polynomials for very efficient computation. However, there are more recent reviews of possibly superior formulations, notably Wexler (1976, 1977), [ 5 ] [ 6 ] reported by Flatau et al. (1992). [ 7 ]
Examples of modern use of these formulae can additionally be found in NASA's GISS Model-E and Seinfeld and Pandis (2006). The former is an extremely simple Antoine equation, while the latter is a polynomial. [ 8 ]
In 2018 a new physics-inspired approximation formula was devised and tested by Huang [ 9 ] who also reviews other recent attempts.
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In mathematics , Varadhan's lemma is a result from the large deviations theory named after S. R. Srinivasa Varadhan . The result gives information on the asymptotic distribution of a statistic φ ( Z ε ) of a family of random variables Z ε as ε becomes small in terms of a rate function for the variables.
Let X be a regular topological space ; let ( Z ε ) ε >0 be a family of random variables taking values in X ; let μ ε be the law ( probability measure ) of Z ε . Suppose that ( μ ε ) ε >0 satisfies the large deviation principle with good rate function I : X → [0, +∞]. Let ϕ : X → R be any continuous function . Suppose that at least one of the following two conditions holds true: either the tail condition
where 1 ( E ) denotes the indicator function of the event E ; or, for some γ > 1, the moment condition
Then
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https://en.wikipedia.org/wiki/Varadhan's_lemma
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The reflected binary code ( RBC ), also known as reflected binary ( RB ) or Gray code after Frank Gray , is an ordering of the binary numeral system such that two successive values differ in only one bit (binary digit).
For example, the representation of the decimal value "1" in binary would normally be " 001 ", and "2" would be " 010 ". In Gray code, these values are represented as " 001 " and " 011 ". That way, incrementing a value from 1 to 2 requires only one bit to change, instead of two.
Gray codes are widely used to prevent spurious output from electromechanical switches and to facilitate error correction in digital communications such as digital terrestrial television and some cable TV systems. The use of Gray code in these devices helps simplify logic operations and reduce errors in practice. [ 3 ]
Many devices indicate position by closing and opening switches. If that device uses natural binary codes , positions 3 and 4 are next to each other but all three bits of the binary representation differ:
The problem with natural binary codes is that physical switches are not ideal: it is very unlikely that physical switches will change states exactly in synchrony. In the transition between the two states shown above, all three switches change state. In the brief period while all are changing, the switches will read some spurious position. Even without keybounce , the transition might look like 011 — 001 — 101 — 100 . When the switches appear to be in position 001 , the observer cannot tell if that is the "real" position 1, or a transitional state between two other positions. If the output feeds into a sequential system, possibly via combinational logic , then the sequential system may store a false value.
This problem can be solved by changing only one switch at a time, so there is never any ambiguity of position, resulting in codes assigning to each of a contiguous set of integers , or to each member of a circular list, a word of symbols such that no two code words are identical and each two adjacent code words differ by exactly one symbol. These codes are also known as unit-distance , [ 4 ] [ 5 ] [ 6 ] [ 7 ] [ 8 ] single-distance , single-step , monostrophic [ 9 ] [ 10 ] [ 7 ] [ 8 ] or syncopic codes , [ 9 ] in reference to the Hamming distance of 1 between adjacent codes.
In principle, there can be more than one such code for a given word length, but the term Gray code was first applied to a particular binary code for non-negative integers, the binary-reflected Gray code , or BRGC . Bell Labs researcher George R. Stibitz described such a code in a 1941 patent application, granted in 1943. [ 11 ] [ 12 ] [ 13 ] Frank Gray introduced the term reflected binary code in his 1947 patent application, remarking that the code had "as yet no recognized name". [ 14 ] He derived the name from the fact that it "may be built up from the conventional binary code by a sort of reflection process".
In the standard encoding of the Gray code the least significant bit follows a repetitive pattern of 2 on, 2 off (... 11001100 ...); the next digit a pattern of 4 on, 4 off; the i -th least significant bit a pattern of 2 i on 2 i off. The most significant digit is an exception to this: for an n -bit Gray code, the most significant digit follows the pattern 2 n −1 on, 2 n −1 off, which is the same (cyclic) sequence of values as for the second-most significant digit, but shifted forwards 2 n −2 places. The four-bit version of this is shown below:
For decimal 15 the code rolls over to decimal 0 with only one switch change. This is called the cyclic or adjacency property of the code. [ 15 ]
In modern digital communications , Gray codes play an important role in error correction . For example, in a digital modulation scheme such as QAM where data is typically transmitted in symbols of 4 bits or more, the signal's constellation diagram is arranged so that the bit patterns conveyed by adjacent constellation points differ by only one bit. By combining this with forward error correction capable of correcting single-bit errors, it is possible for a receiver to correct any transmission errors that cause a constellation point to deviate into the area of an adjacent point. This makes the transmission system less susceptible to noise .
Despite the fact that Stibitz described this code [ 11 ] [ 12 ] [ 13 ] before Gray, the reflected binary code was later named after Gray by others who used it. Two different 1953 patent applications use "Gray code" as an alternative name for the "reflected binary code"; [ 16 ] [ 17 ] one of those also lists "minimum error code" and "cyclic permutation code" among the names. [ 17 ] A 1954 patent application refers to "the Bell Telephone Gray code". [ 18 ] Other names include "cyclic binary code", [ 12 ] "cyclic progression code", [ 19 ] [ 12 ] "cyclic permuting binary" [ 20 ] or "cyclic permuted binary" (CPB). [ 21 ] [ 22 ]
The Gray code is sometimes misattributed to 19th century electrical device inventor Elisha Gray . [ 13 ] [ 23 ] [ 24 ] [ 25 ]
Reflected binary codes were applied to mathematical puzzles before they became known to engineers.
The binary-reflected Gray code represents the underlying scheme of the classical Chinese rings puzzle , a sequential mechanical puzzle mechanism described by the French Louis Gros in 1872. [ 26 ] [ 13 ]
It can serve as a solution guide for the Towers of Hanoi problem, based on a game by the French Édouard Lucas in 1883. [ 27 ] [ 28 ] [ 29 ] [ 30 ] Similarly, the so-called Towers of Bucharest and Towers of Klagenfurt game configurations yield ternary and pentary Gray codes. [ 31 ]
Martin Gardner wrote a popular account of the Gray code in his August 1972 "Mathematical Games" column in Scientific American . [ 32 ]
The code also forms a Hamiltonian cycle on a hypercube , where each bit is seen as one dimension.
When the French engineer Émile Baudot changed from using a 6-unit (6-bit) code to 5-unit code for his printing telegraph system, in 1875 [ 33 ] or 1876, [ 34 ] [ 35 ] he ordered the alphabetic characters on his print wheel using a reflected binary code, and assigned the codes using only three of the bits to vowels. With vowels and consonants sorted in their alphabetical order, [ 36 ] [ 37 ] [ 38 ] and other symbols appropriately placed, the 5-bit character code has been recognized as a reflected binary code. [ 13 ] This code became known as Baudot code [ 39 ] and, with minor changes, was eventually adopted as International Telegraph Alphabet No. 1 (ITA1, CCITT-1) in 1932. [ 40 ] [ 41 ] [ 38 ]
About the same time, the German-Austrian Otto Schäffler [ de ] [ 42 ] demonstrated another printing telegraph in Vienna using a 5-bit reflected binary code for the same purpose, in 1874. [ 43 ] [ 13 ]
Frank Gray , who became famous for inventing the signaling method that came to be used for compatible color television, invented a method to convert analog signals to reflected binary code groups using vacuum tube -based apparatus. Filed in 1947, the method and apparatus were granted a patent in 1953, [ 14 ] and the name of Gray stuck to the codes. The " PCM tube " apparatus that Gray patented was made by Raymond W. Sears of Bell Labs, working with Gray and William M. Goodall, who credited Gray for the idea of the reflected binary code. [ 44 ]
Gray was most interested in using the codes to minimize errors in converting analog signals to digital; his codes are still used today for this purpose.
Gray codes are used in linear and rotary position encoders ( absolute encoders and quadrature encoders ) in preference to weighted binary encoding. This avoids the possibility that, when multiple bits change in the binary representation of a position, a misread will result from some of the bits changing before others.
For example, some rotary encoders provide a disk which has an electrically conductive Gray code pattern on concentric rings (tracks). Each track has a stationary metal spring contact that provides electrical contact to the conductive code pattern. Together, these contacts produce output signals in the form of a Gray code. Other encoders employ non-contact mechanisms based on optical or magnetic sensors to produce the Gray code output signals.
Regardless of the mechanism or precision of a moving encoder, position measurement error can occur at specific positions (at code boundaries) because the code may be changing at the exact moment it is read (sampled). A binary output code could cause significant position measurement errors because it is impossible to make all bits change at exactly the same time. If, at the moment the position is sampled, some bits have changed and others have not, the sampled position will be incorrect. In the case of absolute encoders, the indicated position may be far away from the actual position and, in the case of incremental encoders, this can corrupt position tracking.
In contrast, the Gray code used by position encoders ensures that the codes for any two consecutive positions will differ by only one bit and, consequently, only one bit can change at a time. In this case, the maximum position error will be small, indicating a position adjacent to the actual position.
Due to the Hamming distance properties of Gray codes, they are sometimes used in genetic algorithms . [ 15 ] They are very useful in this field, since mutations in the code allow for mostly incremental changes, but occasionally a single bit-change can cause a big leap and lead to new properties.
Gray codes are also used in labelling the axes of Karnaugh maps since 1953 [ 45 ] [ 46 ] [ 47 ] as well as in Händler circle graphs since 1958, [ 48 ] [ 49 ] [ 50 ] [ 51 ] both graphical methods for logic circuit minimization .
In modern digital communications , 1D- and 2D-Gray codes play an important role in error prevention before applying an error correction . For example, in a digital modulation scheme such as QAM where data is typically transmitted in symbols of 4 bits or more, the signal's constellation diagram is arranged so that the bit patterns conveyed by adjacent constellation points differ by only one bit. By combining this with forward error correction capable of correcting single-bit errors, it is possible for a receiver to correct any transmission errors that cause a constellation point to deviate into the area of an adjacent point. This makes the transmission system less susceptible to noise .
Digital logic designers use Gray codes extensively for passing multi-bit count information between synchronous logic that operates at different clock frequencies. The logic is considered operating in different "clock domains". It is fundamental to the design of large chips that operate with many different clocking frequencies.
If a system has to cycle sequentially through all possible combinations of on-off states of some set of controls, and the changes of the controls require non-trivial expense (e.g. time, wear, human work), a Gray code minimizes the number of setting changes to just one change for each combination of states. An example would be testing a piping system for all combinations of settings of its manually operated valves.
A balanced Gray code can be constructed, [ 52 ] that flips every bit equally often. Since bit-flips are evenly distributed, this is optimal in the following way: balanced Gray codes minimize the maximal count of bit-flips for each digit.
George R. Stibitz utilized a reflected binary code in a binary pulse counting device in 1941 already. [ 11 ] [ 12 ] [ 13 ]
A typical use of Gray code counters is building a FIFO (first-in, first-out) data buffer that has read and write ports that exist in different clock domains. The input and output counters inside such a dual-port FIFO are often stored using Gray code to prevent invalid transient states from being captured when the count crosses clock domains. [ 53 ] The updated read and write pointers need to be passed between clock domains when they change, to be able to track FIFO empty and full status in each domain. Each bit of the pointers is sampled non-deterministically for this clock domain transfer. So for each bit, either the old value or the new value is propagated. Therefore, if more than one bit in the multi-bit pointer is changing at the sampling point, a "wrong" binary value (neither new nor old) can be propagated. By guaranteeing only one bit can be changing, Gray codes guarantee that the only possible sampled values are the new or old multi-bit value. Typically Gray codes of power-of-two length are used.
Sometimes digital buses in electronic systems are used to convey quantities that can only increase or decrease by one at a time, for example the output of an event counter which is being passed between clock domains or to a digital-to-analog converter. The advantage of Gray codes in these applications is that differences in the propagation delays of the many wires that represent the bits of the code cannot cause the received value to go through states that are out of the Gray code sequence. This is similar to the advantage of Gray codes in the construction of mechanical encoders, however the source of the Gray code is an electronic counter in this case. The counter itself must count in Gray code, or if the counter runs in binary then the output value from the counter must be reclocked after it has been converted to Gray code, because when a value is converted from binary to Gray code, [ nb 1 ] it is possible that differences in the arrival times of the binary data bits into the binary-to-Gray conversion circuit will mean that the code could go briefly through states that are wildly out of sequence. Adding a clocked register after the circuit that converts the count value to Gray code may introduce a clock cycle of latency, so counting directly in Gray code may be advantageous. [ 54 ]
To produce the next count value in a Gray-code counter, it is necessary to have some combinational logic that will increment the current count value that is stored. One way to increment a Gray code number is to convert it into ordinary binary code, [ 55 ] add one to it with a standard binary adder, and then convert the result back to Gray code. [ 56 ] Other methods of counting in Gray code are discussed in a report by Robert W. Doran , including taking the output from the first latches of the master-slave flip flops in a binary ripple counter. [ 57 ]
As the execution of program code typically causes an instruction memory access pattern of locally consecutive addresses, bus encodings using Gray code addressing instead of binary addressing can reduce the number of state changes of the address bits significantly, thereby reducing the CPU power consumption in some low-power designs. [ 58 ] [ 59 ]
The binary-reflected Gray code list for n bits can be generated recursively from the list for n − 1 bits by reflecting the list (i.e. listing the entries in reverse order), prefixing the entries in the original list with a binary 0 , prefixing the entries in the reflected list with a binary 1 , and then concatenating the original list with the reversed list. [ 13 ] For example, generating the n = 3 list from the n = 2 list:
The one-bit Gray code is G 1 = ( 0,1 ). This can be thought of as built recursively as above from a zero-bit Gray code G 0 = ( Λ ) consisting of a single entry of zero length. This iterative process of generating G n +1 from G n makes the following properties of the standard reflecting code clear:
These characteristics suggest a simple and fast method of translating a binary value into the corresponding Gray code. Each bit is inverted if the next higher bit of the input value is set to one. This can be performed in parallel by a bit-shift and exclusive-or operation if they are available: the n th Gray code is obtained by computing n ⊕ ⌊ n 2 ⌋ {\displaystyle n\oplus \left\lfloor {\tfrac {n}{2}}\right\rfloor } . Prepending a 0 bit leaves the order of the code words unchanged, prepending a 1 bit reverses the order of the code words. If the bits at position i {\displaystyle i} of codewords are inverted, the order of neighbouring blocks of 2 i {\displaystyle 2^{i}} codewords is reversed. For example, if bit 0 is inverted in a 3 bit codeword sequence, the order of two neighbouring codewords is reversed
If bit 1 is inverted, blocks of 2 codewords change order:
If bit 2 is inverted, blocks of 4 codewords reverse order:
Thus, performing an exclusive or on a bit b i {\displaystyle b_{i}} at position i {\displaystyle i} with the bit b i + 1 {\displaystyle b_{i+1}} at position i + 1 {\displaystyle i+1} leaves the order of codewords intact if b i + 1 = 0 {\displaystyle b_{i+1}={\mathtt {0}}} , and reverses the order of blocks of 2 i + 1 {\displaystyle 2^{i+1}} codewords if b i + 1 = 1 {\displaystyle b_{i+1}={\mathtt {1}}} . Now, this is exactly the same operation as the reflect-and-prefix method to generate the Gray code.
A similar method can be used to perform the reverse translation, but the computation of each bit depends on the computed value of the next higher bit so it cannot be performed in parallel. Assuming g i {\displaystyle g_{i}} is the i {\displaystyle i} th Gray-coded bit ( g 0 {\displaystyle g_{0}} being the most significant bit), and b i {\displaystyle b_{i}} is the i {\displaystyle i} th binary-coded bit ( b 0 {\displaystyle b_{0}} being the most-significant bit), the reverse translation can be given recursively: b 0 = g 0 {\displaystyle b_{0}=g_{0}} , and b i = g i ⊕ b i − 1 {\displaystyle b_{i}=g_{i}\oplus b_{i-1}} . Alternatively, decoding a Gray code into a binary number can be described as a prefix sum of the bits in the Gray code, where each individual summation operation in the prefix sum is performed modulo two.
To construct the binary-reflected Gray code iteratively, at step 0 start with the c o d e 0 = 0 {\displaystyle \mathrm {code} _{0}={\mathtt {0}}} , and at step i > 0 {\displaystyle i>0} find the bit position of the least significant 1 in the binary representation of i {\displaystyle i} and flip the bit at that position in the previous code c o d e i − 1 {\displaystyle \mathrm {code} _{i-1}} to get the next code c o d e i {\displaystyle \mathrm {code} _{i}} . The bit positions start 0, 1, 0, 2, 0, 1, 0, 3, ... [ nb 2 ] See find first set for efficient algorithms to compute these values.
The following functions in C convert between binary numbers and their associated Gray codes. While it may seem that Gray-to-binary conversion requires each bit to be handled one at a time, faster algorithms exist. [ 60 ] [ 55 ] [ nb 1 ]
On newer processors, the number of ALU instructions in the decoding step can be reduced by taking advantage of the CLMUL instruction set . If MASK is the constant binary string of ones ended with a single zero digit, then carryless multiplication of MASK with the grey encoding of x will always give either x or its bitwise negation.
In practice, "Gray code" almost always refers to a binary-reflected Gray code (BRGC). However, mathematicians have discovered other kinds of Gray codes. Like BRGCs, each consists of a list of words, where each word differs from the next in only one digit (each word has a Hamming distance of 1 from the next word).
It is possible to construct binary Gray codes with n bits with a length of less than 2 n , if the length is even. One possibility is to start with a balanced Gray code and remove pairs of values at either the beginning and the end, or in the middle. [ 61 ] OEIS sequence A290772 [ 62 ] gives the number of possible Gray sequences of length 2 n that include zero and use the minimum number of bits.
0 → 000 1 → 001 2 → 002 10 → 012 11 → 011 12 → 010 20 → 020 21 → 021 22 → 022 100 → 122 101 → 121 102 → 120 110 → 110 111 → 111 112 → 112 120 → 102 121 → 101 122 → 100 200 → 200 201 → 201 202 → 202 210 → 212 211 → 211 212 → 210 220 → 220 221 → 221
There are many specialized types of Gray codes other than the binary-reflected Gray code. One such type of Gray code is the n -ary Gray code , also known as a non-Boolean Gray code . As the name implies, this type of Gray code uses non- Boolean values in its encodings.
For example, a 3-ary ( ternary ) Gray code would use the values 0,1,2. [ 31 ] The ( n , k )- Gray code is the n -ary Gray code with k digits. [ 63 ] The sequence of elements in the (3, 2)-Gray code is: 00,01,02,12,11,10,20,21,22. The ( n , k )-Gray code may be constructed recursively, as the BRGC, or may be constructed iteratively . An algorithm to iteratively generate the ( N , k )-Gray code is presented (in C ):
There are other Gray code algorithms for ( n , k )-Gray codes. The ( n , k )-Gray code produced by the above algorithm is always cyclical; some algorithms, such as that by Guan, [ 63 ] lack this property when k is odd. On the other hand, while only one digit at a time changes with this method, it can change by wrapping (looping from n − 1 to 0). In Guan's algorithm, the count alternately rises and falls, so that the numeric difference between two Gray code digits is always one.
Gray codes are not uniquely defined, because a permutation of the columns of such a code is a Gray code too. The above procedure produces a code in which the lower the significance of a digit, the more often it changes, making it similar to normal counting methods.
See also Skew binary number system , a variant ternary number system where at most two digits change on each increment, as each increment can be done with at most one digit carry operation.
Although the binary reflected Gray code is useful in many scenarios, it is not optimal in certain cases because of a lack of "uniformity". [ 52 ] In balanced Gray codes , the number of changes in different coordinate positions are as close as possible. To make this more precise, let G be an R -ary complete Gray cycle having transition sequence ( δ k ) {\displaystyle (\delta _{k})} ; the transition counts ( spectrum ) of G are the collection of integers defined by
λ k = | { j ∈ Z R n : δ j = k } | , for k ∈ Z n {\displaystyle \lambda _{k}=|\{j\in \mathbb {Z} _{R^{n}}:\delta _{j}=k\}|\,,{\text{ for }}k\in \mathbb {Z} _{n}}
A Gray code is uniform or uniformly balanced if its transition counts are all equal, in which case we have λ k = R n n {\displaystyle \lambda _{k}={\tfrac {R^{n}}{n}}} for all k . Clearly, when R = 2 {\displaystyle R=2} , such codes exist only if n is a power of 2. [ 64 ] If n is not a power of 2, it is possible to construct well-balanced binary codes where the difference between two transition counts is at most 2; so that (combining both cases) every transition count is either 2 ⌊ 2 n 2 n ⌋ {\displaystyle 2\left\lfloor {\tfrac {2^{n}}{2n}}\right\rfloor } or 2 ⌈ 2 n 2 n ⌉ {\displaystyle 2\left\lceil {\tfrac {2^{n}}{2n}}\right\rceil } . [ 52 ] Gray codes can also be exponentially balanced if all of their transition counts are adjacent powers of two, and such codes exist for every power of two. [ 65 ]
For example, a balanced 4-bit Gray code has 16 transitions, which can be evenly distributed among all four positions (four transitions per position), making it uniformly balanced: [ 52 ]
whereas a balanced 5-bit Gray code has a total of 32 transitions, which cannot be evenly distributed among the positions. In this example, four positions have six transitions each, and one has eight: [ 52 ]
We will now show a construction [ 66 ] and implementation [ 67 ] for well-balanced binary Gray codes which allows us to generate an n -digit balanced Gray code for every n . The main principle is to inductively construct an ( n + 2)-digit Gray code G ′ {\displaystyle G'} given an n -digit Gray code G in such a way that the balanced property is preserved. To do this, we consider partitions of G = g 0 , … , g 2 n − 1 {\displaystyle G=g_{0},\ldots ,g_{2^{n}-1}} into an even number L of non-empty blocks of the form
{ g 0 } , { g 1 , … , g k 2 } , { g k 2 + 1 , … , g k 3 } , … , { g k L − 2 + 1 , … , g − 2 } , { g − 1 } {\displaystyle \left\{g_{0}\right\},\left\{g_{1},\ldots ,g_{k_{2}}\right\},\left\{g_{k_{2}+1},\ldots ,g_{k_{3}}\right\},\ldots ,\left\{g_{k_{L-2}+1},\ldots ,g_{-2}\right\},\left\{g_{-1}\right\}}
where k 1 = 0 {\displaystyle k_{1}=0} , k L − 1 = − 2 {\displaystyle k_{L-1}=-2} , and k L ≡ − 1 ( mod 2 n ) {\displaystyle k_{L}\equiv -1{\pmod {2^{n}}}} ). This partition induces an ( n + 2 ) {\displaystyle (n+2)} -digit Gray code given by
If we define the transition multiplicities
m i = | { j : δ k j = i , 1 ≤ j ≤ L } | {\displaystyle m_{i}=\left|\left\{j:\delta _{k_{j}}=i,1\leq j\leq L\right\}\right|}
to be the number of times the digit in position i changes between consecutive blocks in a partition, then for the ( n + 2)-digit Gray code induced by this partition the transition spectrum λ i ′ {\displaystyle \lambda '_{i}} is
λ i ′ = { 4 λ i − 2 m i , if 0 ≤ i < n L , otherwise {\displaystyle \lambda '_{i}={\begin{cases}4\lambda _{i}-2m_{i},&{\text{if }}0\leq i<n\\L,&{\text{ otherwise }}\end{cases}}}
The delicate part of this construction is to find an adequate partitioning of a balanced n -digit Gray code such that the code induced by it remains balanced, but for this only the transition multiplicities matter; joining two consecutive blocks over a digit i {\displaystyle i} transition and splitting another block at another digit i {\displaystyle i} transition produces a different Gray code with exactly the same transition spectrum λ i ′ {\displaystyle \lambda '_{i}} , so one may for example [ 65 ] designate the first m i {\displaystyle m_{i}} transitions at digit i {\displaystyle i} as those that fall between two blocks. Uniform codes can be found when R ≡ 0 ( mod 4 ) {\displaystyle R\equiv 0{\pmod {4}}} and R n ≡ 0 ( mod n ) {\displaystyle R^{n}\equiv 0{\pmod {n}}} , and this construction can be extended to the R -ary case as well. [ 66 ]
Long run (or maximum gap ) Gray codes maximize the distance between consecutive changes of digits in the same position. That is, the minimum run-length of any bit remains unchanged for as long as possible. [ 68 ]
Monotonic codes are useful in the theory of interconnection networks, especially for minimizing dilation for linear arrays of processors. [ 69 ] If we define the weight of a binary string to be the number of 1s in the string, then although we clearly cannot have a Gray code with strictly increasing weight, we may want to approximate this by having the code run through two adjacent weights before reaching the next one.
We can formalize the concept of monotone Gray codes as follows: consider the partition of the hypercube Q n = ( V n , E n ) {\displaystyle Q_{n}=(V_{n},E_{n})} into levels of vertices that have equal weight, i.e.
V n ( i ) = { v ∈ V n : v has weight i } {\displaystyle V_{n}(i)=\{v\in V_{n}:v{\text{ has weight }}i\}}
for 0 ≤ i ≤ n {\displaystyle 0\leq i\leq n} . These levels satisfy | V n ( i ) | = ( n i ) {\displaystyle |V_{n}(i)|=\textstyle {\binom {n}{i}}} . Let Q n ( i ) {\displaystyle Q_{n}(i)} be the subgraph of Q n {\displaystyle Q_{n}} induced by V n ( i ) ∪ V n ( i + 1 ) {\displaystyle V_{n}(i)\cup V_{n}(i+1)} , and let E n ( i ) {\displaystyle E_{n}(i)} be the edges in Q n ( i ) {\displaystyle Q_{n}(i)} . A monotonic Gray code is then a Hamiltonian path in Q n {\displaystyle Q_{n}} such that whenever δ 1 ∈ E n ( i ) {\displaystyle \delta _{1}\in E_{n}(i)} comes before δ 2 ∈ E n ( j ) {\displaystyle \delta _{2}\in E_{n}(j)} in the path, then i ≤ j {\displaystyle i\leq j} .
An elegant construction of monotonic n -digit Gray codes for any n is based on the idea of recursively building subpaths P n , j {\displaystyle P_{n,j}} of length 2 ( n j ) {\displaystyle 2\textstyle {\binom {n}{j}}} having edges in E n ( j ) {\displaystyle E_{n}(j)} . [ 69 ] We define P 1 , 0 = ( 0 , 1 ) {\displaystyle P_{1,0}=({\mathtt {0}},{\mathtt {1}})} , P n , j = ∅ {\displaystyle P_{n,j}=\emptyset } whenever j < 0 {\displaystyle j<0} or j ≥ n {\displaystyle j\geq n} , and
P n + 1 , j = 1 P n , j − 1 π n , 0 P n , j {\displaystyle P_{n+1,j}={\mathtt {1}}P_{n,j-1}^{\pi _{n}},{\mathtt {0}}P_{n,j}}
otherwise. Here, π n {\displaystyle \pi _{n}} is a suitably defined permutation and P π {\displaystyle P^{\pi }} refers to the path P with its coordinates permuted by π {\displaystyle \pi } . These paths give rise to two monotonic n -digit Gray codes G n ( 1 ) {\displaystyle G_{n}^{(1)}} and G n ( 2 ) {\displaystyle G_{n}^{(2)}} given by
G n ( 1 ) = P n , 0 P n , 1 R P n , 2 P n , 3 R ⋯ and G n ( 2 ) = P n , 0 R P n , 1 P n , 2 R P n , 3 ⋯ {\displaystyle G_{n}^{(1)}=P_{n,0}P_{n,1}^{R}P_{n,2}P_{n,3}^{R}\cdots {\text{ and }}G_{n}^{(2)}=P_{n,0}^{R}P_{n,1}P_{n,2}^{R}P_{n,3}\cdots }
The choice of π n {\displaystyle \pi _{n}} which ensures that these codes are indeed Gray codes turns out to be π n = E − 1 ( π n − 1 2 ) {\displaystyle \pi _{n}=E^{-1}\left(\pi _{n-1}^{2}\right)} . The first few values of P n , j {\displaystyle P_{n,j}} are shown in the table below.
These monotonic Gray codes can be efficiently implemented in such a way that each subsequent element can be generated in O ( n ) time. The algorithm is most easily described using coroutines .
Monotonic codes have an interesting connection to the Lovász conjecture , which states that every connected vertex-transitive graph contains a Hamiltonian path. The "middle-level" subgraph Q 2 n + 1 ( n ) {\displaystyle Q_{2n+1}(n)} is vertex-transitive (that is, its automorphism group is transitive, so that each vertex has the same "local environment" and cannot be differentiated from the others, since we can relabel the coordinates as well as the binary digits to obtain an automorphism ) and the problem of finding a Hamiltonian path in this subgraph is called the "middle-levels problem", which can provide insights into the more general conjecture. The question has been answered affirmatively for n ≤ 15 {\displaystyle n\leq 15} , and the preceding construction for monotonic codes ensures a Hamiltonian path of length at least 0.839 N , where N is the number of vertices in the middle-level subgraph. [ 70 ]
Another type of Gray code, the Beckett–Gray code , is named for Irish playwright Samuel Beckett , who was interested in symmetry . His play " Quad " features four actors and is divided into sixteen time periods. Each period ends with one of the four actors entering or leaving the stage. The play begins and ends with an empty stage, and Beckett wanted each subset of actors to appear on stage exactly once. [ 71 ] Clearly the set of actors currently on stage can be represented by a 4-bit binary Gray code. Beckett, however, placed an additional restriction on the script: he wished the actors to enter and exit so that the actor who had been on stage the longest would always be the one to exit. The actors could then be represented by a first in, first out queue , so that (of the actors onstage) the actor being dequeued is always the one who was enqueued first. [ 71 ] Beckett was unable to find a Beckett–Gray code for his play, and indeed, an exhaustive listing of all possible sequences reveals that no such code exists for n = 4. It is known today that such codes do exist for n = 2, 5, 6, 7, and 8, and do not exist for n = 3 or 4. An example of an 8-bit Beckett–Gray code can be found in Donald Knuth 's Art of Computer Programming . [ 13 ] According to Sawada and Wong, the search space for n = 6 can be explored in 15 hours, and more than 9500 solutions for the case n = 7 have been found. [ 72 ]
Snake-in-the-box codes, or snakes , are the sequences of nodes of induced paths in an n -dimensional hypercube graph , and coil-in-the-box codes, [ 73 ] or coils , are the sequences of nodes of induced cycles in a hypercube. Viewed as Gray codes, these sequences have the property of being able to detect any single-bit coding error. Codes of this type were first described by William H. Kautz in the late 1950s; [ 5 ] since then, there has been much research on finding the code with the largest possible number of codewords for a given hypercube dimension.
Yet another kind of Gray code is the single-track Gray code (STGC) developed by Norman B. Spedding [ 74 ] [ 75 ] and refined by Hiltgen, Paterson and Brandestini in Single-track Gray Codes (1996). [ 76 ] [ 77 ] The STGC is a cyclical list of P unique binary encodings of length n such that two consecutive words differ in exactly one position, and when the list is examined as a P × n matrix , each column is a cyclic shift of the first column. [ 78 ]
The name comes from their use with rotary encoders , where a number of tracks are being sensed by contacts, resulting for each in an output of 0 or 1 . To reduce noise due to different contacts not switching at exactly the same moment in time, one preferably sets up the tracks so that the data output by the contacts are in Gray code. To get high angular accuracy, one needs lots of contacts; in order to achieve at least 1° accuracy, one needs at least 360 distinct positions per revolution, which requires a minimum of 9 bits of data, and thus the same number of contacts.
If all contacts are placed at the same angular position, then 9 tracks are needed to get a standard BRGC with at least 1° accuracy. However, if the manufacturer moves a contact to a different angular position (but at the same distance from the center shaft), then the corresponding "ring pattern" needs to be rotated the same angle to give the same output. If the most significant bit (the inner ring in Figure 1) is rotated enough, it exactly matches the next ring out. Since both rings are then identical, the inner ring can be cut out, and the sensor for that ring moved to the remaining, identical ring (but offset at that angle from the other sensor on that ring). Those two sensors on a single ring make a quadrature encoder. That reduces the number of tracks for a "1° resolution" angular encoder to 8 tracks. Reducing the number of tracks still further cannot be done with BRGC.
For many years, Torsten Sillke [ 79 ] and other mathematicians believed that it was impossible to encode position on a single track such that consecutive positions differed at only a single sensor, except for the 2-sensor, 1-track quadrature encoder. So for applications where 8 tracks were too bulky, people used single-track incremental encoders (quadrature encoders) or 2-track "quadrature encoder + reference notch" encoders.
Norman B. Spedding, however, registered a patent in 1994 with several examples showing that it was possible. [ 74 ] Although it is not possible to distinguish 2 n positions with n sensors on a single track, it is possible to distinguish close to that many. Etzion and Paterson conjecture that when n is itself a power of 2, n sensors can distinguish at most 2 n − 2 n positions and that for prime n the limit is 2 n − 2 positions. [ 80 ] The authors went on to generate a 504-position single track code of length 9 which they believe is optimal. Since this number is larger than 2 8 = 256, more than 8 sensors are required by any code, although a BRGC could distinguish 512 positions with 9 sensors.
An STGC for P = 30 and n = 5 is reproduced here:
Each column is a cyclic shift of the first column, and from any row to the next row only one bit changes. [ 81 ] The single-track nature (like a code chain) is useful in the fabrication of these wheels (compared to BRGC), as only one track is needed, thus reducing their cost and size.
The Gray code nature is useful (compared to chain codes , also called De Bruijn sequences ), as only one sensor will change at any one time, so the uncertainty during a transition between two discrete states will only be plus or minus one unit of angular measurement the device is capable of resolving. [ 82 ]
Since this 30 degree example was added, there has been a lot of interest in examples with higher angular resolution. In 2008, Gary Williams, [ 83 ] [ user-generated source? ] based on previous work, [ 80 ] discovered a 9-bit single track Gray code that gives a 1 degree resolution. This Gray code was used to design an actual device which was published on the site Thingiverse . This device [ 84 ] was designed by etzenseep (Florian Bauer) in September 2022.
An STGC for P = 360 and n = 9 is reproduced here:
Two-dimensional Gray codes are used in communication to minimize the number of bit errors in quadrature amplitude modulation (QAM) adjacent points in the constellation . In a typical encoding the horizontal and vertical adjacent constellation points differ by a single bit, and diagonal adjacent points differ by 2 bits. [ 85 ]
Two-dimensional Gray codes also have uses in location identifications schemes, where the code would be applied to area maps such as a Mercator projection of the earth's surface and an appropriate cyclic two-dimensional distance function such as the Mannheim metric be used to calculate the distance between two encoded locations, thereby combining the characteristics of the Hamming distance with the cyclic continuation of a Mercator projection. [ 86 ]
If a subsection of a specific codevalue is extracted from that value, for example the last 3 bits of a 4-bit Gray code, the resulting code will be an "excess Gray code". This code shows the property of counting backwards in those extracted bits if the original value is further increased. Reason for this is that Gray-encoded values do not show the behaviour of overflow, known from classic binary encoding, when increasing past the "highest" value.
Example: The highest 3-bit Gray code, 7, is encoded as (0)100. Adding 1 results in number 8, encoded in Gray as 1100. The last 3 bits do not overflow and count backwards if you further increase the original 4 bit code.
When working with sensors that output multiple, Gray-encoded values in a serial fashion, one should therefore pay attention whether the sensor produces those multiple values encoded in 1 single Gray code or as separate ones, as otherwise the values might appear to be counting backwards when an "overflow" is expected.
The bijective mapping { 0 ↔ 00 , 1 ↔ 01 , 2 ↔ 11 , 3 ↔ 10 } establishes an isometry between the metric space over the finite field Z 2 2 {\displaystyle \mathbb {Z} _{2}^{2}} with the metric given by the Hamming distance and the metric space over the finite ring Z 4 {\displaystyle \mathbb {Z} _{4}} (the usual modular arithmetic ) with the metric given by the Lee distance . The mapping is suitably extended to an isometry of the Hamming spaces Z 2 2 m {\displaystyle \mathbb {Z} _{2}^{2m}} and Z 4 m {\displaystyle \mathbb {Z} _{4}^{m}} . Its importance lies in establishing a correspondence between various "good" but not necessarily linear codes as Gray-map images in Z 2 2 {\displaystyle \mathbb {Z} _{2}^{2}} of ring-linear codes from Z 4 {\displaystyle \mathbb {Z} _{4}} . [ 87 ] [ 88 ]
There are a number of binary codes similar to Gray codes, including:
The following binary-coded decimal (BCD) codes are Gray code variants as well:
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Vargulin , [ 1 ] also called Cypridinid luciferin , [ 2 ] Cypridina luciferin , or Vargula luciferin , is the luciferin found in the ostracod Cypridina hilgendorfii , also named Vargula hilgendorfii . [ 3 ] These bottom dwelling ostracods emit a light stream into water when disturbed presumably to deter predation. Vargulin is also used by the midshipman fish, Porichthys .
A partial extraction procedure was developed in 1935 which involved reacting the compound with benzoyl chloride to allow it to be separated from the water-soluble components. [ 4 ] The compound was first isolated and purified to crystals by Osamu Shimomura . [ 5 ] The structure of the compound was confirmed some years later. [ 6 ] Feeding experiments suggest that the compound is synthesized in the animal from three amino-acids: tryptophan , isoleucine , and arginine . [ 7 ]
Vargulin is oxidized by the Vargula luciferase , [ 8 ] a 62 kDa enzyme , to produce blue light at 462 nm (max emission, detected with a 425 to 525 nm filter).
The vargulin does not cross react with luciferases using coelenterazine or Firefly luciferin .
Vargulin (with the associated luciferase) has applications in biotechnology :
Although less stable, the Cypridina system is useful because can be used in multiplex assays with other (red-emitting) luciferin assays.
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The variability hypothesis , also known as the greater male variability hypothesis , is the hypothesis that human males generally display greater variability in traits than human females do.
It has often been discussed in relation to human cognitive ability , where some studies appear to show that males are more likely than females to have either very high or very low IQ test scores . In this context, there is controversy over whether such sex-based differences in the variability of intelligence exist, and if so, whether they are caused by genetic differences, environmental conditioning, or a mixture of both.
Sex-differences in variability have been observed in many abilities and traits – including physical, psychological and genetic ones – across a wide range of sexually dimorphic species. On the genetic level, the greater phenotype variability in males is likely to be associated with human males being a heterogametic sex, while females are homogametic and thus are more likely to display averaged traits in their phenotype. [ 1 ]
The notion of greater male variability—at least in respect to physical characteristics—can be traced back to the writings of Charles Darwin . [ 2 ] When he expounded his theory of sexual selection in The Descent of Man and Selection in Relation to Sex , Darwin cites some observations made by his contemporaries. For example, he highlights findings from the Novara Expedition of 1861–1867 where "a vast number of measurements of various parts of the body in different races were made, and the men were found in almost every case to present a greater range of variation than the women" (p. 275). To Darwin, the evidence from the medical community at the time, which suggested a greater prevalence of physical abnormalities among men than women, was also indicative of men's greater physical variability.
Although Darwin was curious about sex differences in variability throughout the animal kingdom, variability in humans was not a chief concern of his research. The first scholar to carry out a detailed empirical investigation on the question of human sex differences in variability in both physical and mental faculties, was the sexologist Havelock Ellis . In his 1894 publication Man and Woman: A Study of Human Secondary Sexual Characters , Ellis dedicated an entire chapter to the subject, entitled "The Variational Tendency of Men". [ 3 ] In this chapter he posits that "both the physical and mental characters of men show wider limits of variation than do the physical and mental characters of women" (p. 358). Ellis documents several studies that support this assertion (see pp. 360–367), and
The publication of Ellis's Man and Woman led to an intellectual dispute about the variability hypothesis between Ellis and the statistician Karl Pearson , whose critique of Ellis's work was both theoretical and methodological. After Pearson dismissed Ellis's conclusions, he then "presented his own data to show that it was the female who was more variable than the male" [ 4 ] Ellis wrote a letter to Pearson thanking him for the criticisms which would allow him to present his arguments "more clearly & precisely than before", but did not yield his position regarding greater male variability. [ 4 ]
Support for the greater male variability hypothesis grew during the early part of the 20th century. [ 2 ] During this period, the attention of researchers shifted towards studying variability in mental abilities partly due to the advent of standardised mental tests (see the history of the Intelligence quotient ), which made it possible to examine intelligence with greater objectivity and precision.
One advocate of greater male variability during this time was the American psychologist Edward Thorndike , one of the leading exponents of mental testing who played an instrumental role in the development of today's Armed Services Vocational Aptitude Battery ASVAB . In his 1906 publication Sex in Education , Thorndike argued that while mean level sex differences in intellectual ability appeared to be negligible, sex differences in variability were clear. [ 2 ] Other influential proponents of the hypothesis at this time were psychologists G. Stanley Hall and James McKeen Cattell . [ 5 ] [ 6 ] [ 7 ] Thorndike believed that variability in intelligence could have a biological basis and suggested that this could have important implications for achievement and pedagogy. For example, he postulated that greater male variation could mean "eminence and leadership of the world's affairs of whatever sort will inevitably belong oftener to men." [ 8 ] In addition, since the number of women that fall within the extreme top-end of the intelligence distribution would be inherently smaller, he suggested that educational resources should be invested in preparing women for roles and occupations that require only a mediocre level of cognitive ability. [ 9 ]
By examining the case records of 1,000 patients at the Clearing House for Mental Defectives, Leta Hollingworth determined that, although men outnumbered women in the clearing house, the ratio of men to women decreased with age. Hollingworth explained this to be the result of men facing greater societal expectations than women. Consequently, deficiencies in men were often detected at an earlier age, while similar deficiencies in women might not be detected because less was expected of them. Therefore, deficiencies in women would be required to be more pronounced than those in men in order to be detected at similar ages. [ 5 ] [ 6 ] [ 9 ] [ 10 ] [ 7 ]
Hollingworth also attacked the variability hypothesis theoretically, criticizing the underlying logic of the hypothesis. [ 5 ] [ 6 ] [ 9 ] [ 11 ] Hollingworth argued that the variability hypothesis was flawed because: (1) it had not been empirically established that men were more anatomically variable than women, (2) even if greater anatomical variability in men were established this would not necessarily mean that men were also more variable in mental traits, (3) even if it were established that men were more variable in mental traits this would not automatically mean that men were innately more variable, (4) variability is not significant in and of itself, but rather depends on what the variability consists of, and (5) that any possible differences in variability between men and women must also be understood with reference to the fact that women lack the opportunity to achieve eminence because of their prescribed societal and cultural roles at the time. [ 5 ] [ 6 ] [ 9 ] Additionally, the argument that great variability automatically meant greater range was criticized by Hollingworth. [ 9 ] [ 12 ] [ how? ]
In an attempt to examine the validity of the variability hypothesis, while avoiding intervening social and cultural factors, Hollingworth gathered data on birth weight and length of 1,000 male and 1,000 female newborns. This research found virtually no difference in the variability of male and female infants birth weight, and it was concluded that if variability favoured any sex it was the female sex. [ 5 ] [ 6 ] [ 9 ] [ 10 ] The general consensus today is that boys have a higher birth weight than girls, and are more responsive to their mother's diet while in the womb, thus causing an even greater variation if nutritional needs are met. [ 13 ]
The 21st century has witnessed a resurgence of research on gender differences in variability, with most of the emphasis on humans. The results vary based on the type of problem, but some recent studies have found that the variability hypothesis is true for parts of IQ tests, with more men falling at the extremes of the distribution. [ 14 ] [ 15 ] Publications differ as to the extent and distribution of male variability, including on whether variability can be shown across various cultural and social factors. [ 16 ] [ 17 ]
A 2007 meta-analysis found that males are more variable on most measures of quantitative and visuospatial ability, making no conclusions of its causation. [ 18 ]
A 2008 analysis of test scores across 41 countries published in Science concluded that "data shows a higher variance in boys' than girls' results on mathematics and reading tests in most OECD countries", the results implying that "gender differences in the variance of test scores are an international phenomenon". However, it also found that several countries failed to exhibit a gender difference in variance. [ 14 ]
A 2008 study reviewed the history of the hypothesis that general intelligence is more biologically variable in males than in females and presented data which the authors claim "in many ways are the most complete that have ever been compiled [and which] substantially support the hypothesis". [ 2 ]
A 2009 study in developmental psychology examined non-cognitive traits including blood parameters and birth weight as well as certain cognitive traits, and concluded that "greater intrasex phenotype variability in males than in females is a fundamental aspect of the gender differences in humans". [ 19 ]
Recent studies indicate that greater male variability in mathematics persists in the U.S., although the ratio of boys to girls at the top end of the distribution is reversed in Asian Americans. [ 20 ] A 2010 meta-analysis of 242 studies found that males have an 8% greater variance in mathematical abilities than females, which the authors indicate is not meaningfully different from an equal variance. Additionally, they find several datasets indicate no or a reversed variance ratio. [ 21 ]
Looking for genetic evidence of variability between the sexes, a 2013 study used the body sizes of different species that either had heterogametic males ( mammals and insects ) or heterogametic females ( birds and butterflies ). The study found that heterogametic males had higher variability in body size than homogametic females, while heterogametic females had higher variability in body size than homogametic males, thus showing a relation between the chromosomes and variability between the sexes in traits. [ 22 ]
A 2014 review found that males tend to have higher variance on mathematical and verbal abilities but females tend to have higher variance on fear and emotionality; however, the differences in variance are small and without much practical significance and the causes remain unknown. [ 23 ] A 2005 meta-analysis found greater female variability on the standard Raven's Progressive Matrices , and no difference in variability on the advanced progressive matrices, but also found that males had a higher average general intelligence. [ 24 ] This meta analysis, however, was criticized for bias by the authors and for poor methodology. [ 25 ] [ 26 ] [ 27 ]
A 2016 study by Baye and Monseur examining twelve databases from the International Association for the Evaluation of Educational Achievement and the Program for International Student Assessment, were used to analyse gender differences within an international perspective from 1995 to 2015, and concluded, "The 'greater male variability hypothesis' is confirmed." [ 28 ] This study found that on average, boys showed 14% greater variance than girls in science, reading, and math test scores. In reading, boys were significantly represented at the bottom of score distribution, whereas for maths and science they featured more at the top.
The results of Baye and Monseur have been both replicated and criticized in a 2019 meta-analytical extension published by Helen Gray and her associates, which broadly confirmed that variability is greater for males internationally but that there is significant heterogeneity between countries. They also found that policies leading to greater female participation in the workforce tended to increase female variability and, therefore, decrease the variability gap. They also point out that Baye and Monseur had themselves observed a lack of international consistency, leading more support to a cultural hypothesis. [ 29 ]
A 2018 meta analysis of over 1 million school-aged children found strong evidence for higher variability in boys' grades, but for girls to receive higher grades on average, both of which the authors describe as "in line with previous studies". Due in part to the combination of these factors, they conclude that differences in variability are insufficient to explain disparities in STEM college admissions. They note: "Simulations of these differences suggest the top 10% of a class contains equal numbers of girls and boys in STEM, but more girls in non-STEM subjects." [ 30 ]
In October 2020, with respect to brain morphometry , researchers reported "the largest-ever mega-analysis of sex differences in variability of brain structure"; they stated that they "observed significant patterns of greater male than female between-subject variance for all subcortical volumetric measures, all cortical surface area measures, and 60% of cortical thickness measures. This pattern was stable across the lifespan for 50% of the subcortical structures, 70% of the regional area measures, and nearly all regions for thickness." The authors emphasize, however, that this has of yet no practical interpretive meaning, says nothing on causation, and requires further examination and replication. [ 31 ]
In 2021, two meta-analyses on preference measurement in experimental economics find strong evidence for greater male variability for cooperation (variance ratio: 1.30, 95% CI [1.22, 1.38]), [ 32 ] time preferences (1.15, [1.08, 1.22]), risk preferences (1.25 [1.13, 1.37]), dictator game offers (1.18 [1.12, 1.25]) and transfers in the trust game (1.28 [1.18, 1.39]). [ 33 ]
A 2021 study of 10 million AP calculus and statistics students from 1997 to 2019, found that although female participation in these courses has increased significantly, the proportion of males to females at the top scores in the AP math exams is still substantial, though the proportion of males to females at the top scores has been slowly decreasing. [ 34 ]
A 2021 review investigating different hypotheses behind the discrepancy of sexes in STEM jobs summarizes the greater variability research with respect to this question. Given that research finds greater variability in males within quantitative and nonverbal reasoning, [ 35 ] they hold that this can explain some, but not all of the difference seen in STEM occupations. [ 36 ] With regard to the question of whether these results are due to societal influences or of biological origins, they hold that the results showing greater variance at a very young age (for instance IQ differences in variability between the sexes is visible from a young age on [ 37 ] ) lend credence to the theory that biological factors might explain a large part of the observed data.
A 2022 analysis of a large database on energy expenditure in adult humans found that "even when statistically comparing males and females of the same age, height, and body composition, there is much more variation in total, activity, and basal energy expenditure among males". [ 38 ]
In a 1992 paper titled "Variability: A Pernicious Hypothesis," Stanford Professor Nel Noddings discussed the social history which she argued explains "the revulsion with which many feminists react to the variability hypothesis." [ 39 ]
In 2005, then Harvard President, Larry Summers , addressed the National Bureau of Economic Research Conference on the subject of gender diversity in the science and engineering professions, saying: "It does appear that on many, many different human attributes—height, weight, propensity for criminality, overall IQ, mathematical ability, scientific ability—there is relatively clear evidence that whatever the difference in means—which can be debated—there is a difference in the standard deviation, and variability of a male and a female population." [ 40 ] [ 41 ] His remarks caused a backlash; Summers faced a no-confidence vote from the Harvard faculty , prompting his resignation as President. [ 42 ] [ 43 ]
In 2017, a mathematics research paper presenting a possible evolutionary explanation for the variability hypothesis was peer-reviewed, accepted, and formally published in The New York Journal of Mathematics . Three days later, that article was removed without explanation and replaced by an unrelated article by different authors. This caused debate within the scientific community and international publicity. [ 44 ] [ 45 ] [ 46 ] A revised version was subsequently peer reviewed again and published in the Journal of Interdisciplinary Mathematics . [ 47 ]
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A variable-buoyancy pressure vessel system is a type of rigid buoyancy control device for diving systems that retains a constant volume and varies its density by changing the weight (mass) of the contents, either by moving the ambient fluid into and out of a rigid pressure vessel, or by moving a stored liquid between internal and external variable-volume containers. A pressure vessel is used to withstand the hydrostatic pressure of the underwater environment. A variable-buoyancy pressure vessel can have an internal pressure greater or less than ambient pressure , and the pressure difference can vary from positive to negative within the operational depth range, or remain either positive or negative throughout the pressure range, depending on design choices.
Variable buoyancy is a useful characteristic of any mobile underwater system that operates in mid-water without external support. [ 1 ] Examples include submarines , submersibles , benthic landers , remotely operated and autonomous underwater vehicles , and underwater divers .
Several applications only need one cycle from positive to negative and back to get down to depth and return to the surface between deployments; others may need tens to hundreds of cycles over several months during a single deployment, or continual but very small adjustments in both directions to maintain a constant depth or neutral buoyancy at changing depths. Several mechanisms are available for this function; some are suitable for multiple cycles between positive and negative buoyancy, and others must be replenished between uses. Their suitability depends on the required characteristics for the specific application.
Mobile underwater systems that operate in mid-water without external support need variable buoyancy, [ 1 ] and as such these systems are a major research topic in the field of underwater vehicles . [ 2 ] Examples include submarines, submersibles, benthic landers, remotely operated and autonomous underwater vehicles, [ 3 ] and ambient-pressure and single-atmosphere underwater divers. [ 4 ]
A submarine can closely approach equilibrium when submerged but have no inherent stability in depth. The sealed pressure hull structure is usually slightly more compressible than water and will consequently lose buoyancy with increased depth. [ 5 ] For precise and quick control of buoyancy and trim at depth, submarines use depth control tanks ( DCT )—also called hard tanks (due to their ability to withstand higher pressure) or trim tanks . These are variable-buoyancy pressure vessels. The amount of water in depth control tanks can be controlled to change the buoyancy of the vessel so that it moves up or down in the water column, or to maintain a constant depth as outside conditions (mainly water density) change, and water can be pumped between trim tanks to control longitudinal or transverse trim without affecting buoyancy. [ 6 ]
The operating depth of underwater vehicles can be controlled by controlling the buoyancy—by changing either the overall weight or the displaced volume —or by vectored thrust. Buoyancy can be controlled by changing the overall weight of the vehicle at constant volume, [ 7 ] or by changing the displaced volume at a constant vehicle weight. The resulting buoyancy is used to control heave velocity and hovering depth , [ 7 ] and in underwater gliders a positive or negative net buoyancy is used to drive forward motion.
The Avelo scuba system uses a variable-buoyancy pressure vessel, which is both the primary breathing gas cylinder and the scuba buoyancy compensator , with a rechargeable-battery–powered pump and dump valve unit which is demountable from the cylinder. [ 4 ] [ 8 ]
Variable-buoyancy systems have been considered for depth control of tethered ocean current turbine electrical generation . [ 9 ]
The type of variable-buoyancy system best suited to an application depends on the precision of control required, the amount of change needed, and the number of cycles of buoyancy change necessary during a deployment. [ 10 ]
Several types of variable-buoyancy systems have been used, and are briefly described here. Some are based on a relatively incompressible pressure vessel and are nearly stable with variation of hydrostatic pressure.
A buoyancy tank that is within the pressure hull of the vehicle, as in a submarine, will be exposed to the internal pressure of the vehicle, so external pressure loads on the tank may be relatively low. In this case the ballast water transfer into the tank may not require pumping, though a positive-displacement pump may still be useful to accurately control the volume of water admitted. Discharge of ballast water is against the external pressure, which will depend on depth, and will generally require significant work. [ 6 ]
If the buoyancy tank is directly exposed to the ambient hydrostatic pressure, the external load due to depth can be high, but if the internal gas pressure is high enough, the pressure difference will be lower, and the pressure vessel is not subjected to high net external pressure loads which can cause buckling instability, which can allow a lower structural weight. In the extreme case the internal pressure is high enough to rapidly eject the water ballast at maximum operational depth, as in the case of the Avelo integrated diving cylinder and buoyancy control device. A pump is used to move ambient water into the pressure vessel against the internal pressure, compressing the gas further in proportion to volume decrease, so the entire internal volume is not available to hold ballast, as although the gas will decrease in volume, there will always be some gas volume remaining. The water and air in the pressure vessel may be separated by a membrane or free piston to prevent pumping out air in some orientations, and to prevent the air from dissolving in the ballast water under high pressure. [ 10 ] [ 4 ]
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A variable frequency oscillator ( VFO ) in electronics is an oscillator whose frequency can be tuned (i.e., varied) over some range. [ 1 ] It is a necessary component in any tunable radio transmitter and in receivers that work by the superheterodyne principle. The oscillator controls the frequency to which the apparatus is tuned.
In a simple superheterodyne receiver , the incoming radio frequency signal (at frequency f I N {\displaystyle f_{IN}} ) from the antenna is mixed with the VFO output signal tuned to f L O {\displaystyle f_{LO}} , producing an intermediate frequency (IF) signal that can be processed downstream to extract the modulated information. Depending on the receiver design, the IF signal frequency is chosen to be either the sum of the two frequencies at the mixer inputs ( up-conversion ), f I N + f L O {\displaystyle f_{IN}+f_{LO}} or more commonly, the difference frequency (down-conversion), f I N − f L O {\displaystyle f_{IN}-f_{LO}} .
In addition to the desired IF signal and its unwanted image (the mixing product of opposite sign above), the mixer output will also contain the two original frequencies, f I N {\displaystyle f_{IN}} and f L O {\displaystyle f_{LO}} and various harmonic combinations of the input signals. These undesired signals are rejected by the IF filter . If a double balanced mixer is employed, the input signals appearing at the mixer outputs are greatly attenuated, reducing the required complexity of the IF filter.
The advantage of using a VFO as a heterodyning oscillator is that only a small portion of the radio receiver (the sections before the mixer such as the preamplifier) need to have a wide bandwidth. The rest of the receiver can be finely tuned to the IF frequency. [ 2 ]
In a direct-conversion receiver , the VFO is tuned to the same frequency as the incoming radio frequency and f I F = 0 {\displaystyle f_{IF}=0} Hz. Demodulation takes place at baseband using low-pass filters and amplifiers .
In a radio frequency (RF) transmitter , VFOs are often used to tune the frequency of the output signal, often indirectly through a heterodyning process similar to that described above. [ 1 ] Other uses include chirp generators for radar systems where the VFO is swept rapidly through a range of frequencies, [ 3 ] timing signal generation for oscilloscopes and time domain reflectometers , and variable frequency audio generators used in musical instruments and audio test equipment.
There are two main types of VFO in use: analog and digital .
An analog VFO is an electronic oscillator where the value of at least one of the passive components is adjustable under user control so as to alter its output frequency.
The passive component whose value is adjustable is usually a capacitor , but could be a variable inductor .
The variable capacitor is a mechanical device in which the separation of a series of interleaved metal plates is physically altered to vary its capacitance . Adjustment of this capacitor is sometimes facilitated by a mechanical step-down gearbox to achieve fine tuning. [ 2 ]
A reversed-biased semiconductor diode exhibits capacitance. Since the width of its non-conducting depletion region depends on the magnitude of the reverse bias voltage, this voltage can be used to control the junction capacitance. The varactor bias voltage may be generated in a number of ways and there may need to be no significant moving parts in the final design. [ 4 ] Varactors have a number of disadvantages including temperature drift and aging, electronic noise, low Q factor and non-linearity.
Modern radio receivers and transmitters usually use some form of digital frequency synthesis to generate their VFO signal.
The advantages include smaller designs, lack of moving parts, the higher stability of set frequency reference oscillators, and the ease with which preset frequencies can be stored and manipulated in the digital computer that is usually embedded in the design in any case.
It is also possible for the radio to become extremely frequency-agile in that the control computer could alter the radio's tuned frequency many tens, thousands or even millions of times a second.
This capability allows communications receivers effectively to monitor many channels at once, perhaps using digital selective calling ( DSC ) techniques to decide when to open an audio output channel and alert users to incoming communications.
Pre-programmed frequency agility also forms the basis of some military radio encryption and stealth techniques.
Extreme frequency agility lies at the heart of spread spectrum techniques that have gained mainstream acceptance in computer wireless networking such as Wi-Fi .
There are disadvantages to digital synthesis such as the inability of a digital synthesiser to tune smoothly through all frequencies, but with the channelisation of many radio bands, this can also be seen as an advantage in that it prevents radios from operating in between two recognised channels.
Digital frequency synthesis relies on stable crystal controlled reference frequency sources. Crystal-controlled oscillators are more stable than inductively and capacitively controlled oscillators. Their disadvantage is that changing frequency (more than a small amount) requires changing the crystal, but frequency synthesizer techniques have made this unnecessary in modern designs.
The electronic and digital techniques involved in this include:
The quality metrics for a VFO include frequency stability, phase noise and spectral purity. All of these factors tend to be inversely proportional to the tuning circuit's Q factor . Since in general the tuning range is also inversely proportional to Q, these performance factors generally degrade as the VFO's frequency range is increased. [ 5 ]
Stability is the measure of how far a VFO's output frequency drifts with time and temperature. [ 5 ] To mitigate this problem, VFOs are generally "phase locked" to a stable reference oscillator. PLLs use negative feedback to correct for the frequency drift of the VFO allowing for both wide tuning range and good frequency stability. [ 6 ]
Ideally, for the same control input to the VFO, the oscillator should generate exactly the same frequency. A change in the calibration of the VFO can change receiver tuning calibration; periodic re-alignment of a receiver may be needed. VFO's used as part of a phase-locked loop frequency synthesizer have less stringent requirements since the system is as stable as the crystal-controlled reference frequency.
A plot of a VFO's amplitude vs. frequency may show several peaks, probably harmonically related. Each of these peaks can potentially mix with some other incoming signal and produce a spurious response. These spurii (sometimes spelled spuriae ) can result in increased noise or two signals detected where there should only be one. [ 1 ] Additional components can be added to a VFO to suppress high-frequency parasitic oscillations, should these be present.
In a transmitter, these spurious signals are generated along with the one desired signal. Filtering may be required to ensure the transmitted signal meets regulations for bandwidth and spurious emissions.
When examined with very sensitive equipment, the pure sine-wave peak in a VFO's frequency graph will most likely turn out not to be sitting on a flat noise-floor . Slight random ' jitters ' in the signal's timing will mean that the peak is sitting on 'skirts' of phase noise at frequencies either side of the desired one.
These are also troublesome in crowded bands. They allow through unwanted signals that are fairly close to the expected one, but because of the random quality of these phase-noise 'skirts', the signals are usually unintelligible, appearing just as extra noise in the received signal. The effect is that what should be a clean signal in a crowded band can appear to be a very noisy signal, because of the effects of strong signals nearby.
The effect of VFO phase noise on a transmitter is that random noise is actually transmitted either side of the required signal. Again, this must be avoided for legal reasons in many cases.
Digital or digitally controlled oscillators typically rely on constant single frequency references, which can be made to a higher standard than semiconductor and LC circuit -based alternatives. Most commonly a quartz crystal based oscillator is used, although in high accuracy applications such as TDMA cellular networks , atomic clocks such as the Rubidium standard are as of 2018 also common.
Because of the stability of the reference used, digital oscillators themselves tend to be more stable and more repeatable in the long term. This in part explains their huge popularity in low-cost and computer-controlled VFOs. In the shorter term the imperfections introduced by digital frequency division and multiplication ( jitter ), and the susceptibility of the common quartz standard to acoustic shocks, temperature variation, aging, and even radiation, limit the applicability of a naïve digital oscillator.
This is why higher end VFO's like RF transmitters locked to atomic time , tend to combine multiple different references, and in complex ways. Some references like rubidium or cesium clocks provide higher long term stability, while others like hydrogen masers yield lower short term phase noise. Then lower frequency (and so lower cost) oscillators phase locked to a digitally divided version of the master clock deliver the eventual VFO output, smoothing out the noise induced by the division algorithms. Such an arrangement can then give all of the longer term stability and repeatability of an exact reference, the benefits of exact digital frequency selection, and the short term stability, imparted even onto an arbitrary frequency analogue waveform—the best of all worlds.
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https://en.wikipedia.org/wiki/Variable-frequency_oscillator
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In mechanics , a variable-mass system is a collection of matter whose mass varies with time . It can be confusing to try to apply Newton's second law of motion directly to such a system. [ 1 ] [ 2 ] Instead, the time dependence of the mass m can be calculated by rearranging Newton's second law and adding a term to account for the momentum carried by mass entering or leaving the system. The general equation of variable-mass motion is written as
where F ext is the net external force on the body, v rel is the relative velocity of the escaping or incoming mass with respect to the center of mass of the body, and v is the velocity of the body. [ 3 ] In astrodynamics , which deals with the mechanics of rockets , the term v rel is often called the effective exhaust velocity and denoted v e . [ 4 ]
There are different derivations for the variable-mass system motion equation, depending on whether the mass is entering or leaving a body (in other words, whether the moving body's mass is increasing or decreasing, respectively). To simplify calculations, all bodies are considered as particles . It is also assumed that the mass is unable to apply external forces on the body outside of accretion/ablation events.
The following derivation is for a body that is gaining mass ( accretion ). A body of time-varying mass m moves at a velocity v at an initial time t . In the same instant, a particle of mass dm moves with velocity u with respect to ground. The initial momentum can be written as [ 5 ]
Now at a time t + d t , let both the main body and the particle accrete into a body of velocity v + d v . Thus the new momentum of the system can be written as
Since d m d v is the product of two small values, it can be ignored, meaning during d t the momentum of the system varies for
Therefore, by Newton's second law
Noting that u - v is the velocity of d m relative to m , symbolized as v rel , this final equation can be arranged as [ 6 ]
In a system where mass is being ejected or ablated from a main body, the derivation is slightly different. At time t , let a mass m travel at a velocity v , meaning the initial momentum of the system is
Assuming u to be the velocity of the ablated mass d m with respect to the ground, at a time t + d t the momentum of the system becomes
where u is the velocity of the ejected mass with respect to ground, and is negative because the ablated mass moves in opposite direction to the mass. Thus during d t the momentum of the system varies for
Relative velocity v rel of the ablated mass with respect to the mass m is written as
Therefore, change in momentum can be written as
Therefore, by Newton's second law
Therefore, the final equation can be arranged as
By the definition of acceleration , a = d v /d t , so the variable-mass system motion equation can be written as
In bodies that are not treated as particles a must be replaced by a cm , the acceleration of the center of mass of the system, meaning
Often the force due to thrust is defined as F t h r u s t = v r e l d m d t {\displaystyle \mathbf {F} _{\mathrm {thrust} }=\mathbf {v} _{\mathrm {rel} }{\frac {\mathrm {d} m}{\mathrm {d} t}}} so that
This form shows that a body can have acceleration due to thrust even if no external forces act on it ( F ext = 0). Note finally that if one lets F net be the sum of F ext and F thrust then the equation regains the usual form of Newton's second law:
The ideal rocket equation , or the Tsiolkovsky rocket equation, can be used to study the motion of vehicles that behave like a rocket (where a body accelerates itself by ejecting part of its mass, a propellant , with high speed). It can be derived from the general equation of motion for variable-mass systems as follows: when no external forces act on a body ( F ext = 0) the variable-mass system motion equation reduces to [ 2 ]
If the velocity of the ejected propellant, v rel , is assumed to have the opposite direction as the rocket's acceleration, d v /d t , the scalar equivalent of this equation can be written as
from which d t can be canceled out to give
Integration by separation of variables gives
By rearranging and letting Δ v = v 1 - v 0 , one arrives at the standard form of the ideal rocket equation:
where m 0 is the initial total mass, including propellant, m 1 is the final total mass, v rel is the effective exhaust velocity (often denoted as v e ), and Δ v is the maximum change of speed of the vehicle (when no external forces are acting).
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https://en.wikipedia.org/wiki/Variable-mass_system
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In mathematics , a variable (from Latin variabilis ' changeable ' ) is a symbol , typically a letter, that refers to an unspecified mathematical object . [ 1 ] [ 2 ] [ 3 ] One says colloquially that the variable represents or denotes the object, and that any valid candidate for the object is the value of the variable. The values a variable can take are usually of the same kind, often numbers. More specifically, the values involved may form a set , such as the set of real numbers .
The object may not always exist, or it might be uncertain whether any valid candidate exists or not. For example, one could represent two integers by the variables p and q and require that the value of the square of p is twice the square of q , which in algebraic notation can be written p 2 = 2 q 2 . A definitive proof that this relationship is impossible to satisfy when p and q are restricted to integer numbers isn't obvious, but it has been known since ancient times and has had a big influence on mathematics ever since.
Originally, the term variable was used primarily for the argument of a function , in which case its value could be thought of as varying within the domain of the function . This is the motivation for the choice of the term. Also, variables are used for denoting values of functions, such as the symbol y in the equation y = f ( x ) , where x is the argument and f denotes the function itself.
A variable may represent an unspecified number that remains fixed during the resolution of a problem; in which case, it is often called a parameter . A variable may denote an unknown number that has to be determined; in which case, it is called an unknown ; for example, in the quadratic equation ax 2 + bx + c = 0 , the variables a , b , c are parameters, and x is the unknown.
Sometimes the same symbol can be used to denote both a variable and a constant , that is a well defined mathematical object. For example, the Greek letter π generally represents the number π , but has also been used to denote a projection . Similarly, the letter e often denotes Euler's number , but has been used to denote an unassigned coefficient for quartic function and higher degree polynomials . Even the symbol 1 has been used to denote an identity element of an arbitrary field . These two notions are used almost identically, therefore one usually must be told whether a given symbol denotes a variable or a constant. [ 4 ]
Variables are often used for representing matrices , functions , their arguments, sets and their elements , vectors , spaces , etc. [ 5 ]
In mathematical logic , a variable is a symbol that either represents an unspecified constant of the theory, or is being quantified over. [ 6 ] [ 7 ] [ 8 ]
The earliest uses of an "unknown quantity" date back to at least the Ancient Egyptians with the Moscow Mathematical Papyrus (c. 1500 BC) which described problems with unknowns rhetorically, called the "Aha problems". The "Aha problems" involve finding unknown quantities (referred to as aha , "stack") if the sum of the quantity and part(s) of it are given (The Rhind Mathematical Papyrus also contains four of these type of problems). For example, problem 19 asks one to calculate a quantity taken 1 + 1 ⁄ 2 times and added to 4 to make 10. [ 9 ] In modern mathematical notation: 3 / 2 x + 4 = 10 . Around the same time in Mesopotamia, mathematics of the Old Babylonian period (c. 2000 BC – 1500 BC) was more advanced, also studying quadratic and cubic equations . [ 10 ]
In works of ancient greece such as Euclid's Elements (c. 300 BC), mathematics was described geometrically . For example, The Elements , proposition 1 of Book II, Euclid includes the proposition:
"If there be two straight lines, and one of them be cut into any number of segments whatever, the rectangle contained by the two straight lines is equal to the rectangles contained by the uncut straight line and each of the segments."
This corresponds to the algebraic identity a ( b + c ) = ab + ac ( distributivity ), but is described entirely geometrically. Euclid, and other greek geometers, also used single letters refer to geometric points and shapes. This kind of algebra is now sometimes called Greek geometric algebra . [ 10 ]
Diophantus of Alexandria , [ 11 ] pioneered a form of syncopated algebra in his Arithmetica (c. 200 AD), which introduced symbolic manipulation of expressions with unknowns and powers, but without modern symbols for relations (such as equality or inequality ) or exponents . [ 12 ] An unknown number was called ζ {\displaystyle \zeta } . [ 13 ] The square of ζ {\displaystyle \zeta } was Δ v {\displaystyle \Delta ^{v}} ; the cube was K v {\displaystyle K^{v}} ; the fourth power was Δ v Δ {\displaystyle \Delta ^{v}\Delta } ; and the fifth power was Δ K v {\displaystyle \Delta K^{v}} . [ 14 ] So for example, what would be written in modern notation as: x 3 − 2 x 2 + 10 x − 1 , {\displaystyle x^{3}-2x^{2}+10x-1,} would be written in Diophantus's syncopated notation as:
In the 7th century BC, Brahmagupta used different colours to represent the unknowns in algebraic equations in the Brāhmasphuṭasiddhānta . One section of this book is called "Equations of Several Colours". [ 15 ] Greek and other ancient mathematical advances, were often trapped in long periods of stagnation, and so there were few revolutions in notation, but this began to change by the early modern period .
At the end of the 16th century, François Viète introduced the idea of representing known and unknown numbers by letters, nowadays called variables, and the idea of computing with them as if they were numbers—in order to obtain the result by a simple replacement. Viète's convention was to use consonants for known values, and vowels for unknowns. [ 16 ]
In 1637, René Descartes "invented the convention of representing unknowns in equations by x , y , and z , and knowns by a , b , and c ". [ 17 ] Contrarily to Viète's convention, Descartes' is still commonly in use. The history of the letter x in math was discussed in an 1887 Scientific American article. [ 18 ]
Starting in the 1660s, Isaac Newton and Gottfried Wilhelm Leibniz independently developed the infinitesimal calculus , which essentially consists of studying how an infinitesimal variation of a time-varying quantity, called a Fluent , induces a corresponding variation of another quantity which is a function of the first variable. Almost a century later, Leonhard Euler fixed the terminology of infinitesimal calculus, and introduced the notation y = f ( x ) for a function f , its variable x and its value y . Until the end of the 19th century, the word variable referred almost exclusively to the arguments and the values of functions.
In the second half of the 19th century, it appeared that the foundation of infinitesimal calculus was not formalized enough to deal with apparent paradoxes such as a nowhere differentiable continuous function . To solve this problem, Karl Weierstrass introduced a new formalism consisting of replacing the intuitive notion of limit by a formal definition. The older notion of limit was "when the variable x varies and tends toward a , then f ( x ) tends toward L ", without any accurate definition of "tends". Weierstrass replaced this sentence by the formula
in which none of the five variables is considered as varying.
This static formulation led to the modern notion of variable, which is simply a symbol representing a mathematical object that either is unknown, or may be replaced by any element of a given set (e.g., the set of real numbers ).
Variables are generally denoted by a single letter, most often from the Latin alphabet and less often from the Greek , which may be lowercase or capitalized. The letter may be followed by a subscript: a number (as in x 2 ), another variable ( x i ), a word or abbreviation of a word as a label ( x total ) or a mathematical expression ( x 2 i +1 ). Under the influence of computer science , some variable names in pure mathematics consist of several letters and digits. Following René Descartes (1596–1650), letters at the beginning of the alphabet such as a , b , c are commonly used for known values and parameters, and letters at the end of the alphabet such as x , y , z are commonly used for unknowns and variables of functions. [ 19 ] In printed mathematics, the norm is to set variables and constants in an italic typeface. [ 20 ]
For example, a general quadratic function is conventionally written as ax 2 + bx + c , where a , b and c are parameters (also called constants , because they are constant functions ), while x is the variable of the function. A more explicit way to denote this function is x ↦ ax 2 + bx + c , which clarifies the function-argument status of x and the constant status of a , b and c . Since c occurs in a term that is a constant function of x , it is called the constant term . [ 21 ]
Specific branches and applications of mathematics have specific naming conventions for variables. Variables with similar roles or meanings are often assigned consecutive letters or the same letter with different subscripts. For example, the three axes in 3D coordinate space are conventionally called x , y , and z . In physics, the names of variables are largely determined by the physical quantity they describe, but various naming conventions exist. A convention often followed in probability and statistics is to use X , Y , Z for the names of random variables , keeping x , y , z for variables representing corresponding better-defined values.
It is common for variables to play different roles in the same mathematical formula, and names or qualifiers have been introduced to distinguish them. For example, the general cubic equation
is interpreted as having five variables: four, a , b , c , d , which are taken to be given numbers and the fifth variable, x , is understood to be an unknown number. To distinguish them, the variable x is called an unknown , and the other variables are called parameters or coefficients , or sometimes constants , although this last terminology is incorrect for an equation, and should be reserved for the function defined by the left-hand side of this equation.
In the context of functions, the term variable refers commonly to the arguments of the functions. This is typically the case in sentences like " function of a real variable ", " x is the variable of the function f : x ↦ f ( x ) ", " f is a function of the variable x " (meaning that the argument of the function is referred to by the variable x ).
In the same context, variables that are independent of x define constant functions and are therefore called constant . For example, a constant of integration is an arbitrary constant function that is added to a particular antiderivative to obtain the other antiderivatives. Because of the strong relationship between polynomials and polynomial functions , the term "constant" is often used to denote the coefficients of a polynomial, which are constant functions of the indeterminates.
Other specific names for variables are:
All these denominations of variables are of semantic nature, and the way of computing with them ( syntax ) is the same for all.
In calculus and its application to physics and other sciences, it is rather common to consider a variable, say y , whose possible values depend on the value of another variable, say x . In mathematical terms, the dependent variable y represents the value of a function of x . To simplify formulas, it is often useful to use the same symbol for the dependent variable y and the function mapping x onto y . For example, the state of a physical system depends on measurable quantities such as the pressure , the temperature , the spatial position, ..., and all these quantities vary when the system evolves, that is, they are function of the time. In the formulas describing the system, these quantities are represented by variables which are dependent on the time, and thus considered implicitly as functions of the time.
Therefore, in a formula, a dependent variable is a variable that is implicitly a function of another (or several other) variables. An independent variable is a variable that is not dependent. [ 23 ]
The property of a variable to be dependent or independent depends often of the point of view and is not intrinsic. For example, in the notation f ( x , y , z ) , the three variables may be all independent and the notation represents a function of three variables. On the other hand, if y and z depend on x (are dependent variables ) then the notation represents a function of the single independent variable x . [ 24 ]
If one defines a function f from the real numbers to the real numbers by
then x is a variable standing for the argument of the function being defined, which can be any real number.
In the identity
the variable i is a summation variable which designates in turn each of the integers 1, 2, ..., n (it is also called index because its variation is over a discrete set of values) while n is a parameter (it does not vary within the formula).
In the theory of polynomials , a polynomial of degree 2 is generally denoted as ax 2 + bx + c , where a , b and c are called coefficients (they are assumed to be fixed, i.e., parameters of the problem considered) while x is called a variable. When studying this polynomial for its polynomial function this x stands for the function argument. When studying the polynomial as an object in itself, x is taken to be an indeterminate, and would often be written with a capital letter instead to indicate this status.
Consider the equation describing the ideal gas law, P V = N k B T . {\displaystyle PV=Nk_{\text{B}}T.} This equation would generally be interpreted to have four variables, and one constant. The constant is k B , the Boltzmann constant . One of the variables, N , the number of particles, is a positive integer (and therefore a discrete variable), while the other three, P , V and T , for pressure, volume and temperature, are continuous variables.
One could rearrange this equation to obtain P as a function of the other variables, P ( V , N , T ) = N k B T V . {\displaystyle P(V,N,T)={\frac {Nk_{\text{B}}T}{V}}.} Then P , as a function of the other variables, is the dependent variable, while its arguments, V , N and T , are independent variables. One could approach this function more formally and think about its domain and range: in function notation, here P is a function P : R > 0 × N × R > 0 → R {\displaystyle P:\mathbb {R} _{>0}\times \mathbb {N} \times \mathbb {R} _{>0}\rightarrow \mathbb {R} } .
However, in an experiment, in order to determine the dependence of pressure on a single one of the independent variables, it is necessary to fix all but one of the variables, say T . This gives a function P ( T ) = N k B T V , {\displaystyle P(T)={\frac {Nk_{\text{B}}T}{V}},} where now N and V are also regarded as constants. Mathematically, this constitutes a partial application of the earlier function P .
This illustrates how independent variables and constants are largely dependent on the point of view taken. One could even regard k B as a variable to obtain a function P ( V , N , T , k B ) = N k B T V . {\displaystyle P(V,N,T,k_{\text{B}})={\frac {Nk_{\text{B}}T}{V}}.}
Considering constants and variables can lead to the concept of moduli spaces. For illustration, consider the equation for a parabola , y = a x 2 + b x + c , {\displaystyle y=ax^{2}+bx+c,} where a , b , c , x and y are all considered to be real. The set of points ( x , y ) in the 2D plane satisfying this equation trace out the graph of a parabola. Here, a , b and c are regarded as constants, which specify the parabola, while x and y are variables.
Then instead regarding a , b and c as variables, we observe that each set of 3-tuples ( a , b , c ) corresponds to a different parabola. That is, they specify coordinates on the 'space of parabolas': this is known as a moduli space of parabolas .
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https://en.wikipedia.org/wiki/Variable_(mathematics)
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Variable air volume ( VAV ) is a type of heating, ventilating, and/or air-conditioning ( HVAC ) system. Unlike constant air volume (CAV) systems, which supply a constant airflow at a variable temperature, VAV systems vary the airflow at a constant or varying temperature. [ 1 ] [ 2 ] The advantages of VAV systems over constant-volume systems include more precise temperature control, reduced compressor wear, lower energy consumption by system fans, less fan noise, and additional passive dehumidification. [ 3 ]
The most simple form of a VAV box is the single duct terminal configuration, which is connected to a single supply air duct that delivers treated air from an air-handling unit (AHU) to the space the box is serving. [ 2 ] This configuration can deliver air at variable temperatures or air volumes to meet the heating and cooling loads as well as the ventilation rates required by the space. [ 2 ]
Most commonly, VAV boxes are pressure independent, meaning the VAV box uses controls to deliver a constant flow rate regardless of variations in system pressures experienced at the VAV inlet. [ 2 ] This is accomplished by an airflow sensor that is placed at the VAV inlet which opens or closes the damper within the VAV box to adjust the airflow. [ 2 ] The difference between a CAV and VAV box is that a VAV box can be programmed to modulate between different flowrate setpoints depending on the conditions of the space. The VAV box is programmed to operate between a minimum and maximum airflow setpoint and can modulate the flow of air depending on occupancy, temperature, or other control parameters. [ 4 ] A CAV box can only operate between a constant, maximum value, or an “off” state. [ 5 ] This difference means the VAV box can provide tighter space temperature control while using much less energy. Another reason why VAV boxes save more energy is that they are coupled with variable-speed drives on fans , so the fans can ramp down when the VAV boxes are experiencing part load conditions. [ 6 ] [ 7 ]
It is common for VAV boxes to include a form of reheat, either electric or hydronic heating coils. [ 4 ] While electric coils operate on the principle of electric resistance heating, whereby electrical energy is converted to heat via electric resistance, hydronic heating uses hot water to transfer heat from the coil to the air. The addition of reheat coils allows the box to adjust the supply air temperature to meet the heating loads in the space while delivering the required ventilation rates. [ 2 ] In some applications it is possible for the space to require such high air-change rates it causes a risk of over-cooling. [ 5 ] In this scenario, the reheat coils could increase the air temperature to maintain the temperature setpoint in the space. [ 2 ] This scenario tends to happen during cooling seasons in buildings which have perimeter and interior zones. The perimeter zones, with more sun exposure, require a lower supply air temperature from the air-handling unit than the interior zones, which have less sun exposure and tend to stay cooler than the perimeter zones when left un-conditioned. With the same supply air temperature being delivered to both zones, the reheat coils must heat the air for the interior zone to avoid over-cooling. [ 8 ]
The air blower's flow rate is variable. For a single VAV air handler that serves multiple thermal zones, the flow rate to each zone must be varied as well.
A VAV terminal unit , [ 9 ] often called a VAV box , is the zone-level flow control device. It is basically a calibrated air damper with an automatic actuator . The VAV terminal unit is connected to either a local or a central control system. Historically, pneumatic control was commonplace, but electronic direct digital control systems are popular especially for mid- to large-size applications. Hybrid control, for example having pneumatic actuators with digital data collection, is popular as well. [ 10 ]
A common commercial application is shown in the diagram. This VAV system consists of a VAV box, ductwork, and four air terminals.
Control of the system's fan capacity is critical in VAV systems. Without proper and rapid flow rate control, the system's ductwork, or its sealing, can easily be damaged by overpressurization. In the cooling mode of operation, as the temperature in the space is satisfied, a VAV box closes to limit the flow of cool air into the space. As the temperature increases in the space, the box opens to bring the temperature back down. The fan maintains a constant static pressure in the discharge duct regardless of the position of the VAV box. Therefore, as the box closes, the fan slows down or restricts the amount of air going into the supply duct. As the box opens, the fan speeds up and allows more air flow into the duct, maintaining a constant static pressure. [ 11 ]
One of the challenges for VAV systems is providing adequate temperature control for multiple zones with different environmental conditions, such as an office on the glass perimeter of a building vs. an interior office down the hall. Dual duct systems provide cool air in one duct and warm air in a second duct to provide an appropriate temperature of mixed supply air for any zone. An extra duct, however, is cumbersome and expensive. Reheating the air from a single duct, using electric or hot water heating, is often a more cost-effective solution. [ 12 ]
Traditional VAV reheat systems use minimum airflow rates of 30% to 50% the design airflow. These airflow minimums are selected to avoid the risk of under-ventilation and thermal comfort issues. However, published research supporting the efficacy of this approach is scarce. Systems operating at lower minimum airflow ranges (10% to 20% of design airflow) stand to use less fan and reheat coil energy relative to a traditional system, and recent research has shown that thermal comfort and adequate ventilation can still be attained at these lower minimums. [ 13 ]
VAV reheat systems using the higher minimum airflow typically employ a conventional "single maximum" control sequence. Under this control sequence, a single cooling maximum airflow setpoint is selected for design cooling conditions. The cooling airflow is gradually lowered to the minimum airflow setpoint, where it remains as the space temperature lowers beyond the cooling temperature setpoint. When the heating setpoint is reached, the electric or hydronic heating coil is activated and gradually provides more heat until the maximum heating capacity is reached at the design heating temperature. [ 14 ]
Research has shown that using a different, "dual maximum" control sequence can save substantial amounts of energy relative to the conventional "single maximum" control sequence. This is accomplished due to the "dual maximum" sequence’s use of lower minimum airflow rates. [ 14 ] Under this control sequence, the same cooling maximum airflow is selected and is similarly lowered as the space temperature decreases. By the time the space temperature drops to the cooling temperature setpoint, the airflow reaches a lower minimum value than that used in the "single maximum" sequence (10% - 20% vs. 30% - 50% of maximum cooling airflow). When the space temperature reaches the heating temperature setpoint, the heating coil is activated and increases its electrical power (for electric coils) or hot water valve position (for hydronic coils) while the airflow remains at the minimum setpoint. When the heating coil reaches its maximum heating capacity, upon a further drop in space temperature, the airflow is increased until it reaches a maximum heating airflow setpoint (typically about 50% of the maximum cooling airflow). [ 5 ]
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https://en.wikipedia.org/wiki/Variable_air_volume
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In science and research , an attribute is a quality of an object (person, thing, etc.). [ 1 ] Attributes are closely related to variables. A variable is a logical set of attributes. [ 1 ] Variables can "vary" – for example, be high or low. [ 1 ] How high, or how low, is determined by the value of the attribute (and in fact, an attribute could be just the word "low" or "high"). [ 1 ] (For example see: Binary option )
While an attribute is often intuitive, the variable is the operationalized way in which the attribute is represented for further data processing . In data processing data are often represented by a combination of items (objects organized in rows), and multiple variables (organized in columns).
Values of each variable statistically "vary" (or are distributed) across the variable's domain. A domain is a set of all possible values that a variable is allowed to have. The values are ordered in a logical way and must be defined for each variable. Domains can be bigger or smaller. The smallest possible domains have those variables that can only have two values, also called binary (or dichotomous) variables. Bigger domains have non-dichotomous variables and the ones with a higher level of measurement . (See also domain of discourse .)
Semantically, greater precision can be obtained when considering an object's characteristics by distinguishing 'attributes' (characteristics that are attributed to an object) from ' traits ' (characteristics that are inherent to the object).
Age is an attribute that can be operationalized in many ways. It can be dichotomized so that only two values – "old" and "young" – are allowed for further data processing. In this case the attribute "age" is operationalized as a binary variable. If more than two values are possible and they can be ordered, the attribute is represented by ordinal variable, such as "young", "middle age", and "old". Next it can be made of rational values, such as 1, 2, 3.... 99. [ 1 ]
The "social class" attribute can be operationalized in similar ways as age, including "lower", "middle" and "upper class" and each class could be differentiated between upper and lower, transforming thus changing the three attributes into six (see the model proposed by William Lloyd Warner ) or it could use different terminology (such as the working class as in the model by Gilbert and Kahl ).
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https://en.wikipedia.org/wiki/Variable_and_attribute_(research)
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Variable bitrate ( VBR ) is a term used in telecommunications and computing that relates to the bitrate used in sound or video encoding. As opposed to constant bitrate (CBR), VBR files vary the amount of output data per time segment. VBR allows a higher bitrate (and therefore more storage space) to be allocated to the more complex segments of media files while less space is allocated to less complex segments. The average of these rates can be calculated to produce an average bitrate for the file.
MP3 , WMA and AAC audio files can optionally be encoded in VBR, while Opus and Vorbis are encoded in VBR by default. [ 1 ] [ 2 ] [ 3 ] Variable bit rate encoding is also commonly used on MPEG-2 video, MPEG-4 Part 2 video ( Xvid , DivX , etc.), MPEG-4 Part 10 /H.264 video, Theora , Dirac and other video compression formats. [ citation needed ] Additionally, variable rate encoding is inherent in lossless compression schemes such as FLAC and Apple Lossless . [ citation needed ]
The advantages of VBR are that it produces a better quality-to-space ratio compared to a CBR file of the same data. The bits available are used more flexibly to encode the sound or video data more accurately, with fewer bits used in less demanding passages and more bits used in difficult-to-encode passages. [ 2 ] [ 4 ]
The disadvantages are that it may take more time to encode, as the process is more complex, and that some hardware might not be compatible with VBR files. [ 2 ]
VBR is created using so-called single-pass encoding or multi-pass encoding . Single-pass encoding analyzes and encodes the data "on the fly" and it is also used in constant bitrate encoding. Single-pass encoding is used when the encoding speed is most important — e.g. for real-time encoding. Single-pass VBR encoding is usually controlled by the fixed quality setting or by the bitrate range (minimum and maximum allowed bitrate) or by the average bitrate setting. Multi-pass encoding is used when the encoding quality is most important. Multi-pass encoding cannot be used in real-time encoding, live broadcast or live streaming . Multi-pass encoding takes much longer than single-pass encoding, because every pass means one pass through the input data (usually through the whole input file). Multi-pass encoding is used only for VBR encoding, because CBR encoding doesn't offer any flexibility to change the bitrate. The most common multi-pass encoding is two-pass encoding. In the first pass of two-pass encoding, the input data is being analyzed and the result is stored in a log file. In the second pass, the collected data from the first pass is used to achieve the best encoding quality. In a video encoding, two-pass encoding is usually controlled by the average bitrate setting or by the bitrate range setting (minimal and maximal allowed bitrate) or by the target video file size setting. [ 5 ] [ 6 ]
This VBR encoding method allows the user to specify a bitrate range — a minimum and/or maximum allowed bitrate. [ 7 ] Some encoders extend this method with an average bitrate. The minimum and maximum allowed bitrate set bounds in which the bitrate may vary. The disadvantage of this method is that the average bitrate (and hence file size) will not be known ahead of time. The bitrate range is also used in some fixed quality encoding methods, but usually without permission to change a particular bitrate. [ 8 ]
The disadvantage of single pass ABR encoding (with or without Constrained Variable Bitrate) is the opposite of fixed quantizer VBR — the size of the output is known ahead of time, but the resulting quality is unknown, although still better than CBR. [ 9 ]
The multi-pass ABR encoding is more similar to fixed quantizer VBR, because a higher average will really increase the quality. [ 10 ]
VBR encoding using the file size setting is usually multi-pass encoding. It allows the user to specify a specific target file size. In the first pass, the encoder analyzes the input file and automatically calculates possible bitrate range and/or average bitrate. In the last pass, the encoder distributes the available bits among the entire video to achieve uniform quality. [ 10 ]
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https://en.wikipedia.org/wiki/Variable_bitrate
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A variable cycle engine ( VCE ), also referred to as adaptive cycle engine ( ACE ), is an aircraft jet engine that is designed to operate efficiently under mixed flight conditions, such as subsonic , transonic and supersonic .
An advanced technology engine is a turbine engine that allows different turbines to spin at different, individually optimum speeds, instead of at one speed for all. [ 1 ] It emerged on larger airplanes, before finding other applications.
The next generation of supersonic transport (SST) may require some form of VCE. To reduce aircraft drag at supercruise , SST engines require a high specific thrust (net thrust/airflow) to minimize the powerplant's cross-sectional area. This implies a high jet velocity supersonic cruise and at take-off, which makes the aircraft noisy.
A high specific thrust engine has a high jet velocity by definition, as implied by the approximate equation for net thrust: [ 2 ]
where:
Rearranging the equation, specific thrust is given by:
So for zero flight velocity, specific thrust is directly proportional to jet velocity.
The Rolls-Royce/Snecma Olympus 593 in Concorde had a high specific thrust in supersonic cruise and at dry take-off. This made the engines noisy. The problem was compounded by the need for a modest amount of afterburning (reheat) at take-off (and transonic acceleration).
One SST VCE concept is the tandem fan engine. The engine has two fans, both mounted on the low-pressure shaft, separated by a significant axial gap. The engine operates in series mode while cruising and parallel mode take-off, climb-out, approach, and final-descent.
In series mode, air enters in the front of the engine. After passing through the front fan, the air passes directly into the second fan, so that the engine behaves much like a turbofan .
In parallel mode, air leaving the front fan exits the engine through an auxiliary nozzle on the underside of the nacelle , skipping the rear fan. Intakes on each side of the engine open to capture air and send it directly to the rear fan and the rest of the engine. Parallel mode substantially increases the total air accelerated by the engine, lowering the velocity of the air and accompanying noise.
In the 1970s, Boeing modified a Pratt & Whitney JT8D to use a tandem fan configuration and successfully demonstrated the switch from series to parallel operation (and vice versa) with the engine running, albeit at partial power.
In the mid-tandem fan concept, a high specific flow single stage fan is located between the high pressure (HP) and low pressure (LP) compressors of a turbojet core. Only bypass air passes through the fan. The LP compressor exit flow passes through passages within the fan disc, directly underneath the fan blades. Some bypass air enters the engine via an auxiliary intake. During take-off and approach the engine behaves much like a conventional turbofan, with an acceptable jet noise level (i.e., low specific thrust). However, for supersonic cruise , the fan variable inlet guide vanes and auxiliary intake close to minimize bypass flow and increase specific thrust. In this mode the engine acts more like a 'leaky' turbojet (e.g. the F404 ).
In the mixed-flow turbofan with ejector concept, a low bypass ratio engine is mounted in front of a long tube, called an ejector. The ejector reduces noise. It is deployed during take-off and approach. Turbofan exhaust gases send air into the ejector via an auxiliary air intake, thereby reducing the specific thrust/mean jet velocity of the final exhaust. The mixed-flow design is not particularly efficient at low speed, but is considerably simpler.
The three-stream architecture adds a third, directable air stream. This stream bypasses the core when fuel efficiency is required or through the core for greater power. Under the Versatile Affordable Advanced Turbine Engines (VAATE) program, the U.S Air Force and industry partners developed this concept under the Adaptive Versatile Engine Technology (ADVENT) and the follow-on Adaptive Engine Technology Demonstrator (AETD) and Adaptive Engine Transition Program (AETP) programs. [ 3 ] Examples include the General Electric XA100 and the Pratt & Whitney XA101 , as well as the propulsion system for the Next Generation Air Dominance (NGAD) fighter. [ 4 ]
General Electric developed a variable cycle engine, known as the GE37 or General Electric YF120 , for the YF-22 / YF-23 fighter aircraft competition, in the late 1980s. GE used a double bypass/hybrid fan arrangement, but never disclosed how they exploited the concept. The Air Force instead selected the conventional Pratt & Whitney F119 for what became the Lockheed Martin F-22 Raptor .
Geared turbofans are also used in the following engines, some still in development: Garrett TFE731 , Lycoming ALF 502 / LF 507 , Pratt & Whitney PW1000G , Turbomeca Astafan , and Turbomeca Aspin , and Aviadvigatel PD-18R.
The Rolls Royce Ultrafan is the largest and most efficient engine to allow multiple turbine speeds. The turbines behind the main fan are small and allow more air to pass straight through, while a planetary gearbox "allows the main fan to spin slower and the compressors to spin faster, putting each in their optimal zones." [ 5 ]
Startup Astro Mechanica is developing what it calls a turboelectric-adaptive jet engine that shifts from turbofan to turbojet to ramjet mode as it accelerates from a standing start to a projected Mach 6 . This is achieved by using a dual turbine approach. One turbine acts as an turbogenerator . The second turbine acts as the propulsion unit. The turbogenerator powers an electric motor that controls the compressor of the second turbine. The motor can change speeds to keep the fan turning at the ideal RPM for a specific flight mode. In turbojet and ramjet modes, the inlet is narrowed to compress the air and eliminate bypass. The turbogenerator is commercially available, while the propulsion unit is built by the company. A key innovation is that electric motors have dramatically increased their power density so that the weight of the motor is no longer prohibitive. [ 6 ] [ 7 ] [ 8 ]
Instead of a fixed gearbox, it uses an electric motor to turn the turbine(s) behind the fan at an ideal speed for each phase of flight. The company claimed it would support efficient take-off, subsonic, supersonic, and hypersonic speeds. The electric motor is powered by a generator in turn powered by a turbine. The approach relies on the improved power density of novel electric motors such as yokeless dual-rotor axial flux motors that offer far more kw/kg than conventional designs that were too heavy for such an application. [ 7 ]
Air flows in through a turbogenerator to produce electric power to power an electric motor. The electric motor adaptively controls the propulsion unit, allowing it to behave like a turbofan, turbojet, or ramjet depending on airspeed. In effect the engine can operate at any point along the specific impulse (Isp) curve - high Isp at low speed or low Isp at high speed. [ 9 ] [ 7 ] [ 10 ]
It is in some respects similar to turbo-electric marine engines that allow propellers to turn at a different speed than the steam turbines that power them.
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https://en.wikipedia.org/wiki/Variable_cycle_engine
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A variable electro-precipitator (VEP) is a waste water remediation unit using electrocoagulation . The differences between a standard electrocoagulation (EC) unit and a variable Electro-precipitation unit are in the enhanced flow path and the unit electrode connections. The variable electro-precipitator's flow path has been designed to maximize retention time and to increase the turbulence of the water within the unit. This design aids in increasing the amount of effective treatment per gallon of water .
A major design weakness of the electrocoagulation units is the method used in connecting the electrode to the power source . These designs cause overheating, resulting in premature failure of the electrocoagulation reaction chamber. VEP reaction chambers are designed to resolve these performance issues by changing all electrode connections from the standard wet connection (inside the chamber) to an external dry connection. The VEP is cooler-operating, and has a longer chamber life than an electrocoagulation unit.
This water supply –related article is a stub . You can help Wikipedia by expanding it .
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https://en.wikipedia.org/wiki/Variable_electro-precipitator
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A variable geometry turbomachine uses movable vanes to optimize its efficiency at different operating conditions. This article refers to movable vanes as used in liquid pumps and turbocharger turbines. It does not cover the widespread use of movable vanes in gas turbine compressors.
If all fluid velocities at corresponding points within the turbomachine are in the same direction and proportional to the blade speed, then the operating condition of a turbomachine at two different rotational speeds will be dynamically similar.
If two points, each on dissimilar head-flow characteristics curve, represent similar dynamic operation of the turbo machine, then the non-dimensional variables (ignoring Reynolds number effects) will have same values.
Head coefficient
Efficiency
Power coefficient
Where,
N {\displaystyle N} is speed of rotation.
Q {\displaystyle Q} is flow rate.
D {\displaystyle D} is impeller diameter.
Thus non-dimensional representation is highly advantageous for converging to single performance curve that would otherwise result in multiple curves if plotted dimensionally.
Figure 1 shows head characteristics [ 1 ] of centrifugal pump versus flow coefficient. Within the normal operating range of this pump, 0.03 <Q/(ND 3 ) < 0.06 , the head characteristic curves approximately coincide for different values of speed (2500 <N<5000 rev/min) and little scatter appears may be due to the effect of Reynolds number. For smaller flow co-efficient, Q/(ND 3 ) < 0.025 , the flow became unsteady but dynamically similar conditions still appear i.e. head characteristic curves still coincides for different values of speed. But at high flow rates deviation from the single-curve are noticed for higher values of speed. This effect is due to cavitation , [ 2 ] a high speed phenomenon of hydraulic machines caused by the release of vapour bubbles at low pressures.
Thus during off-design operating conditions, i.e. Q/(ND 3 ) < 0.03 and Q/(ND 3 ) > 0.06 , the flow become unsteady and cavitations occurs . So to avoid cavitation increase efficiency at high flow rates we resort to Variable Geometry Turbomachine.
Fixed geometry machines are designed to operate at maximum efficiency condition. The efficiency of a fixed geometry machine depends on the flow coefficient and Reynolds number . For a constant Reynolds number as flow coefficient increases, efficiency also increases, reaches a maximum value and then decreases. Thus off-design operation is completely inefficient and may result in cavitation at higher flow rates.
A variable geometry turbomachine uses movable vanes to regulate the flow. Vane angles are varied using cams driven by servo motor ( actuator ). In large installations involving many thousands of kilowatts and where operating conditions fluctuate, sophisticated systems of control are incorporated. Thus variable geometry turbomachine offer a better match of efficiency with changing flow conditions.
Figure 2 describes the envelope of optimum efficiency [ 1 ] for a variable geometry turbomachine. In the figure each of curves ( a , b a n d c ) {\displaystyle (a,b\,and\,c)} represents different fixed geometry machines. The efficiency of the variable geometry turbomachine intersects the point of maximum efficiency for each of the curves ( a , b a n d c ) {\displaystyle (a,b\,and\,c)} .
As the vane angles are variable in Variable Geometry Turbomachine, therefore we introduce an additional variable β {\displaystyle \beta } into equation 1 and 2 to represent the setting of the vanes. We can write:
ψ = f 4 ( ϕ , β ) , {\displaystyle \psi \ =f_{4}(\phi ,\beta ),\,}
η = f 5 ( ϕ , β ) , {\displaystyle \eta \ =f_{5}(\phi ,\beta ),\,}
Where, flow coefficient, ϕ = ( Q N D 3 ) . {\displaystyle \phi \ =\!\left({Q \over {ND^{3}}}\right).\,}
Alternatively, with
β = f 6 ( ψ , ϕ ) , {\displaystyle \beta \ =f_{6}(\psi ,\phi ),\,}
β = f 7 ( η , ϕ ) , {\displaystyle \beta \ =f_{7}(\eta ,\phi ),\,}
β {\displaystyle \beta } can be eliminated to give a new functional dependence:
η = f 8 ( ϕ , ψ ) = f 8 ( Q N D 3 , g H N 2 D 2 ) . {\displaystyle \eta \ =f_{8}(\phi ,\psi )=f_{8}\!\left({Q \over {ND^{3}}},{gH \over {N^{2}D^{2}}}\right).\,} [ 3 ]
Thus, efficiency in a variable geometry pump is a function of both flow coefficient and energy transfer coefficient.
Variable Geometry Turbomachine technology is used in turbocharger of diesel engines, where the turbo has variable vanes which control the flow of exhaust onto the turbine blades. A Variable Geometry Turbocharger [ 4 ] has movable vanes which direct the flow of exhaust onto the turbine blades. Actuators are used to adjust the vane angles. Angle of vanes vary throughout the range of RPM to optimize turbine behaviour. At high engine speed, vanes are fully open and the exhaust is fully directed onto the turbine blades. At low engine speeds vanes are almost closed creating a narrow passage for the exhaust. This accelerates the exhaust towards the turbine blades, making them spin faster.
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https://en.wikipedia.org/wiki/Variable_geometry_turbomachine
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A variable rate feeder (often shortened to belt feeder , or simply feeder ) is a piece of industrial control equipment used to deliver solid material at a known rate into some process.
In both form and function, belt feeders are essentially short belt conveyors . There are a few key areas in which the two differ, however:
A number of different techniques and transport mechanisms are used, but these broadly split into these categories, in descending order of accuracy:
Any of these may be combined with subsequent checks from beltweighers or (in the case of grains, powders and liquids) flowmeters to provide a more accurate flow of material.
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https://en.wikipedia.org/wiki/Variable_rate_feeder
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A variable star is a star whose brightness as seen from Earth (its apparent magnitude ) changes systematically with time. This variation may be caused by a change in emitted light or by something partly blocking the light, so variable stars are classified as either: [ 1 ]
Many, possibly most, stars exhibit at least some oscillation in luminosity: the energy output of the Sun , for example, varies by about 0.1% over an 11-year solar cycle . [ 2 ]
An ancient Egyptian calendar of lucky and unlucky days composed some 3,200 years ago may be the oldest preserved historical document of the discovery of a variable star,
the eclipsing binary Algol . [ 3 ] [ 4 ] [ 5 ] Aboriginal Australians are also known to have observed the variability of Betelgeuse and Antares , incorporating these brightness changes into narratives that are passed down through oral tradition. [ 6 ] [ 7 ] [ 8 ]
Of the modern astronomers, the first variable star was identified in 1638 when Johannes Holwarda noticed that Omicron Ceti (later named Mira) pulsated in a cycle taking 11 months; the star had previously been described as a nova by David Fabricius in 1596. This discovery, combined with supernovae observed in 1572 and 1604, proved that the starry sky was not eternally invariable as Aristotle and other ancient philosophers had taught. In this way, the discovery of variable stars contributed to the astronomical revolution of the sixteenth and early seventeenth centuries.
The second variable star to be described was the eclipsing variable Algol, by Geminiano Montanari in 1669; John Goodricke gave the correct explanation of its variability in 1784. Chi Cygni was identified in 1686 by G. Kirch , then R Hydrae in 1704 by G. D. Maraldi . By 1786, ten variable stars were known. John Goodricke himself discovered Delta Cephei and Beta Lyrae . Since 1850, the number of known variable stars has increased rapidly, especially after 1890 when it became possible to identify variable stars by means of photography.
In 1930, astrophysicist Cecilia Payne published the book The Stars of High Luminosity, [ 9 ] in which she made numerous observations of variable stars, paying particular attention to Cepheid variables . [ 10 ] Her analyses and observations of variable stars, carried out with her husband, Sergei Gaposchkin, laid the basis for all subsequent work on the subject. [ 11 ]
The latest edition of the General Catalogue of Variable Stars [ 12 ] (2008) lists more than 46,000 variable stars in the Milky Way, as well as 10,000 in other galaxies, and over 10,000 'suspected' variables.
The most common kinds of variability involve changes in brightness, but other types of variability also occur, in particular changes in the spectrum . By combining light curve data with observed spectral changes, astronomers are often able to explain why a particular star is variable.
Variable stars are generally analysed using photometry , spectrophotometry and spectroscopy . Measurements of their changes in brightness can be plotted to produce light curves . For regular variables, the period of variation and its amplitude can be very well established; for many variable stars, though, these quantities may vary slowly over time, or even from one period to the next. Peak brightnesses in the light curve are known as maxima, while troughs are known as minima.
Amateur astronomers can do useful scientific study of variable stars by visually comparing the star with other stars within the same telescopic field of view of which the magnitudes are known and constant. By estimating the variable's magnitude and noting the time of observation a visual lightcurve can be constructed. The American Association of Variable Star Observers collects such observations from participants around the world and shares the data with the scientific community.
From the light curve the following data are derived:
From the spectrum the following data are derived:
In very few cases it is possible to make pictures of a stellar disk. These may show darker spots on its surface.
Combining light curves with spectral data often gives a clue as to the changes that occur in a variable star. [ 13 ] For example, evidence for a pulsating star is found in its shifting spectrum because its surface periodically moves toward and away from us, with the same frequency as its changing brightness. [ 14 ]
About two-thirds of all variable stars appear to be pulsating. [ 15 ] In the 1930s astronomer Arthur Stanley Eddington showed that the mathematical equations that describe the interior of a star may lead to instabilities that cause a star to pulsate. [ 16 ] The most common type of instability is related to oscillations in the degree of ionization in outer, convective layers of the star. [ 17 ]
When the star is in the swelling phase, its outer layers expand, causing them to cool. Because of the decreasing temperature the degree of ionization also decreases. This makes the gas more transparent, and thus makes it easier for the star to radiate its energy. This in turn makes the star start to contract. As the gas is thereby compressed, it is heated and the degree of ionization again increases. This makes the gas more opaque, and radiation temporarily becomes captured in the gas. This heats the gas further, leading it to expand once again. Thus a cycle of expansion and compression (swelling and shrinking) is maintained. [ citation needed ]
The pulsation of cepheids is known to be driven by oscillations in the ionization of helium (from He ++ to He + and back to He ++ ). [ 18 ]
In a given constellation, the first variable stars discovered were designated with letters R through Z, e.g. R Andromedae . This system of nomenclature was developed by Friedrich W. Argelander , who gave the first previously unnamed variable in a constellation the letter R, the first letter not used by Bayer . Letters RR through RZ, SS through SZ, up to ZZ are used for the next discoveries, e.g. RR Lyrae . Later discoveries used letters AA through AZ, BB through BZ, and up to QQ through QZ (with J omitted). Once those 334 combinations are exhausted, variables are numbered in order of discovery, starting with the prefixed V335 onwards.
Variable stars may be either intrinsic or extrinsic .
These subgroups themselves are further divided into specific types of variable stars that are usually named after their prototype. For example, dwarf novae are designated U Geminorum stars after the first recognized star in the class, U Geminorum .
Examples of types within these divisions are given below.
Pulsating stars swell and shrink, affecting their brightness and spectrum. Pulsations are generally split into: radial , where the entire star expands and shrinks as a whole; and non-radial, where one part of the star expands while another part shrinks.
Depending on the type of pulsation and its location within the star, there is a natural or fundamental frequency which determines the period of the star. Stars may also pulsate in a harmonic or overtone which is a higher frequency, corresponding to a shorter period. Pulsating variable stars sometimes have a single well-defined period, but often they pulsate simultaneously with multiple frequencies and complex analysis is required to determine the separate interfering periods. In some cases, the pulsations do not have a defined frequency, causing a random variation, referred to as stochastic . The study of stellar interiors using their pulsations is known as asteroseismology .
The expansion phase of a pulsation is caused by the blocking of the internal energy flow by material with a high opacity, but this must occur at a particular depth of the star to create visible pulsations. If the expansion occurs below a convective zone then no variation will be visible at the surface. If the expansion occurs too close to the surface the restoring force will be too weak to create a pulsation. The restoring force to create the contraction phase of a pulsation can be pressure if the pulsation occurs in a non-degenerate layer deep inside a star, and this is called an acoustic or pressure mode of pulsation, abbreviated to p-mode . In other cases, the restoring force is gravity and this is called a g-mode . Pulsating variable stars typically pulsate in only one of these modes.
This group consists of several kinds of pulsating stars, all found on the instability strip , that swell and shrink very regularly caused by the star's own mass resonance , generally by the fundamental frequency . Generally the Eddington valve mechanism for pulsating variables is believed to account for cepheid-like pulsations. Each of the subgroups on the instability strip has a fixed relationship between period and absolute magnitude, as well as a relation between period and mean density of the star. The period-luminosity relationship was first established for Delta Cepheids by Henrietta Leavitt , and makes these high luminosity Cepheids very useful for determining distances to galaxies within the Local Group and beyond. Edwin Hubble used this method to prove that the so-called spiral nebulae are in fact distant galaxies.
The Cepheids are named only for Delta Cephei , while a completely separate class of variables is named after Beta Cephei .
Classical Cepheids (or Delta Cephei variables) are population I (young, massive, and luminous) yellow supergiants which undergo pulsations with very regular periods on the order of days to months. On September 10, 1784, Edward Pigott detected the variability of Eta Aquilae , the first known representative of the class of Cepheid variables. However, the namesake for classical Cepheids is the star Delta Cephei , discovered to be variable by John Goodricke a few months later.
Type II Cepheids (historically termed W Virginis stars) have extremely regular light pulsations and a luminosity relation much like the δ Cephei variables, so initially they were confused with the latter category. Type II Cepheids stars belong to older Population II stars, than do the type I Cepheids. The Type II have somewhat lower metallicity , much lower mass, somewhat lower luminosity, and a slightly offset period versus luminosity relationship, so it is always important to know which type of star is being observed.
These stars are somewhat similar to Cepheids, but are not as luminous and have shorter periods. They are older than type I Cepheids, belonging to Population II , but of lower mass than type II Cepheids. Due to their common occurrence in globular clusters , they are occasionally referred to as cluster Cepheids . They also have a well established period-luminosity relationship, and so are also useful as distance indicators. These A-type stars vary by about 0.2–2 magnitudes (20% to over 500% change in luminosity) over a period of several hours to a day or more.
Delta Scuti (δ Sct) variables are similar to Cepheids but much fainter and with much shorter periods. They were once known as Dwarf Cepheids . They often show many superimposed periods, which combine to form an extremely complex light curve. The typical δ Scuti star has an amplitude of 0.003–0.9 magnitudes (0.3% to about 130% change in luminosity) and a period of 0.01–0.2 days. Their spectral type is usually between A0 and F5.
These stars of spectral type A2 to F5, similar to δ Scuti variables, are found mainly in globular clusters. They exhibit fluctuations in their brightness in the order of 0.7 magnitude (about 100% change in luminosity) or so every 1 to 2 hours.
These stars of spectral type A or occasionally F0, a sub-class of δ Scuti variables found on the main sequence. They have extremely rapid variations with periods of a few minutes and amplitudes of a few thousandths of a magnitude.
The long period variables are cool evolved stars that pulsate with periods in the range of weeks to several years.
Mira variables are Asymptotic giant branch (AGB) red giants. Over periods of many months they fade and brighten by between 2.5 and 11 magnitudes , a 6 fold to 30,000 fold change in luminosity. Mira itself, also known as Omicron Ceti (ο Cet), varies in brightness from almost 2nd magnitude to as faint as 10th magnitude with a period of roughly 332 days. The very large visual amplitudes are mainly due to the shifting of energy output between visual and infra-red as the temperature of the star changes. In a few cases, Mira variables show dramatic period changes over a period of decades, thought to be related to the thermal pulsing cycle of the most advanced AGB stars.
These are red giants or supergiants . Semiregular variables may show a definite period on occasion, but more often show less well-defined variations that can sometimes be resolved into multiple periods. A well-known example of a semiregular variable is Betelgeuse , which varies from about magnitudes +0.2 to +1.2 (a factor 2.5 change in luminosity). At least some of the semi-regular variables are very closely related to Mira variables, possibly the only difference being pulsating in a different harmonic.
These are red giants or supergiants with little or no detectable periodicity. Some are poorly studied semiregular variables, often with multiple periods, but others may simply be chaotic.
Many variable red giants and supergiants show variations over several hundred to several thousand days. The brightness may change by several magnitudes although it is often much smaller, with the more rapid primary variations are superimposed. The reasons for this type of variation are not clearly understood, being variously ascribed to pulsations, binarity, and stellar rotation. [ 19 ] [ 20 ] [ 21 ]
Beta Cephei (β Cep) variables (sometimes called Beta Canis Majoris variables, especially in Europe) [ 22 ] undergo short period pulsations in the order of 0.1–0.6 days with an amplitude of 0.01–0.3 magnitudes (1% to 30% change in luminosity). They are at their brightest during minimum contraction. Many stars of this kind exhibits multiple pulsation periods. [ 23 ]
Slowly pulsating B (SPB) stars are hot main-sequence stars slightly less luminous than the Beta Cephei stars, with longer periods and larger amplitudes. [ 24 ]
The prototype of this rare class is V361 Hydrae , a 15th magnitude subdwarf B star . They pulsate with periods of a few minutes and may simultaneous pulsate with multiple periods. They have amplitudes of a few hundredths of a magnitude and are given the GCVS acronym RPHS. They are p-mode pulsators. [ 25 ]
Stars in this class are type Bp supergiants with a period of 0.1–1 day and an amplitude of 0.1 magnitude on average. Their spectra are peculiar by having weak hydrogen while on the other hand carbon and helium lines are extra strong, a type of extreme helium star .
These are yellow supergiant stars (actually low mass post-AGB stars at the most luminous stage of their lives) which have alternating deep and shallow minima. This double-peaked variation typically has periods of 30–100 days and amplitudes of 3–4 magnitudes. Superimposed on this variation, there may be long-term variations over periods of several years. Their spectra are of type F or G at maximum light and type K or M at minimum brightness. They lie near the instability strip, cooler than type I Cepheids more luminous than type II Cepheids. Their pulsations are caused by the same basic mechanisms related to helium opacity, but they are at a very different stage of their lives.
Alpha Cygni (α Cyg) variables are nonradially pulsating supergiants of spectral classes B ep to A ep Ia. Their periods range from several days to several weeks, and their amplitudes of variation are typically of the order of 0.1 magnitudes. The light changes, which often seem irregular, are caused by the superposition of many oscillations with close periods. Deneb , in the constellation of Cygnus is the prototype of this class.
Gamma Doradus (γ Dor) variables are non-radially pulsating main-sequence stars of spectral classes F to late A. Their periods are around one day and their amplitudes typically of the order of 0.1 magnitudes.
These non-radially pulsating stars have short periods of hundreds to thousands of seconds with tiny fluctuations of 0.001 to 0.2 magnitudes. Known types of pulsating white dwarf (or pre-white dwarf) include the DAV , or ZZ Ceti , stars, with hydrogen-dominated atmospheres and the spectral type DA; [ 26 ] DBV , or V777 Her , stars, with helium-dominated atmospheres and the spectral type DB; [ 27 ] and GW Vir stars, with atmospheres dominated by helium, carbon, and oxygen. GW Vir stars may be subdivided into DOV and PNNV stars. [ 28 ] [ 29 ]
The Sun oscillates with very low amplitude in a large number of modes having periods around 5 minutes. The study of these oscillations is known as helioseismology . Oscillations in the Sun are driven stochastically by convection in its outer layers. The term solar-like oscillations is used to describe oscillations in other stars that are excited in the same way and the study of these oscillations is one of the main areas of active research in the field of asteroseismology .
A Blue Large-Amplitude Pulsator (BLAP) is a pulsating star characterized by changes of 0.2 to 0.4 magnitudes with typical periods of 20 to 40 minutes.
A fast yellow pulsating supergiant (FYPS) is a luminous yellow supergiant with pulsations shorter than a day. They are thought to have evolved beyond a red supergiant phase, but the mechanism for the pulsations is unknown. The class was named in 2020 through analysis of TESS observations. [ 30 ]
Eruptive variable stars show irregular or semi-regular brightness variations caused by material being lost from the star, or in some cases being accreted to it. Despite the name, these are not explosive events.
Protostars are young objects that have not yet completed the process of contraction from a gas nebula to a veritable star. Most protostars exhibit irregular brightness variations.
Variability of more massive (2–8 solar mass) Herbig Ae/Be stars is thought to be due to gas-dust clumps, orbiting in the circumstellar disks.
Orion variables are young, hot pre–main-sequence stars usually embedded in nebulosity. They have irregular periods with amplitudes of several magnitudes. A well-known subtype of Orion variables are the T Tauri variables. Variability of T Tauri stars is due to spots on the stellar surface and gas-dust clumps, orbiting in the circumstellar disks.
These stars reside in reflection nebulae and show gradual increases in their luminosity in the order of 6 magnitudes followed by a lengthy phase of constant brightness. They then dim by 2 magnitudes (six times dimmer) or so over a period of many years. V1057 Cygni for example dimmed by 2.5 magnitude (ten times dimmer) during an eleven-year period. FU Orionis variables are of spectral type A through G and are possibly an evolutionary phase in the life of T Tauri stars.
Large stars lose their matter relatively easily. For this reason variability due to eruptions and mass loss is fairly common among giants and supergiants.
Also known as the S Doradus variables, the most luminous stars known belong to this class. Examples include the hypergiants η Carinae and P Cygni . They have permanent high mass loss, but at intervals of years internal pulsations cause the star to exceed its Eddington limit and the mass loss increases hugely. Visual brightness increases although the overall luminosity is largely unchanged. Giant eruptions observed in a few LBVs do increase the luminosity, so much so that they have been tagged supernova impostors , and may be a different type of event.
These massive evolved stars are unstable due to their high luminosity and position above the instability strip, and they exhibit slow but sometimes large photometric and spectroscopic changes due to high mass loss and occasional larger eruptions, combined with secular variation on an observable timescale. The best known example is Rho Cassiopeiae .
While classed as eruptive variables, these stars do not undergo periodic increases in brightness. Instead they spend most of their time at maximum brightness, but at irregular intervals they suddenly fade by 1–9 magnitudes (2.5 to 4000 times dimmer) before recovering to their initial brightness over months to years. Most are classified as yellow supergiants by luminosity, although they are actually post-AGB stars, but there are both red and blue giant R CrB stars. R Coronae Borealis (R CrB) is the prototype star. DY Persei variables are a subclass of R CrB variables that have a periodic variability in addition to their eruptions.
Classic population I Wolf–Rayet stars are massive hot stars that sometimes show variability, probably due to several different causes including binary interactions and rotating gas clumps around the star. They exhibit broad emission line spectra with helium , nitrogen , carbon and oxygen lines. Variations in some stars appear to be stochastic while others show multiple periods.
Gamma Cassiopeiae (γ Cas) variables are non-supergiant fast-rotating B class emission line-type stars that fluctuate irregularly by up to 1.5 magnitudes (4 fold change in luminosity) due to the ejection of matter at their equatorial regions caused by the rapid rotational velocity.
In main-sequence stars major eruptive variability is exceptional. It is common only among the flare stars , also known as the UV Ceti variables, very faint main-sequence stars which undergo regular flares. They increase in brightness by up to two magnitudes (six times brighter) in just a few seconds, and then fade back to normal brightness in half an hour or less. Several nearby red dwarfs are flare stars, including Proxima Centauri and Wolf 359 .
These are close binary systems with highly active chromospheres, including huge sunspots and flares, believed to be enhanced by the close companion. Variability scales ranges from days, close to the orbital period and sometimes also with eclipses, to years as sunspot activity varies.
Supernovae are the most dramatic type of cataclysmic variable, being some of the most energetic events in the universe. A supernova can briefly emit as much energy as an entire galaxy , brightening by more than 20 magnitudes (over one hundred million times brighter). The supernova explosion is caused by a white dwarf or a star core reaching a certain mass/density limit, the Chandrasekhar limit , causing the object to collapse in a fraction of a second. This collapse "bounces" and causes the star to explode and emit this enormous energy quantity. The outer layers of these stars are blown away at speeds of many thousands of kilometers per second. The expelled matter may form nebulae called supernova remnants . A well-known example of such a nebula is the Crab Nebula , left over from a supernova that was observed in China and elsewhere in 1054. The progenitor object may either disintegrate completely in the explosion, or, in the case of a massive star, the core can become a neutron star (generally a pulsar ) or a black hole .
Supernovae can result from the death of an extremely massive star, many times heavier than the Sun. At the end of the life of this massive star, a non-fusible iron core is formed from fusion ashes. This iron core is pushed towards the Chandrasekhar limit till it surpasses it and therefore collapses. One of the most studied supernovae of this type is SN 1987A in the Large Magellanic Cloud .
A supernova may also result from mass transfer onto a white dwarf from a star companion in a double star system. The Chandrasekhar limit is surpassed from the infalling matter. The absolute luminosity of this latter type is related to properties of its light curve, so that these supernovae can be used to establish the distance to other galaxies.
Luminous red novae are stellar explosions caused by the merger of two stars. They are not related to classical novae . They have a characteristic red appearance and very slow decline following the initial outburst.
Novae are also the result of dramatic explosions, but unlike supernovae do not result in the destruction of the progenitor star. Also unlike supernovae, novae ignite from the sudden onset of thermonuclear fusion, which under certain high pressure conditions ( degenerate matter ) accelerates explosively. They form in close binary systems , one component being a white dwarf accreting matter from the other ordinary star component, and may recur over periods of decades to centuries or millennia. Novae are categorised as fast , slow or very slow , depending on the behaviour of their light curve. Several naked eye novae have been recorded, Nova Cygni 1975 being the brightest in the recent history, reaching 2nd magnitude.
Dwarf novae are double stars involving a white dwarf in which matter transfer between the component gives rise to regular outbursts. There are three types of dwarf nova:
DQ Herculis systems are interacting binaries in which a low-mass star transfers mass to a highly magnetic white dwarf. The white dwarf spin period is significantly shorter than the binary orbital period and can sometimes be detected as a photometric periodicity. An accretion disk usually forms around the white dwarf, but its innermost regions are magnetically truncated by the white dwarf. Once captured by the white dwarf's magnetic field, the material from the inner disk travels along the magnetic field lines until it accretes. In extreme cases, the white dwarf's magnetism prevents the formation of an accretion disk.
In these cataclysmic variables, the white dwarf's magnetic field is so strong that it synchronizes the white dwarf's spin period with the binary orbital period. Instead of forming an accretion disk, the accretion flow is channeled along the white dwarf's magnetic field lines until it impacts the white dwarf near a magnetic pole. Cyclotron radiation beamed from the accretion region can cause orbital variations of several magnitudes.
These symbiotic binary systems are composed of a red giant and a hot blue star enveloped in a cloud of gas and dust. They undergo nova-like outbursts with amplitudes of up to 4 magnitudes. The prototype for this class is Z Andromedae .
AM CVn variables are symbiotic binaries where a white dwarf is accreting helium-rich material from either another white dwarf, a helium star, or an evolved main-sequence star. They undergo complex variations, or at times no variations, with ultrashort periods.
There are two main groups of extrinsic variables: rotating stars and eclipsing stars.
Stars with sizeable sunspots may show significant variations in brightness as they rotate, and brighter areas of the surface are brought into view. Bright spots also occur at the magnetic poles of magnetic stars. Stars with ellipsoidal shapes may also show changes in brightness as they present varying areas of their surfaces to the observer. [ 31 ]
These are very close binaries, the components of which are non-spherical due to their tidal interaction. As the stars rotate the area of their surface presented towards the observer changes and this in turn affects their brightness as seen from Earth.
The surface of the star is not uniformly bright, but has darker and brighter areas (like the sun's solar spots ). The star's chromosphere too may vary in brightness. As the star rotates we observe brightness variations of a few tenths of magnitudes.
These stars rotate extremely rapidly (~100 km/s at the equator ); hence they are ellipsoidal in shape. They are (apparently) single giant stars with spectral types G and K and show strong chromospheric emission lines . Examples are FK Com , V1794 Cygni and UZ Librae . A possible explanation for the rapid rotation of FK Comae stars is that they are the result of the merger of a (contact) binary . [ 34 ]
BY Draconis stars are of spectral class K or M and vary by less than 0.5 magnitudes (70% change in luminosity).
Alpha 2 Canum Venaticorum (α 2 CVn) variables are main-sequence stars of spectral class B8–A7 that show fluctuations of 0.01 to 0.1 magnitudes (1% to 10%) due to changes in their magnetic fields.
Stars in this class exhibit brightness fluctuations of some 0.1 magnitude caused by changes in their magnetic fields due to high rotation speeds.
Few pulsars have been detected in visible light . These neutron stars change in brightness as they rotate. Because of the rapid rotation, brightness variations are extremely fast, from milliseconds to a few seconds. The first and the best known example is the Crab Pulsar .
Extrinsic variables have variations in their brightness, as seen by terrestrial observers, due to some external source. One of the most common reasons for this is the presence of a binary companion star, so that the two together form a binary star . When seen from certain angles, one star may eclipse the other, causing a reduction in brightness. One of the most famous eclipsing binaries is Algol , or Beta Persei (β Per).
Algol variables undergo eclipses with one or two minima separated by periods of nearly constant light. The prototype of this class is Algol in the constellation Perseus .
Double periodic variables exhibit cyclical mass exchange which causes the orbital period to vary predictably over a very long period. The best known example is V393 Scorpii .
Beta Lyrae (β Lyr) variables are extremely close binaries, named after the star Sheliak . The light curves of this class of eclipsing variables are constantly changing, making it almost impossible to determine the exact onset and end of each eclipse.
W Serpentis is the prototype of a class of semi-detached binaries including a giant or supergiant transferring material to a massive more compact star. They are characterised, and distinguished from the similar β Lyr systems, by strong UV emission from accretions hotspots on a disc of material.
The stars in this group show periods of less than a day. The stars are so closely situated to each other that their surfaces are almost in contact with each other.
Stars with planets may also show brightness variations if their planets pass between Earth and the star. These variations are much smaller than those seen with stellar companions and are only detectable with extremely accurate observations. Examples include HD 209458 and GSC 02652-01324 , and all of the planets and planet candidates detected by the Kepler Mission .
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A variable structure system , or VSS , is a discontinuous nonlinear system of the form
where x ≜ [ x 1 , x 2 , … , x n ] T ∈ R n {\displaystyle \mathbf {x} \triangleq [x_{1},x_{2},\ldots ,x_{n}]^{\operatorname {T} }\in \mathbb {R} ^{n}} is the state vector, t ∈ R {\displaystyle t\in \mathbb {R} } is the time variable, and φ ( x , t ) ≜ [ φ 1 ( x , t ) , φ 2 ( x , t ) , … , φ n ( x , t ) ] T : R n + 1 ↦ R n {\displaystyle \varphi (\mathbf {x} ,t)\triangleq [\varphi _{1}(\mathbf {x} ,t),\varphi _{2}(\mathbf {x} ,t),\ldots ,\varphi _{n}(\mathbf {x} ,t)]^{\operatorname {T} }:\mathbb {R} ^{n+1}\mapsto \mathbb {R} ^{n}} is a piecewise continuous function. [ 1 ] Due to the piecewise continuity of these systems, they behave like different continuous nonlinear systems in different regions of their state space . At the boundaries of these regions, their dynamics switch abruptly. Hence, their structure varies over different parts of their state space.
The development of variable structure control depends upon methods of analyzing variable structure systems, which are special cases of hybrid dynamical systems .
2. Emelyanov, S.V., ed. (1967). Variable Structure Control Systems. Moscow: Nauka.
3. Emelyanov S, Utkin V, Tarin V, Kostyleva N, Shubladze A, Ezerov V, Dubrovsky E. 1970. Theory of Variable Structure Control Systems (in Russian). Moscow: Nauka.
4. Variable Structure Systems: From Principles to Implementation. A. Sabanovic, L. Fridman and S. Spurgeon (eds.), IEE, London, 2004, ISBN 0863413501.
5. Advances in Variable Structure Systems and Sliding Mode Control—Theory and Applications. Li, S., Yu, X., Fridman, L., Man, Z., Wang, X.(Eds.), Studies in Systems, Decision and Control, v. 115, Springer, 2017, ISBN 978-3-319-62895-0
6.Variable-Structure Systems and Sliding-Mode Control. M. Steinberger, M. Horn, L. Fridman.(eds.), Studies in Systems, Decision and Control, v.271, Springer International Publishing, Cham, 2020, ISBN 978-3-030-36620-9.
Y. Shtessel, C. Edwards, L. Fridman, A. Levant. Sliding Mode Control and Observation, Series: Control Engineering, Birkhauser: Basel, 2014, ISBN 978-0-81764-8923
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In probability theory and statistics , variance is the expected value of the squared deviation from the mean of a random variable . The standard deviation (SD) is obtained as the square root of the variance. Variance is a measure of dispersion , meaning it is a measure of how far a set of numbers is spread out from their average value. It is the second central moment of a distribution , and the covariance of the random variable with itself, and it is often represented by σ 2 {\displaystyle \sigma ^{2}} , s 2 {\displaystyle s^{2}} , Var ( X ) {\displaystyle \operatorname {Var} (X)} , V ( X ) {\displaystyle V(X)} , or V ( X ) {\displaystyle \mathbb {V} (X)} . [ 1 ]
An advantage of variance as a measure of dispersion is that it is more amenable to algebraic manipulation than other measures of dispersion such as the expected absolute deviation ; for example, the variance of a sum of uncorrelated random variables is equal to the sum of their variances. A disadvantage of the variance for practical applications is that, unlike the standard deviation, its units differ from the random variable, which is why the standard deviation is more commonly reported as a measure of dispersion once the calculation is finished. Another disadvantage is that the variance is not finite for many distributions.
There are two distinct concepts that are both called "variance". One, as discussed above, is part of a theoretical probability distribution and is defined by an equation. The other variance is a characteristic of a set of observations. When variance is calculated from observations, those observations are typically measured from a real-world system. If all possible observations of the system are present, then the calculated variance is called the population variance. Normally, however, only a subset is available, and the variance calculated from this is called the sample variance. The variance calculated from a sample is considered an estimate of the full population variance. There are multiple ways to calculate an estimate of the population variance, as discussed in the section below.
The two kinds of variance are closely related. To see how, consider that a theoretical probability distribution can be used as a generator of hypothetical observations. If an infinite number of observations are generated using a distribution, then the sample variance calculated from that infinite set will match the value calculated using the distribution's equation for variance. Variance has a central role in statistics, where some ideas that use it include descriptive statistics , statistical inference , hypothesis testing , goodness of fit , and Monte Carlo sampling .
The variance of a random variable X {\displaystyle X} is the expected value of the squared deviation from the mean of X {\displaystyle X} , μ = E [ X ] {\displaystyle \mu =\operatorname {E} [X]} : Var ( X ) = E [ ( X − μ ) 2 ] . {\displaystyle \operatorname {Var} (X)=\operatorname {E} \left[(X-\mu )^{2}\right].} This definition encompasses random variables that are generated by processes that are discrete , continuous , neither , or mixed. The variance can also be thought of as the covariance of a random variable with itself:
Var ( X ) = Cov ( X , X ) . {\displaystyle \operatorname {Var} (X)=\operatorname {Cov} (X,X).} The variance is also equivalent to the second cumulant of a probability distribution that generates X {\displaystyle X} . The variance is typically designated as Var ( X ) {\displaystyle \operatorname {Var} (X)} , or sometimes as V ( X ) {\displaystyle V(X)} or V ( X ) {\displaystyle \mathbb {V} (X)} , or symbolically as σ X 2 {\displaystyle \sigma _{X}^{2}} or simply σ 2 {\displaystyle \sigma ^{2}} (pronounced " sigma squared"). The expression for the variance can be expanded as follows: Var ( X ) = E [ ( X − E [ X ] ) 2 ] = E [ X 2 − 2 X E [ X ] + E [ X ] 2 ] = E [ X 2 ] − 2 E [ X ] E [ X ] + E [ X ] 2 = E [ X 2 ] − 2 E [ X ] 2 + E [ X ] 2 = E [ X 2 ] − E [ X ] 2 {\displaystyle {\begin{aligned}\operatorname {Var} (X)&=\operatorname {E} \left[{\left(X-\operatorname {E} [X]\right)}^{2}\right]\\[4pt]&=\operatorname {E} \left[X^{2}-2X\operatorname {E} [X]+\operatorname {E} [X]^{2}\right]\\[4pt]&=\operatorname {E} \left[X^{2}\right]-2\operatorname {E} [X]\operatorname {E} [X]+\operatorname {E} [X]^{2}\\[4pt]&=\operatorname {E} \left[X^{2}\right]-2\operatorname {E} [X]^{2}+\operatorname {E} [X]^{2}\\[4pt]&=\operatorname {E} \left[X^{2}\right]-\operatorname {E} [X]^{2}\end{aligned}}}
In other words, the variance of X is equal to the mean of the square of X minus the square of the mean of X . This equation should not be used for computations using floating-point arithmetic , because it suffers from catastrophic cancellation if the two components of the equation are similar in magnitude. For other numerically stable alternatives, see algorithms for calculating variance .
If the generator of random variable X {\displaystyle X} is discrete with probability mass function x 1 ↦ p 1 , x 2 ↦ p 2 , … , x n ↦ p n {\displaystyle x_{1}\mapsto p_{1},x_{2}\mapsto p_{2},\ldots ,x_{n}\mapsto p_{n}} , then
Var ( X ) = ∑ i = 1 n p i ⋅ ( x i − μ ) 2 , {\displaystyle \operatorname {Var} (X)=\sum _{i=1}^{n}p_{i}\cdot {\left(x_{i}-\mu \right)}^{2},}
where μ {\displaystyle \mu } is the expected value. That is,
μ = ∑ i = 1 n p i x i . {\displaystyle \mu =\sum _{i=1}^{n}p_{i}x_{i}.}
(When such a discrete weighted variance is specified by weights whose sum is not 1, then one divides by the sum of the weights.)
The variance of a collection of n {\displaystyle n} equally likely values can be written as
Var ( X ) = 1 n ∑ i = 1 n ( x i − μ ) 2 {\displaystyle \operatorname {Var} (X)={\frac {1}{n}}\sum _{i=1}^{n}(x_{i}-\mu )^{2}}
where μ {\displaystyle \mu } is the average value. That is,
μ = 1 n ∑ i = 1 n x i . {\displaystyle \mu ={\frac {1}{n}}\sum _{i=1}^{n}x_{i}.}
The variance of a set of n {\displaystyle n} equally likely values can be equivalently expressed, without directly referring to the mean, in terms of squared deviations of all pairwise squared distances of points from each other: [ 2 ]
Var ( X ) = 1 n 2 ∑ i = 1 n ∑ j = 1 n 1 2 ( x i − x j ) 2 = 1 n 2 ∑ i ∑ j > i ( x i − x j ) 2 . {\displaystyle \operatorname {Var} (X)={\frac {1}{n^{2}}}\sum _{i=1}^{n}\sum _{j=1}^{n}{\frac {1}{2}}{\left(x_{i}-x_{j}\right)}^{2}={\frac {1}{n^{2}}}\sum _{i}\sum _{j>i}{\left(x_{i}-x_{j}\right)}^{2}.}
If the random variable X {\displaystyle X} has a probability density function f ( x ) {\displaystyle f(x)} , and F ( x ) {\displaystyle F(x)} is the corresponding cumulative distribution function , then
Var ( X ) = σ 2 = ∫ R ( x − μ ) 2 f ( x ) d x = ∫ R x 2 f ( x ) d x − 2 μ ∫ R x f ( x ) d x + μ 2 ∫ R f ( x ) d x = ∫ R x 2 d F ( x ) − 2 μ ∫ R x d F ( x ) + μ 2 ∫ R d F ( x ) = ∫ R x 2 d F ( x ) − 2 μ ⋅ μ + μ 2 ⋅ 1 = ∫ R x 2 d F ( x ) − μ 2 , {\displaystyle {\begin{aligned}\operatorname {Var} (X)=\sigma ^{2}&=\int _{\mathbb {R} }{\left(x-\mu \right)}^{2}f(x)\,dx\\[4pt]&=\int _{\mathbb {R} }x^{2}f(x)\,dx-2\mu \int _{\mathbb {R} }xf(x)\,dx+\mu ^{2}\int _{\mathbb {R} }f(x)\,dx\\[4pt]&=\int _{\mathbb {R} }x^{2}\,dF(x)-2\mu \int _{\mathbb {R} }x\,dF(x)+\mu ^{2}\int _{\mathbb {R} }\,dF(x)\\[4pt]&=\int _{\mathbb {R} }x^{2}\,dF(x)-2\mu \cdot \mu +\mu ^{2}\cdot 1\\[4pt]&=\int _{\mathbb {R} }x^{2}\,dF(x)-\mu ^{2},\end{aligned}}}
or equivalently,
Var ( X ) = ∫ R x 2 f ( x ) d x − μ 2 , {\displaystyle \operatorname {Var} (X)=\int _{\mathbb {R} }x^{2}f(x)\,dx-\mu ^{2},}
where μ {\displaystyle \mu } is the expected value of X {\displaystyle X} given by
μ = ∫ R x f ( x ) d x = ∫ R x d F ( x ) . {\displaystyle \mu =\int _{\mathbb {R} }xf(x)\,dx=\int _{\mathbb {R} }x\,dF(x).}
In these formulas, the integrals with respect to d x {\displaystyle dx} and d F ( x ) {\displaystyle dF(x)} are Lebesgue and Lebesgue–Stieltjes integrals, respectively.
If the function x 2 f ( x ) {\displaystyle x^{2}f(x)} is Riemann-integrable on every finite interval [ a , b ] ⊂ R , {\displaystyle [a,b]\subset \mathbb {R} ,} then
Var ( X ) = ∫ − ∞ + ∞ x 2 f ( x ) d x − μ 2 , {\displaystyle \operatorname {Var} (X)=\int _{-\infty }^{+\infty }x^{2}f(x)\,dx-\mu ^{2},}
where the integral is an improper Riemann integral .
The exponential distribution with parameter λ > 0 is a continuous distribution whose probability density function is given by f ( x ) = λ e − λ x {\displaystyle f(x)=\lambda e^{-\lambda x}} on the interval [0, ∞) . Its mean can be shown to be E [ X ] = ∫ 0 ∞ x λ e − λ x d x = 1 λ . {\displaystyle \operatorname {E} [X]=\int _{0}^{\infty }x\lambda e^{-\lambda x}\,dx={\frac {1}{\lambda }}.}
Using integration by parts and making use of the expected value already calculated, we have: E [ X 2 ] = ∫ 0 ∞ x 2 λ e − λ x d x = [ − x 2 e − λ x ] 0 ∞ + ∫ 0 ∞ 2 x e − λ x d x = 0 + 2 λ E [ X ] = 2 λ 2 . {\displaystyle {\begin{aligned}\operatorname {E} \left[X^{2}\right]&=\int _{0}^{\infty }x^{2}\lambda e^{-\lambda x}\,dx\\&={\left[-x^{2}e^{-\lambda x}\right]}_{0}^{\infty }+\int _{0}^{\infty }2xe^{-\lambda x}\,dx\\&=0+{\frac {2}{\lambda }}\operatorname {E} [X]\\&={\frac {2}{\lambda ^{2}}}.\end{aligned}}}
Thus, the variance of X is given by Var ( X ) = E [ X 2 ] − E [ X ] 2 = 2 λ 2 − ( 1 λ ) 2 = 1 λ 2 . {\displaystyle \operatorname {Var} (X)=\operatorname {E} \left[X^{2}\right]-\operatorname {E} [X]^{2}={\frac {2}{\lambda ^{2}}}-\left({\frac {1}{\lambda }}\right)^{2}={\frac {1}{\lambda ^{2}}}.}
A fair six-sided die can be modeled as a discrete random variable, X , with outcomes 1 through 6, each with equal probability 1/6. The expected value of X is ( 1 + 2 + 3 + 4 + 5 + 6 ) / 6 = 7 / 2. {\displaystyle (1+2+3+4+5+6)/6=7/2.} Therefore, the variance of X is Var ( X ) = ∑ i = 1 6 1 6 ( i − 7 2 ) 2 = 1 6 ( ( − 5 / 2 ) 2 + ( − 3 / 2 ) 2 + ( − 1 / 2 ) 2 + ( 1 / 2 ) 2 + ( 3 / 2 ) 2 + ( 5 / 2 ) 2 ) = 35 12 ≈ 2.92. {\displaystyle {\begin{aligned}\operatorname {Var} (X)&=\sum _{i=1}^{6}{\frac {1}{6}}\left(i-{\frac {7}{2}}\right)^{2}\\[5pt]&={\frac {1}{6}}\left((-5/2)^{2}+(-3/2)^{2}+(-1/2)^{2}+(1/2)^{2}+(3/2)^{2}+(5/2)^{2}\right)\\[5pt]&={\frac {35}{12}}\approx 2.92.\end{aligned}}}
The general formula for the variance of the outcome, X , of an n -sided die is Var ( X ) = E ( X 2 ) − ( E ( X ) ) 2 = 1 n ∑ i = 1 n i 2 − ( 1 n ∑ i = 1 n i ) 2 = ( n + 1 ) ( 2 n + 1 ) 6 − ( n + 1 2 ) 2 = n 2 − 1 12 . {\displaystyle {\begin{aligned}\operatorname {Var} (X)&=\operatorname {E} \left(X^{2}\right)-(\operatorname {E} (X))^{2}\\[5pt]&={\frac {1}{n}}\sum _{i=1}^{n}i^{2}-\left({\frac {1}{n}}\sum _{i=1}^{n}i\right)^{2}\\[5pt]&={\frac {(n+1)(2n+1)}{6}}-\left({\frac {n+1}{2}}\right)^{2}\\[4pt]&={\frac {n^{2}-1}{12}}.\end{aligned}}}
The following table lists the variance for some commonly used probability distributions.
Variance is non-negative because the squares are positive or zero: Var ( X ) ≥ 0. {\displaystyle \operatorname {Var} (X)\geq 0.}
The variance of a constant is zero. Var ( a ) = 0. {\displaystyle \operatorname {Var} (a)=0.}
Conversely, if the variance of a random variable is 0, then it is almost surely a constant. That is, it always has the same value: Var ( X ) = 0 ⟺ ∃ a : P ( X = a ) = 1. {\displaystyle \operatorname {Var} (X)=0\iff \exists a:P(X=a)=1.}
If a distribution does not have a finite expected value, as is the case for the Cauchy distribution , then the variance cannot be finite either. However, some distributions may not have a finite variance, despite their expected value being finite. An example is a Pareto distribution whose index k {\displaystyle k} satisfies 1 < k ≤ 2. {\displaystyle 1<k\leq 2.}
The general formula for variance decomposition or the law of total variance is: If X {\displaystyle X} and Y {\displaystyle Y} are two random variables, and the variance of X {\displaystyle X} exists, then
Var [ X ] = E ( Var [ X ∣ Y ] ) + Var ( E [ X ∣ Y ] ) . {\displaystyle \operatorname {Var} [X]=\operatorname {E} (\operatorname {Var} [X\mid Y])+\operatorname {Var} (\operatorname {E} [X\mid Y]).}
The conditional expectation E ( X ∣ Y ) {\displaystyle \operatorname {E} (X\mid Y)} of X {\displaystyle X} given Y {\displaystyle Y} , and the conditional variance Var ( X ∣ Y ) {\displaystyle \operatorname {Var} (X\mid Y)} may be understood as follows. Given any particular value y of the random variable Y , there is a conditional expectation E ( X ∣ Y = y ) {\displaystyle \operatorname {E} (X\mid Y=y)} given the event Y = y . This quantity depends on the particular value y ; it is a function g ( y ) = E ( X ∣ Y = y ) {\displaystyle g(y)=\operatorname {E} (X\mid Y=y)} . That same function evaluated at the random variable Y is the conditional expectation E ( X ∣ Y ) = g ( Y ) . {\displaystyle \operatorname {E} (X\mid Y)=g(Y).}
In particular, if Y {\displaystyle Y} is a discrete random variable assuming possible values y 1 , y 2 , y 3 … {\displaystyle y_{1},y_{2},y_{3}\ldots } with corresponding probabilities p 1 , p 2 , p 3 … , {\displaystyle p_{1},p_{2},p_{3}\ldots ,} , then in the formula for total variance, the first term on the right-hand side becomes
E ( Var [ X ∣ Y ] ) = ∑ i p i σ i 2 , {\displaystyle \operatorname {E} (\operatorname {Var} [X\mid Y])=\sum _{i}p_{i}\sigma _{i}^{2},}
where σ i 2 = Var [ X ∣ Y = y i ] {\displaystyle \sigma _{i}^{2}=\operatorname {Var} [X\mid Y=y_{i}]} . Similarly, the second term on the right-hand side becomes
Var ( E [ X ∣ Y ] ) = ∑ i p i μ i 2 − ( ∑ i p i μ i ) 2 = ∑ i p i μ i 2 − μ 2 , {\displaystyle \operatorname {Var} (\operatorname {E} [X\mid Y])=\sum _{i}p_{i}\mu _{i}^{2}-\left(\sum _{i}p_{i}\mu _{i}\right)^{2}=\sum _{i}p_{i}\mu _{i}^{2}-\mu ^{2},}
where μ i = E [ X ∣ Y = y i ] {\displaystyle \mu _{i}=\operatorname {E} [X\mid Y=y_{i}]} and μ = ∑ i p i μ i {\displaystyle \mu =\sum _{i}p_{i}\mu _{i}} . Thus the total variance is given by
Var [ X ] = ∑ i p i σ i 2 + ( ∑ i p i μ i 2 − μ 2 ) . {\displaystyle \operatorname {Var} [X]=\sum _{i}p_{i}\sigma _{i}^{2}+\left(\sum _{i}p_{i}\mu _{i}^{2}-\mu ^{2}\right).}
A similar formula is applied in analysis of variance , where the corresponding formula is
M S total = M S between + M S within ; {\displaystyle {\mathit {MS}}_{\text{total}}={\mathit {MS}}_{\text{between}}+{\mathit {MS}}_{\text{within}};}
here M S {\displaystyle {\mathit {MS}}} refers to the Mean of the Squares. In linear regression analysis the corresponding formula is
M S total = M S regression + M S residual . {\displaystyle {\mathit {MS}}_{\text{total}}={\mathit {MS}}_{\text{regression}}+{\mathit {MS}}_{\text{residual}}.}
This can also be derived from the additivity of variances, since the total (observed) score is the sum of the predicted score and the error score, where the latter two are uncorrelated.
Similar decompositions are possible for the sum of squared deviations (sum of squares, S S {\displaystyle {\mathit {SS}}} ): S S total = S S between + S S within , {\displaystyle {\mathit {SS}}_{\text{total}}={\mathit {SS}}_{\text{between}}+{\mathit {SS}}_{\text{within}},} S S total = S S regression + S S residual . {\displaystyle {\mathit {SS}}_{\text{total}}={\mathit {SS}}_{\text{regression}}+{\mathit {SS}}_{\text{residual}}.}
The population variance for a non-negative random variable can be expressed in terms of the cumulative distribution function F using
2 ∫ 0 ∞ u ( 1 − F ( u ) ) d u − [ ∫ 0 ∞ ( 1 − F ( u ) ) d u ] 2 . {\displaystyle 2\int _{0}^{\infty }u(1-F(u))\,du-{\left[\int _{0}^{\infty }(1-F(u))\,du\right]}^{2}.}
This expression can be used to calculate the variance in situations where the CDF, but not the density , can be conveniently expressed.
The second moment of a random variable attains the minimum value when taken around the first moment (i.e., mean) of the random variable, i.e. a r g m i n m E ( ( X − m ) 2 ) = E ( X ) {\displaystyle \mathrm {argmin} _{m}\,\mathrm {E} \left(\left(X-m\right)^{2}\right)=\mathrm {E} (X)} . Conversely, if a continuous function φ {\displaystyle \varphi } satisfies a r g m i n m E ( φ ( X − m ) ) = E ( X ) {\displaystyle \mathrm {argmin} _{m}\,\mathrm {E} (\varphi (X-m))=\mathrm {E} (X)} for all random variables X , then it is necessarily of the form φ ( x ) = a x 2 + b {\displaystyle \varphi (x)=ax^{2}+b} , where a > 0 . This also holds in the multidimensional case. [ 3 ]
Unlike the expected absolute deviation , the variance of a variable has units that are the square of the units of the variable itself. For example, a variable measured in meters will have a variance measured in meters squared. For this reason, describing data sets via their standard deviation or root mean square deviation is often preferred over using the variance. In the dice example the standard deviation is √ 2.9 ≈ 1.7 , slightly larger than the expected absolute deviation of 1.5.
The standard deviation and the expected absolute deviation can both be used as an indicator of the "spread" of a distribution. The standard deviation is more amenable to algebraic manipulation than the expected absolute deviation, and, together with variance and its generalization covariance , is used frequently in theoretical statistics; however the expected absolute deviation tends to be more robust as it is less sensitive to outliers arising from measurement anomalies or an unduly heavy-tailed distribution .
Variance is invariant with respect to changes in a location parameter . That is, if a constant is added to all values of the variable, the variance is unchanged: Var ( X + a ) = Var ( X ) . {\displaystyle \operatorname {Var} (X+a)=\operatorname {Var} (X).}
If all values are scaled by a constant, the variance is scaled by the square of that constant: Var ( a X ) = a 2 Var ( X ) . {\displaystyle \operatorname {Var} (aX)=a^{2}\operatorname {Var} (X).}
The variance of a sum of two random variables is given by Var ( a X + b Y ) = a 2 Var ( X ) + b 2 Var ( Y ) + 2 a b Cov ( X , Y ) Var ( a X − b Y ) = a 2 Var ( X ) + b 2 Var ( Y ) − 2 a b Cov ( X , Y ) {\displaystyle {\begin{aligned}\operatorname {Var} (aX+bY)&=a^{2}\operatorname {Var} (X)+b^{2}\operatorname {Var} (Y)+2ab\,\operatorname {Cov} (X,Y)\\[1ex]\operatorname {Var} (aX-bY)&=a^{2}\operatorname {Var} (X)+b^{2}\operatorname {Var} (Y)-2ab\,\operatorname {Cov} (X,Y)\end{aligned}}}
where Cov ( X , Y ) {\displaystyle \operatorname {Cov} (X,Y)} is the covariance .
In general, for the sum of N {\displaystyle N} random variables { X 1 , … , X N } {\displaystyle \{X_{1},\dots ,X_{N}\}} , the variance becomes: Var ( ∑ i = 1 N X i ) = ∑ i , j = 1 N Cov ( X i , X j ) = ∑ i = 1 N Var ( X i ) + ∑ i , j = 1 , i ≠ j N Cov ( X i , X j ) , {\displaystyle \operatorname {Var} \left(\sum _{i=1}^{N}X_{i}\right)=\sum _{i,j=1}^{N}\operatorname {Cov} (X_{i},X_{j})=\sum _{i=1}^{N}\operatorname {Var} (X_{i})+\sum _{i,j=1,i\neq j}^{N}\operatorname {Cov} (X_{i},X_{j}),} see also general Bienaymé's identity .
These results lead to the variance of a linear combination as:
Var ( ∑ i = 1 N a i X i ) = ∑ i , j = 1 N a i a j Cov ( X i , X j ) = ∑ i = 1 N a i 2 Var ( X i ) + ∑ i ≠ j a i a j Cov ( X i , X j ) = ∑ i = 1 N a i 2 Var ( X i ) + 2 ∑ 1 ≤ i < j ≤ N a i a j Cov ( X i , X j ) . {\displaystyle {\begin{aligned}\operatorname {Var} \left(\sum _{i=1}^{N}a_{i}X_{i}\right)&=\sum _{i,j=1}^{N}a_{i}a_{j}\operatorname {Cov} (X_{i},X_{j})\\&=\sum _{i=1}^{N}a_{i}^{2}\operatorname {Var} (X_{i})+\sum _{i\neq j}a_{i}a_{j}\operatorname {Cov} (X_{i},X_{j})\\&=\sum _{i=1}^{N}a_{i}^{2}\operatorname {Var} (X_{i})+2\sum _{1\leq i<j\leq N}a_{i}a_{j}\operatorname {Cov} (X_{i},X_{j}).\end{aligned}}}
If the random variables X 1 , … , X N {\displaystyle X_{1},\dots ,X_{N}} are such that Cov ( X i , X j ) = 0 , ∀ ( i ≠ j ) , {\displaystyle \operatorname {Cov} (X_{i},X_{j})=0\ ,\ \forall \ (i\neq j),} then they are said to be uncorrelated . It follows immediately from the expression given earlier that if the random variables X 1 , … , X N {\displaystyle X_{1},\dots ,X_{N}} are uncorrelated, then the variance of their sum is equal to the sum of their variances, or, expressed symbolically:
Var ( ∑ i = 1 N X i ) = ∑ i = 1 N Var ( X i ) . {\displaystyle \operatorname {Var} \left(\sum _{i=1}^{N}X_{i}\right)=\sum _{i=1}^{N}\operatorname {Var} (X_{i}).}
Since independent random variables are always uncorrelated (see Covariance § Uncorrelatedness and independence ), the equation above holds in particular when the random variables X 1 , … , X n {\displaystyle X_{1},\dots ,X_{n}} are independent. Thus, independence is sufficient but not necessary for the variance of the sum to equal the sum of the variances.
Define X {\displaystyle X} as a column vector of n {\displaystyle n} random variables X 1 , … , X n {\displaystyle X_{1},\ldots ,X_{n}} , and c {\displaystyle c} as a column vector of n {\displaystyle n} scalars c 1 , … , c n {\displaystyle c_{1},\ldots ,c_{n}} . Therefore, c T X {\displaystyle c^{\mathsf {T}}X} is a linear combination of these random variables, where c T {\displaystyle c^{\mathsf {T}}} denotes the transpose of c {\displaystyle c} . Also let Σ {\displaystyle \Sigma } be the covariance matrix of X {\displaystyle X} . The variance of c T X {\displaystyle c^{\mathsf {T}}X} is then given by: [ 4 ]
Var ( c T X ) = c T Σ c . {\displaystyle \operatorname {Var} \left(c^{\mathsf {T}}X\right)=c^{\mathsf {T}}\Sigma c.}
This implies that the variance of the mean can be written as (with a column vector of ones)
Var ( x ¯ ) = Var ( 1 n 1 ′ X ) = 1 n 2 1 ′ Σ 1. {\displaystyle \operatorname {Var} \left({\bar {x}}\right)=\operatorname {Var} \left({\frac {1}{n}}1'X\right)={\frac {1}{n^{2}}}1'\Sigma 1.}
One reason for the use of the variance in preference to other measures of dispersion is that the variance of the sum (or the difference) of uncorrelated random variables is the sum of their variances:
Var ( ∑ i = 1 n X i ) = ∑ i = 1 n Var ( X i ) . {\displaystyle \operatorname {Var} \left(\sum _{i=1}^{n}X_{i}\right)=\sum _{i=1}^{n}\operatorname {Var} (X_{i}).}
This statement is called the Bienaymé formula [ 5 ] and was discovered in 1853. [ 6 ] [ 7 ] It is often made with the stronger condition that the variables are independent , but being uncorrelated suffices. So if all the variables have the same variance σ 2 , then, since division by n is a linear transformation, this formula immediately implies that the variance of their mean is
Var ( X ¯ ) = Var ( 1 n ∑ i = 1 n X i ) = 1 n 2 ∑ i = 1 n Var ( X i ) = 1 n 2 n σ 2 = σ 2 n . {\displaystyle \operatorname {Var} \left({\overline {X}}\right)=\operatorname {Var} \left({\frac {1}{n}}\sum _{i=1}^{n}X_{i}\right)={\frac {1}{n^{2}}}\sum _{i=1}^{n}\operatorname {Var} \left(X_{i}\right)={\frac {1}{n^{2}}}n\sigma ^{2}={\frac {\sigma ^{2}}{n}}.}
That is, the variance of the mean decreases when n increases. This formula for the variance of the mean is used in the definition of the standard error of the sample mean, which is used in the central limit theorem .
To prove the initial statement, it suffices to show that
Var ( X + Y ) = Var ( X ) + Var ( Y ) . {\displaystyle \operatorname {Var} (X+Y)=\operatorname {Var} (X)+\operatorname {Var} (Y).}
The general result then follows by induction. Starting with the definition,
Var ( X + Y ) = E [ ( X + Y ) 2 ] − ( E [ X + Y ] ) 2 = E [ X 2 + 2 X Y + Y 2 ] − ( E [ X ] + E [ Y ] ) 2 . {\displaystyle {\begin{aligned}\operatorname {Var} (X+Y)&=\operatorname {E} \left[(X+Y)^{2}\right]-(\operatorname {E} [X+Y])^{2}\\[5pt]&=\operatorname {E} \left[X^{2}+2XY+Y^{2}\right]-(\operatorname {E} [X]+\operatorname {E} [Y])^{2}.\end{aligned}}}
Using the linearity of the expectation operator and the assumption of independence (or uncorrelatedness) of X and Y , this further simplifies as follows:
Var ( X + Y ) = E [ X 2 ] + 2 E [ X Y ] + E [ Y 2 ] − ( E [ X ] 2 + 2 E [ X ] E [ Y ] + E [ Y ] 2 ) = E [ X 2 ] + E [ Y 2 ] − E [ X ] 2 − E [ Y ] 2 = Var ( X ) + Var ( Y ) . {\displaystyle {\begin{aligned}\operatorname {Var} (X+Y)&=\operatorname {E} {\left[X^{2}\right]}+2\operatorname {E} [XY]+\operatorname {E} {\left[Y^{2}\right]}-\left(\operatorname {E} [X]^{2}+2\operatorname {E} [X]\operatorname {E} [Y]+\operatorname {E} [Y]^{2}\right)\\[5pt]&=\operatorname {E} \left[X^{2}\right]+\operatorname {E} \left[Y^{2}\right]-\operatorname {E} [X]^{2}-\operatorname {E} [Y]^{2}\\[5pt]&=\operatorname {Var} (X)+\operatorname {Var} (Y).\end{aligned}}}
In general, the variance of the sum of n variables is the sum of their covariances :
Var ( ∑ i = 1 n X i ) = ∑ i = 1 n ∑ j = 1 n Cov ( X i , X j ) = ∑ i = 1 n Var ( X i ) + 2 ∑ 1 ≤ i < j ≤ n Cov ( X i , X j ) . {\displaystyle \operatorname {Var} \left(\sum _{i=1}^{n}X_{i}\right)=\sum _{i=1}^{n}\sum _{j=1}^{n}\operatorname {Cov} \left(X_{i},X_{j}\right)=\sum _{i=1}^{n}\operatorname {Var} \left(X_{i}\right)+2\sum _{1\leq i<j\leq n}\operatorname {Cov} \left(X_{i},X_{j}\right).}
(Note: The second equality comes from the fact that Cov( X i , X i ) = Var( X i ) .)
Here, Cov ( ⋅ , ⋅ ) {\displaystyle \operatorname {Cov} (\cdot ,\cdot )} is the covariance , which is zero for independent random variables (if it exists). The formula states that the variance of a sum is equal to the sum of all elements in the covariance matrix of the components. The next expression states equivalently that the variance of the sum is the sum of the diagonal of covariance matrix plus two times the sum of its upper triangular elements (or its lower triangular elements); this emphasizes that the covariance matrix is symmetric. This formula is used in the theory of Cronbach's alpha in classical test theory .
So, if the variables have equal variance σ 2 and the average correlation of distinct variables is ρ , then the variance of their mean is
Var ( X ¯ ) = σ 2 n + n − 1 n ρ σ 2 . {\displaystyle \operatorname {Var} \left({\overline {X}}\right)={\frac {\sigma ^{2}}{n}}+{\frac {n-1}{n}}\rho \sigma ^{2}.}
This implies that the variance of the mean increases with the average of the correlations. In other words, additional correlated observations are not as effective as additional independent observations at reducing the uncertainty of the mean . Moreover, if the variables have unit variance, for example if they are standardized, then this simplifies to
Var ( X ¯ ) = 1 n + n − 1 n ρ . {\displaystyle \operatorname {Var} \left({\overline {X}}\right)={\frac {1}{n}}+{\frac {n-1}{n}}\rho .}
This formula is used in the Spearman–Brown prediction formula of classical test theory. This converges to ρ if n goes to infinity, provided that the average correlation remains constant or converges too. So for the variance of the mean of standardized variables with equal correlations or converging average correlation we have
lim n → ∞ Var ( X ¯ ) = ρ . {\displaystyle \lim _{n\to \infty }\operatorname {Var} \left({\overline {X}}\right)=\rho .}
Therefore, the variance of the mean of a large number of standardized variables is approximately equal to their average correlation. This makes clear that the sample mean of correlated variables does not generally converge to the population mean, even though the law of large numbers states that the sample mean will converge for independent variables.
There are cases when a sample is taken without knowing, in advance, how many observations will be acceptable according to some criterion. In such cases, the sample size N is a random variable whose variation adds to the variation of X , such that, [ 8 ] Var ( ∑ i = 1 N X i ) = E [ N ] Var ( X ) + Var ( N ) ( E [ X ] ) 2 {\displaystyle \operatorname {Var} \left(\sum _{i=1}^{N}X_{i}\right)=\operatorname {E} \left[N\right]\operatorname {Var} (X)+\operatorname {Var} (N)(\operatorname {E} \left[X\right])^{2}} which follows from the law of total variance .
If N has a Poisson distribution , then E [ N ] = Var ( N ) {\displaystyle \operatorname {E} [N]=\operatorname {Var} (N)} with estimator n = N . So, the estimator of Var ( ∑ i = 1 n X i ) {\displaystyle \operatorname {Var} \left(\sum _{i=1}^{n}X_{i}\right)} becomes n S x 2 + n X ¯ 2 {\displaystyle n{S_{x}}^{2}+n{\bar {X}}^{2}} , giving SE ( X ¯ ) = S x 2 + X ¯ 2 n {\displaystyle \operatorname {SE} ({\bar {X}})={\sqrt {\frac {{S_{x}}^{2}+{\bar {X}}^{2}}{n}}}} (see standard error of the sample mean ).
The scaling property and the Bienaymé formula, along with the property of the covariance Cov( aX , bY ) = ab Cov( X , Y ) jointly imply that
Var ( a X ± b Y ) = a 2 Var ( X ) + b 2 Var ( Y ) ± 2 a b Cov ( X , Y ) . {\displaystyle \operatorname {Var} (aX\pm bY)=a^{2}\operatorname {Var} (X)+b^{2}\operatorname {Var} (Y)\pm 2ab\,\operatorname {Cov} (X,Y).}
This implies that in a weighted sum of variables, the variable with the largest weight will have a disproportionally large weight in the variance of the total. For example, if X and Y are uncorrelated and the weight of X is two times the weight of Y , then the weight of the variance of X will be four times the weight of the variance of Y .
The expression above can be extended to a weighted sum of multiple variables:
Var ( ∑ i n a i X i ) = ∑ i = 1 n a i 2 Var ( X i ) + 2 ∑ 1 ≤ i ∑ < j ≤ n a i a j Cov ( X i , X j ) {\displaystyle \operatorname {Var} \left(\sum _{i}^{n}a_{i}X_{i}\right)=\sum _{i=1}^{n}a_{i}^{2}\operatorname {Var} (X_{i})+2\sum _{1\leq i}\sum _{<j\leq n}a_{i}a_{j}\operatorname {Cov} (X_{i},X_{j})}
If two variables X and Y are independent , the variance of their product is given by [ 9 ] Var ( X Y ) = [ E ( X ) ] 2 Var ( Y ) + [ E ( Y ) ] 2 Var ( X ) + Var ( X ) Var ( Y ) . {\displaystyle \operatorname {Var} (XY)=[\operatorname {E} (X)]^{2}\operatorname {Var} (Y)+[\operatorname {E} (Y)]^{2}\operatorname {Var} (X)+\operatorname {Var} (X)\operatorname {Var} (Y).}
Equivalently, using the basic properties of expectation, it is given by
Var ( X Y ) = E ( X 2 ) E ( Y 2 ) − [ E ( X ) ] 2 [ E ( Y ) ] 2 . {\displaystyle \operatorname {Var} (XY)=\operatorname {E} \left(X^{2}\right)\operatorname {E} \left(Y^{2}\right)-[\operatorname {E} (X)]^{2}[\operatorname {E} (Y)]^{2}.}
In general, if two variables are statistically dependent, then the variance of their product is given by: Var ( X Y ) = E [ X 2 Y 2 ] − [ E ( X Y ) ] 2 = Cov ( X 2 , Y 2 ) + E ( X 2 ) E ( Y 2 ) − [ E ( X Y ) ] 2 = Cov ( X 2 , Y 2 ) + ( Var ( X ) + [ E ( X ) ] 2 ) ( Var ( Y ) + [ E ( Y ) ] 2 ) − [ Cov ( X , Y ) + E ( X ) E ( Y ) ] 2 {\displaystyle {\begin{aligned}\operatorname {Var} (XY)={}&\operatorname {E} \left[X^{2}Y^{2}\right]-[\operatorname {E} (XY)]^{2}\\[5pt]={}&\operatorname {Cov} \left(X^{2},Y^{2}\right)+\operatorname {E} (X^{2})\operatorname {E} \left(Y^{2}\right)-[\operatorname {E} (XY)]^{2}\\[5pt]={}&\operatorname {Cov} \left(X^{2},Y^{2}\right)+\left(\operatorname {Var} (X)+[\operatorname {E} (X)]^{2}\right)\left(\operatorname {Var} (Y)+[\operatorname {E} (Y)]^{2}\right)\\[5pt]&-[\operatorname {Cov} (X,Y)+\operatorname {E} (X)\operatorname {E} (Y)]^{2}\end{aligned}}}
The delta method uses second-order Taylor expansions to approximate the variance of a function of one or more random variables: see Taylor expansions for the moments of functions of random variables . For example, the approximate variance of a function of one variable is given by
Var [ f ( X ) ] ≈ ( f ′ ( E [ X ] ) ) 2 Var [ X ] {\displaystyle \operatorname {Var} \left[f(X)\right]\approx \left(f'(\operatorname {E} \left[X\right])\right)^{2}\operatorname {Var} \left[X\right]}
provided that f is twice differentiable and that the mean and variance of X are finite.
Real-world observations such as the measurements of yesterday's rain throughout the day typically cannot be complete sets of all possible observations that could be made. As such, the variance calculated from the finite set will in general not match the variance that would have been calculated from the full population of possible observations. This means that one estimates the mean and variance from a limited set of observations by using an estimator equation. The estimator is a function of the sample of n observations drawn without observational bias from the whole population of potential observations. In this example, the sample would be the set of actual measurements of yesterday's rainfall from available rain gauges within the geography of interest.
The simplest estimators for population mean and population variance are simply the mean and variance of the sample, the sample mean and (uncorrected) sample variance – these are consistent estimators (they converge to the value of the whole population as the number of samples increases) but can be improved. Most simply, the sample variance is computed as the sum of squared deviations about the (sample) mean, divided by n as the number of samples . However, using values other than n improves the estimator in various ways. Four common values for the denominator are n, n − 1, n + 1, and n − 1.5: n is the simplest (the variance of the sample), n − 1 eliminates bias, [ 10 ] n + 1 minimizes mean squared error for the normal distribution, [ 11 ] and n − 1.5 mostly eliminates bias in unbiased estimation of standard deviation for the normal distribution. [ 12 ]
Firstly, if the true population mean is unknown, then the sample variance (which uses the sample mean in place of the true mean) is a biased estimator : it underestimates the variance by a factor of ( n − 1) / n ; correcting this factor, resulting in the sum of squared deviations about the sample mean divided by n -1 instead of n , is called Bessel's correction . [ 10 ] The resulting estimator is unbiased and is called the (corrected) sample variance or unbiased sample variance . If the mean is determined in some other way than from the same samples used to estimate the variance, then this bias does not arise, and the variance can safely be estimated as that of the samples about the (independently known) mean.
Secondly, the sample variance does not generally minimize mean squared error between sample variance and population variance. Correcting for bias often makes this worse: one can always choose a scale factor that performs better than the corrected sample variance, though the optimal scale factor depends on the excess kurtosis of the population (see mean squared error: variance ) and introduces bias. This always consists of scaling down the unbiased estimator (dividing by a number larger than n − 1) and is a simple example of a shrinkage estimator : one "shrinks" the unbiased estimator towards zero. For the normal distribution, dividing by n + 1 (instead of n − 1 or n ) minimizes mean squared error. [ 11 ] The resulting estimator is biased, however, and is known as the biased sample variation .
In general, the population variance of a finite population of size N with values x i is given by σ 2 = 1 N ∑ i = 1 N ( x i − μ ) 2 = 1 N ∑ i = 1 N ( x i 2 − 2 μ x i + μ 2 ) = ( 1 N ∑ i = 1 N x i 2 ) − 2 μ ( 1 N ∑ i = 1 N x i ) + μ 2 = E [ x i 2 ] − μ 2 {\displaystyle {\begin{aligned}\sigma ^{2}&={\frac {1}{N}}\sum _{i=1}^{N}{\left(x_{i}-\mu \right)}^{2}={\frac {1}{N}}\sum _{i=1}^{N}\left(x_{i}^{2}-2\mu x_{i}+\mu ^{2}\right)\\[5pt]&=\left({\frac {1}{N}}\sum _{i=1}^{N}x_{i}^{2}\right)-2\mu \left({\frac {1}{N}}\sum _{i=1}^{N}x_{i}\right)+\mu ^{2}\\[5pt]&=\operatorname {E} [x_{i}^{2}]-\mu ^{2}\end{aligned}}}
where the population mean is μ = E [ x i ] = 1 N ∑ i = 1 N x i {\textstyle \mu =\operatorname {E} [x_{i}]={\frac {1}{N}}\sum _{i=1}^{N}x_{i}} and E [ x i 2 ] = ( 1 N ∑ i = 1 N x i 2 ) {\textstyle \operatorname {E} [x_{i}^{2}]=\left({\frac {1}{N}}\sum _{i=1}^{N}x_{i}^{2}\right)} , where E {\textstyle \operatorname {E} } is the expectation value operator.
The population variance can also be computed using [ 13 ]
σ 2 = 1 N 2 ∑ i < j ( x i − x j ) 2 = 1 2 N 2 ∑ i , j = 1 N ( x i − x j ) 2 . {\displaystyle \sigma ^{2}={\frac {1}{N^{2}}}\sum _{i<j}\left(x_{i}-x_{j}\right)^{2}={\frac {1}{2N^{2}}}\sum _{i,j=1}^{N}\left(x_{i}-x_{j}\right)^{2}.}
(The right side has duplicate terms in the sum while the middle side has only unique terms to sum.) This is true because 1 2 N 2 ∑ i , j = 1 N ( x i − x j ) 2 = 1 2 N 2 ∑ i , j = 1 N ( x i 2 − 2 x i x j + x j 2 ) = 1 2 N ∑ j = 1 N ( 1 N ∑ i = 1 N x i 2 ) − ( 1 N ∑ i = 1 N x i ) ( 1 N ∑ j = 1 N x j ) + 1 2 N ∑ i = 1 N ( 1 N ∑ j = 1 N x j 2 ) = 1 2 ( σ 2 + μ 2 ) − μ 2 + 1 2 ( σ 2 + μ 2 ) = σ 2 . {\displaystyle {\begin{aligned}&{\frac {1}{2N^{2}}}\sum _{i,j=1}^{N}{\left(x_{i}-x_{j}\right)}^{2}\\[5pt]={}&{\frac {1}{2N^{2}}}\sum _{i,j=1}^{N}\left(x_{i}^{2}-2x_{i}x_{j}+x_{j}^{2}\right)\\[5pt]={}&{\frac {1}{2N}}\sum _{j=1}^{N}\left({\frac {1}{N}}\sum _{i=1}^{N}x_{i}^{2}\right)-\left({\frac {1}{N}}\sum _{i=1}^{N}x_{i}\right)\left({\frac {1}{N}}\sum _{j=1}^{N}x_{j}\right)+{\frac {1}{2N}}\sum _{i=1}^{N}\left({\frac {1}{N}}\sum _{j=1}^{N}x_{j}^{2}\right)\\[5pt]={}&{\frac {1}{2}}\left(\sigma ^{2}+\mu ^{2}\right)-\mu ^{2}+{\frac {1}{2}}\left(\sigma ^{2}+\mu ^{2}\right)\\[5pt]={}&\sigma ^{2}.\end{aligned}}}
The population variance matches the variance of the generating probability distribution. In this sense, the concept of population can be extended to continuous random variables with infinite populations.
In many practical situations, the true variance of a population is not known a priori and must be computed somehow. When dealing with extremely large populations, it is not possible to count every object in the population, so the computation must be performed on a sample of the population. [ 14 ] This is generally referred to as sample variance or empirical variance . Sample variance can also be applied to the estimation of the variance of a continuous distribution from a sample of that distribution.
We take a sample with replacement of n values Y 1 , ..., Y n from the population of size N , where n < N , and estimate the variance on the basis of this sample. [ 15 ] Directly taking the variance of the sample data gives the average of the squared deviations : [ 16 ]
S ~ Y 2 = 1 n ∑ i = 1 n ( Y i − Y ¯ ) 2 = ( 1 n ∑ i = 1 n Y i 2 ) − Y ¯ 2 = 1 n 2 ∑ i , j : i < j ( Y i − Y j ) 2 . {\displaystyle {\tilde {S}}_{Y}^{2}={\frac {1}{n}}\sum _{i=1}^{n}\left(Y_{i}-{\overline {Y}}\right)^{2}=\left({\frac {1}{n}}\sum _{i=1}^{n}Y_{i}^{2}\right)-{\overline {Y}}^{2}={\frac {1}{n^{2}}}\sum _{i,j\,:\,i<j}\left(Y_{i}-Y_{j}\right)^{2}.}
(See the section Population variance for the derivation of this formula.) Here, Y ¯ {\displaystyle {\overline {Y}}} denotes the sample mean : Y ¯ = 1 n ∑ i = 1 n Y i . {\displaystyle {\overline {Y}}={\frac {1}{n}}\sum _{i=1}^{n}Y_{i}.}
Since the Y i are selected randomly, both Y ¯ {\displaystyle {\overline {Y}}} and S ~ Y 2 {\displaystyle {\tilde {S}}_{Y}^{2}} are random variables . Their expected values can be evaluated by averaging over the ensemble of all possible samples { Y i } of size n from the population. For S ~ Y 2 {\displaystyle {\tilde {S}}_{Y}^{2}} this gives: E [ S ~ Y 2 ] = E [ 1 n ∑ i = 1 n ( Y i − 1 n ∑ j = 1 n Y j ) 2 ] = 1 n ∑ i = 1 n E [ Y i 2 − 2 n Y i ∑ j = 1 n Y j + 1 n 2 ∑ j = 1 n Y j ∑ k = 1 n Y k ] = 1 n ∑ i = 1 n ( E [ Y i 2 ] − 2 n ( ∑ j ≠ i E [ Y i Y j ] + E [ Y i 2 ] ) + 1 n 2 ∑ j = 1 n ∑ k ≠ j n E [ Y j Y k ] + 1 n 2 ∑ j = 1 n E [ Y j 2 ] ) = 1 n ∑ i = 1 n ( n − 2 n E [ Y i 2 ] − 2 n ∑ j ≠ i E [ Y i Y j ] + 1 n 2 ∑ j = 1 n ∑ k ≠ j n E [ Y j Y k ] + 1 n 2 ∑ j = 1 n E [ Y j 2 ] ) = 1 n ∑ i = 1 n [ n − 2 n ( σ 2 + μ 2 ) − 2 n ( n − 1 ) μ 2 + 1 n 2 n ( n − 1 ) μ 2 + 1 n ( σ 2 + μ 2 ) ] = n − 1 n σ 2 . {\displaystyle {\begin{aligned}\operatorname {E} [{\tilde {S}}_{Y}^{2}]&=\operatorname {E} \left[{\frac {1}{n}}\sum _{i=1}^{n}{\left(Y_{i}-{\frac {1}{n}}\sum _{j=1}^{n}Y_{j}\right)}^{2}\right]\\[5pt]&={\frac {1}{n}}\sum _{i=1}^{n}\operatorname {E} \left[Y_{i}^{2}-{\frac {2}{n}}Y_{i}\sum _{j=1}^{n}Y_{j}+{\frac {1}{n^{2}}}\sum _{j=1}^{n}Y_{j}\sum _{k=1}^{n}Y_{k}\right]\\[5pt]&={\frac {1}{n}}\sum _{i=1}^{n}\left(\operatorname {E} \left[Y_{i}^{2}\right]-{\frac {2}{n}}\left(\sum _{j\neq i}\operatorname {E} \left[Y_{i}Y_{j}\right]+\operatorname {E} \left[Y_{i}^{2}\right]\right)+{\frac {1}{n^{2}}}\sum _{j=1}^{n}\sum _{k\neq j}^{n}\operatorname {E} \left[Y_{j}Y_{k}\right]+{\frac {1}{n^{2}}}\sum _{j=1}^{n}\operatorname {E} \left[Y_{j}^{2}\right]\right)\\[5pt]&={\frac {1}{n}}\sum _{i=1}^{n}\left({\frac {n-2}{n}}\operatorname {E} \left[Y_{i}^{2}\right]-{\frac {2}{n}}\sum _{j\neq i}\operatorname {E} \left[Y_{i}Y_{j}\right]+{\frac {1}{n^{2}}}\sum _{j=1}^{n}\sum _{k\neq j}^{n}\operatorname {E} \left[Y_{j}Y_{k}\right]+{\frac {1}{n^{2}}}\sum _{j=1}^{n}\operatorname {E} \left[Y_{j}^{2}\right]\right)\\[5pt]&={\frac {1}{n}}\sum _{i=1}^{n}\left[{\frac {n-2}{n}}\left(\sigma ^{2}+\mu ^{2}\right)-{\frac {2}{n}}(n-1)\mu ^{2}+{\frac {1}{n^{2}}}n(n-1)\mu ^{2}+{\frac {1}{n}}\left(\sigma ^{2}+\mu ^{2}\right)\right]\\[5pt]&={\frac {n-1}{n}}\sigma ^{2}.\end{aligned}}}
Here σ 2 = E [ Y i 2 ] − μ 2 {\textstyle \sigma ^{2}=\operatorname {E} [Y_{i}^{2}]-\mu ^{2}} derived in the section is population variance and E [ Y i Y j ] = E [ Y i ] E [ Y j ] = μ 2 {\textstyle \operatorname {E} [Y_{i}Y_{j}]=\operatorname {E} [Y_{i}]\operatorname {E} [Y_{j}]=\mu ^{2}} due to independency of Y i {\textstyle Y_{i}} and Y j {\textstyle Y_{j}} .
Hence S ~ Y 2 {\textstyle {\tilde {S}}_{Y}^{2}} gives an estimate of the population variance σ 2 {\textstyle \sigma ^{2}} that is biased by a factor of n − 1 n {\textstyle {\frac {n-1}{n}}} because the expectation value of S ~ Y 2 {\textstyle {\tilde {S}}_{Y}^{2}} is smaller than the population variance (true variance) by that factor. For this reason, S ~ Y 2 {\textstyle {\tilde {S}}_{Y}^{2}} is referred to as the biased sample variance .
Correcting for this bias yields the unbiased sample variance , denoted S 2 {\displaystyle S^{2}} :
S 2 = n n − 1 S ~ Y 2 = n n − 1 [ 1 n ∑ i = 1 n ( Y i − Y ¯ ) 2 ] = 1 n − 1 ∑ i = 1 n ( Y i − Y ¯ ) 2 {\displaystyle S^{2}={\frac {n}{n-1}}{\tilde {S}}_{Y}^{2}={\frac {n}{n-1}}\left[{\frac {1}{n}}\sum _{i=1}^{n}\left(Y_{i}-{\overline {Y}}\right)^{2}\right]={\frac {1}{n-1}}\sum _{i=1}^{n}\left(Y_{i}-{\overline {Y}}\right)^{2}}
Either estimator may be simply referred to as the sample variance when the version can be determined by context. The same proof is also applicable for samples taken from a continuous probability distribution.
The use of the term n − 1 is called Bessel's correction , and it is also used in sample covariance and the sample standard deviation (the square root of variance). The square root is a concave function and thus introduces negative bias (by Jensen's inequality ), which depends on the distribution, and thus the corrected sample standard deviation (using Bessel's correction) is biased. The unbiased estimation of standard deviation is a technically involved problem, though for the normal distribution using the term n − 1.5 yields an almost unbiased estimator.
The unbiased sample variance is a U-statistic for the function f ( y 1 , y 2 ) = ( y 1 − y 2 ) 2 /2 , meaning that it is obtained by averaging a 2-sample statistic over 2-element subsets of the population.
For a set of numbers {10, 15, 30, 45, 57, 52, 63, 72, 81, 93, 102, 105}, if this set is the whole data population for some measurement, then variance is the population variance 932.743 as the sum of the squared deviations about the mean of this set, divided by 12 as the number of the set members. If the set is a sample from the whole population, then the unbiased sample variance can be calculated as 1017.538 that is the sum of the squared deviations about the mean of the sample, divided by 11 instead of 12. A function VAR.S in Microsoft Excel gives the unbiased sample variance while VAR.P is for population variance.
Being a function of random variables , the sample variance is itself a random variable, and it is natural to study its distribution. In the case that Y i are independent observations from a normal distribution , Cochran's theorem shows that the unbiased sample variance S 2 follows a scaled chi-squared distribution (see also: asymptotic properties and an elementary proof ): [ 17 ] ( n − 1 ) S 2 σ 2 ∼ χ n − 1 2 {\displaystyle (n-1){\frac {S^{2}}{\sigma ^{2}}}\sim \chi _{n-1}^{2}}
where σ 2 is the population variance . As a direct consequence, it follows that E ( S 2 ) = E ( σ 2 n − 1 χ n − 1 2 ) = σ 2 , {\displaystyle \operatorname {E} \left(S^{2}\right)=\operatorname {E} \left({\frac {\sigma ^{2}}{n-1}}\chi _{n-1}^{2}\right)=\sigma ^{2},}
and [ 18 ]
Var [ S 2 ] = Var ( σ 2 n − 1 χ n − 1 2 ) = σ 4 ( n − 1 ) 2 Var ( χ n − 1 2 ) = 2 σ 4 n − 1 . {\displaystyle \operatorname {Var} \left[S^{2}\right]=\operatorname {Var} \left({\frac {\sigma ^{2}}{n-1}}\chi _{n-1}^{2}\right)={\frac {\sigma ^{4}}{{\left(n-1\right)}^{2}}}\operatorname {Var} \left(\chi _{n-1}^{2}\right)={\frac {2\sigma ^{4}}{n-1}}.}
If Y i are independent and identically distributed, but not necessarily normally distributed, then [ 19 ]
E [ S 2 ] = σ 2 , Var [ S 2 ] = σ 4 n ( κ − 1 + 2 n − 1 ) = 1 n ( μ 4 − n − 3 n − 1 σ 4 ) , {\displaystyle \operatorname {E} \left[S^{2}\right]=\sigma ^{2},\quad \operatorname {Var} \left[S^{2}\right]={\frac {\sigma ^{4}}{n}}\left(\kappa -1+{\frac {2}{n-1}}\right)={\frac {1}{n}}\left(\mu _{4}-{\frac {n-3}{n-1}}\sigma ^{4}\right),}
where κ is the kurtosis of the distribution and μ 4 is the fourth central moment .
If the conditions of the law of large numbers hold for the squared observations, S 2 is a consistent estimator of σ 2 . One can see indeed that the variance of the estimator tends asymptotically to zero. An asymptotically equivalent formula was given in Kenney and Keeping (1951:164), Rose and Smith (2002:264), and Weisstein (n.d.). [ 20 ] [ 21 ] [ 22 ]
Samuelson's inequality is a result that states bounds on the values that individual observations in a sample can take, given that the sample mean and (biased) variance have been calculated. [ 23 ] Values must lie within the limits y ¯ ± σ Y ( n − 1 ) 1 / 2 . {\displaystyle {\bar {y}}\pm \sigma _{Y}(n-1)^{1/2}.}
It has been shown [ 24 ] that for a sample { y i } of positive real numbers,
σ y 2 ≤ 2 y max ( A − H ) , {\displaystyle \sigma _{y}^{2}\leq 2y_{\max }(A-H),}
where y max is the maximum of the sample, A is the arithmetic mean, H is the harmonic mean of the sample and σ y 2 {\displaystyle \sigma _{y}^{2}} is the (biased) variance of the sample.
This bound has been improved, and it is known that variance is bounded by
σ y 2 ≤ y max ( A − H ) ( y max − A ) y max − H , σ y 2 ≥ y min ( A − H ) ( A − y min ) H − y min , {\displaystyle {\begin{aligned}\sigma _{y}^{2}&\leq {\frac {y_{\max }(A-H)(y_{\max }-A)}{y_{\max }-H}},\\[1ex]\sigma _{y}^{2}&\geq {\frac {y_{\min }(A-H)(A-y_{\min })}{H-y_{\min }}},\end{aligned}}}
where y min is the minimum of the sample. [ 25 ]
The F-test of equality of variances and the chi square tests are adequate when the sample is normally distributed. Non-normality makes testing for the equality of two or more variances more difficult.
Several non parametric tests have been proposed: these include the Barton–David–Ansari–Freund–Siegel–Tukey test, the Capon test , Mood test , the Klotz test and the Sukhatme test . The Sukhatme test applies to two variances and requires that both medians be known and equal to zero. The Mood, Klotz, Capon and Barton–David–Ansari–Freund–Siegel–Tukey tests also apply to two variances. They allow the median to be unknown but do require that the two medians are equal.
The Lehmann test is a parametric test of two variances. Of this test there are several variants known. Other tests of the equality of variances include the Box test , the Box–Anderson test and the Moses test .
Resampling methods, which include the bootstrap and the jackknife , may be used to test the equality of variances.
The variance of a probability distribution is analogous to the moment of inertia in classical mechanics of a corresponding mass distribution along a line, with respect to rotation about its center of mass. [ 26 ] It is because of this analogy that such things as the variance are called moments of probability distributions . [ 26 ] The covariance matrix is related to the moment of inertia tensor for multivariate distributions. The moment of inertia of a cloud of n points with a covariance matrix of Σ {\displaystyle \Sigma } is given by [ citation needed ] I = n ( 1 3 × 3 tr ( Σ ) − Σ ) . {\displaystyle I=n\left(\mathbf {1} _{3\times 3}\operatorname {tr} (\Sigma )-\Sigma \right).}
This difference between moment of inertia in physics and in statistics is clear for points that are gathered along a line. Suppose many points are close to the x axis and distributed along it. The covariance matrix might look like Σ = [ 10 0 0 0 0.1 0 0 0 0.1 ] . {\displaystyle \Sigma ={\begin{bmatrix}10&0&0\\0&0.1&0\\0&0&0.1\end{bmatrix}}.}
That is, there is the most variance in the x direction. Physicists would consider this to have a low moment about the x axis so the moment-of-inertia tensor is I = n [ 0.2 0 0 0 10.1 0 0 0 10.1 ] . {\displaystyle I=n{\begin{bmatrix}0.2&0&0\\0&10.1&0\\0&0&10.1\end{bmatrix}}.}
The semivariance is calculated in the same manner as the variance but only those observations that fall below the mean are included in the calculation: Semivariance = 1 n ∑ i : x i < μ ( x i − μ ) 2 {\displaystyle {\text{Semivariance}}={\frac {1}{n}}\sum _{i:x_{i}<\mu }{\left(x_{i}-\mu \right)}^{2}} It is also described as a specific measure in different fields of application. For skewed distributions, the semivariance can provide additional information that a variance does not. [ 27 ]
For inequalities associated with the semivariance, see Chebyshev's inequality § Semivariances .
The term variance was first introduced by Ronald Fisher in his 1918 paper The Correlation Between Relatives on the Supposition of Mendelian Inheritance : [ 28 ]
The great body of available statistics show us that the deviations of a human measurement from its mean follow very closely the Normal Law of Errors , and, therefore, that the variability may be uniformly measured by the standard deviation corresponding to the square root of the mean square error . When there are two independent causes of variability capable of producing in an otherwise uniform population distributions with standard deviations σ 1 {\displaystyle \sigma _{1}} and σ 2 {\displaystyle \sigma _{2}} , it is found that the distribution, when both causes act together, has a standard deviation σ 1 2 + σ 2 2 {\displaystyle {\sqrt {\sigma _{1}^{2}+\sigma _{2}^{2}}}} . It is therefore desirable in analysing the causes of variability to deal with the square of the standard deviation as the measure of variability. We shall term this quantity the Variance...
If x {\displaystyle x} is a scalar complex -valued random variable, with values in C , {\displaystyle \mathbb {C} ,} then its variance is E [ ( x − μ ) ( x − μ ) ∗ ] , {\displaystyle \operatorname {E} \left[(x-\mu )(x-\mu )^{*}\right],} where x ∗ {\displaystyle x^{*}} is the complex conjugate of x . {\displaystyle x.} This variance is a real scalar.
If X {\displaystyle X} is a vector -valued random variable, with values in R n , {\displaystyle \mathbb {R} ^{n},} and thought of as a column vector, then a natural generalization of variance is E [ ( X − μ ) ( X − μ ) T ] , {\displaystyle \operatorname {E} \left[(X-\mu ){(X-\mu )}^{\mathsf {T}}\right],} where μ = E ( X ) {\displaystyle \mu =\operatorname {E} (X)} and X T {\displaystyle X^{\mathsf {T}}} is the transpose of X , and so is a row vector. The result is a positive semi-definite square matrix , commonly referred to as the variance-covariance matrix (or simply as the covariance matrix ).
If X {\displaystyle X} is a vector- and complex-valued random variable, with values in C n , {\displaystyle \mathbb {C} ^{n},} then the covariance matrix is E [ ( X − μ ) ( X − μ ) † ] , {\displaystyle \operatorname {E} \left[(X-\mu ){(X-\mu )}^{\dagger }\right],} where X † {\displaystyle X^{\dagger }} is the conjugate transpose of X . {\displaystyle X.} [ citation needed ] This matrix is also positive semi-definite and square.
Another generalization of variance for vector-valued random variables X {\displaystyle X} , which results in a scalar value rather than in a matrix, is the generalized variance det ( C ) {\displaystyle \det(C)} , the determinant of the covariance matrix. The generalized variance can be shown to be related to the multidimensional scatter of points around their mean. [ 29 ]
A different generalization is obtained by considering the equation for the scalar variance, Var ( X ) = E [ ( X − μ ) 2 ] {\displaystyle \operatorname {Var} (X)=\operatorname {E} \left[(X-\mu )^{2}\right]} , and reinterpreting ( X − μ ) 2 {\displaystyle (X-\mu )^{2}} as the squared Euclidean distance between the random variable and its mean, or, simply as the scalar product of the vector X − μ {\displaystyle X-\mu } with itself. This results in E [ ( X − μ ) T ( X − μ ) ] = tr ( C ) , {\displaystyle \operatorname {E} \left[(X-\mu )^{\mathsf {T}}(X-\mu )\right]=\operatorname {tr} (C),} which is the trace of the covariance matrix.
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In microbiology and virology , the term variant or genetic variant is used to describe a subtype of a microorganism that is genetically distinct from a main strain, but not sufficiently different to be termed a distinct strain . A similar distinction is made in botany between different cultivated varieties of a species of plant, termed cultivars .
It was said in 2013 that "there is no universally accepted definition for the terms 'strain', 'variant', and 'isolate' in the virology community, and most virologists simply copy the usage of terms from others". [ 1 ] The lack of precise definition continued in 2020; in the context of the Variant of Concern 202012/01 version of the SARS-CoV-2 virus, the website of the US Centers for Disease Control and Prevention (CDC) states, "For the time being in the context of this variant, the [terms "variant", "strain", and "lineage"] are generally being used interchangeably by the scientific community". [ 2 ]
This microbiology -related article is a stub . You can help Wikipedia by expanding it .
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https://en.wikipedia.org/wiki/Variant_(biology)
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Variants of PCR represent a diverse array of techniques that have evolved from the basic polymerase chain reaction (PCR) method, each tailored to specific applications in molecular biology , such as genetic analysis, DNA sequencing , and disease diagnosis, by modifying factors like primer design, temperature conditions, and enzyme usage.
Often only a small modification needs to be made to the standard PCR protocol to achieve a desired goal:
Multiplex-PCR uses several pairs of primers annealing to different target sequences. This permits the simultaneous analysis of multiple targets in a single sample. For example, in testing for genetic mutations, six or more amplifications might be combined. In the standard protocol for DNA fingerprinting , the targets assayed are often amplified in groups of 3 or 4. Multiplex Ligation-dependent Probe Amplification ( MLPA ) permits multiple targets to be amplified using only a single pair of primers, avoiding the resolution limitations of multiplex PCR. Multiplex PCR has also been used for analysis of microsatellites and SNPs . [ 1 ]
Variable Number of Tandem Repeats (VNTR) PCR targets repetitive areas of the genome that exhibit length variation . Analysis of the genotypes in the samples usually involves sizing of the amplification products by gel electrophoresis . Analysis of smaller VNTR segments known as short tandem repeats (or STRs) is the basis for DNA fingerprinting databases such as CODIS .
Asymmetric PCR preferentially amplifies one strand of a double-stranded DNA target. It is used in some sequencing methods and hybridization probing to generate one DNA strand as product. Thermocycling is carried out exactly as in conventional PCR, but with a limiting amount or leaving out one of the primers. When the limiting primer becomes depleted, replication increases arithmetically rather than exponentially through extension of the excess primer. [ 2 ] A modification of this process, named L inear- A fter- T he- E xponential-PCR (or LATE-PCR ), uses a limiting primer with a higher melting temperature (T m ) than the excess primer in order to maintain reaction efficiency as the limiting primer concentration decreases mid-reaction. [ 3 ] See also overlap-extension PCR .
Some modifications are needed to perform long PCR . The original Klenow-based PCR process did not generate products that were larger than about 400 bp. Taq polymerase can however amplify targets of up to several thousand bp long. [ 4 ] Since then, modified protocols with Taq enzyme have allowed targets of over 50 kb to be amplified. [ 5 ]
Nested PCR is used to increase the specificity of DNA amplification. Two sets of primers are used in two successive reactions. In the first PCR, one pair of primers is used to generate DNA products, which may contain products amplified from non-target areas. The products from the first PCR are then used as template in a second PCR, using one ('hemi-nesting') or two different primers whose binding sites are located (nested) within the first set, thus increasing specificity. Nested PCR is often more successful in specifically amplifying long DNA products than conventional PCR, but it requires more detailed knowledge of the sequence of the target.
Quantitative PCR ( qPCR ) is used to measure the specific amount of target DNA (or RNA) in a sample. By measuring amplification only within the phase of true exponential increase, the amount of measured product more accurately reflects the initial amount of target. Special thermal cyclers are used that monitor the amount of product during the amplification.
Quantitative Real-Time PCR ( QRT-PCR ), sometimes simply called Real-Time PCR ( RT-PCR ), refers to a collection of methods that use fluorescent dyes, such as Sybr Green, or fluorophore -containing DNA probes, such as TaqMan , to measure the amount of amplified product in real time as the amplification progresses.
Hot-start PCR is a technique performed manually by heating the reaction components to the DNA melting temperature (e.g. 95 °C) before adding the polymerase. In this way, non-specific amplification at lower temperatures is prevented. [ 6 ] Alternatively, specialized reagents inhibit the polymerase's activity at ambient temperature, either by the binding of an antibody , or by the presence of covalently bound inhibitors that only dissociate after a high-temperature activation step. 'Hot-start/cold-finish PCR' is achieved with new hybrid polymerases that are inactive at ambient temperature and are only activated at elevated temperatures.
In touchdown PCR , the annealing temperature is gradually decreased in later cycles. The annealing temperature in the early cycles is usually 3–5 °C above the standard T m of the primers used, while in the later cycles it is a similar amount below the T m . The initial higher annealing temperature leads to greater specificity for primer binding, while the lower temperatures permit more efficient amplification at the end of the reaction. [ 7 ]
Assembly PCR (also known as Polymerase Cycling Assembly or PCA ) is the synthesis of long DNA structures by performing PCR on a pool of long oligonucleotides with short overlapping segments, to assemble two or more pieces of DNA into one piece. It involves an initial PCR with primers that have an overlap and a second PCR using the products as the template that generates the final full-length product. This technique may substitute for ligation -based assembly. [ 8 ]
In colony PCR , bacterial colonies are screened directly by PCR, for example, the screen for correct DNA vector constructs. Colonies are sampled with a sterile pipette tip and a small quantity of cells transferred into a PCR mix. To release the DNA from the cells, the PCR is either started with an extended time at 95 °C (when standard polymerase is used), or with a shortened denaturation step at 100 °C and special chimeric DNA polymerase. [ 9 ]
The digital polymerase chain reaction simultaneously amplifies thousands of samples, each in a separate droplet within an emulsion or partition within an micro-well.
Suicide PCR is typically used in paleogenetics or other studies where avoiding false positives and ensuring the specificity of the amplified fragment is the highest priority. It was originally described in a study to verify the presence of the microbe Yersinia pestis in dental samples obtained from 14th-century graves of people supposedly killed by plague during the medieval Black Death epidemic. [ 10 ] The method prescribes the use of any primer combination only once in a PCR (hence the term "suicide"), which should never have been used in any positive-control PCR reaction, and the primers should always target a genomic region never amplified before in the lab using this or any other set of primers. This ensures that no contaminating DNA from previous PCR reactions is present in the lab, which could otherwise generate false positives.
COLD-PCR ( co -amplification at l ower d enaturation temperature-PCR) is a modified protocol that enriches variant alleles from a mixture of wild-type and mutation-containing DNA samples.
The basic PCR process can sometimes precede or follow another technique.
RT-PCR (or R everse T ranscription PCR ) is used to reverse-transcribe and amplify RNA to cDNA . PCR is preceded by a reaction using reverse transcriptase , an enzyme that converts RNA into cDNA. The two reactions may be combined in a tube, with the initial heating step of PCR being used to inactivate the transcriptase. [ 4 ] The Tth polymerase (described below) has RT activity, and can carry out the entire reaction. RT-PCR is widely used in expression profiling , which detects the expression of a gene. It can also be used to obtain sequence of an RNA transcript, which may aid the determination of the transcription start and termination sites (by RACE-PCR ) and facilitate mapping of the location of exons and introns in a gene sequence. Multiplex qRT-PCR has been developed to enhance SARS-CoV-2 detection efficiency by allowing multiple viral targets to be tested simultaneously. [ 11 ]
Two-tailed PCR uses a single primer that binds to a microRNA target with both 3' and 5' ends, known as hemiprobes. [ 12 ] Both ends must be complementary for binding to occur. The 3'-end is then extended by reverse transcriptase forming a long cDNA. The cDNA is then amplified using two target specific PCR primers. The combination of two hemiprobes, both targeting the short microRNA target, makes the Two-tailed assay exceedingly sensitive and specific.
Ligation-mediated PCR uses small DNA oligonucleotide 'linkers' (or adaptors) that are first ligated to fragments of the target DNA. PCR primers that anneal to the linker sequences are then used to amplify the target fragments. This method is deployed for DNA sequencing, genome walking, and DNA footprinting . [ 13 ] A related technique is amplified fragment length polymorphism , which generates diagnostic fragments of a genome.
Methylation-specific PCR ( MSP ) is used to identify patterns of DNA methylation at cytosine-guanine (CpG) islands in genomic DNA. [ 14 ] Target DNA is first treated with sodium bisulfite , which converts unmethylated cytosine bases to uracil , which is complementary to adenosine in PCR primers. Two amplifications are then carried out on the bisulfite-treated DNA: one primer set anneals to DNA with cytosines (corresponding to methylated cytosine), and the other set anneals to DNA with uracil (corresponding to unmethylated cytosine). MSP used in quantitative PCR provides quantitative information about the methylation state of a given CpG island. [ 15 ]
Adjustments of the components in PCR is commonly used for optimal performance.
The divalent magnesium ion (Mg ++ ) is required for PCR polymerase activity. Lower concentrations Mg ++ will increase replication fidelity, while higher concentrations will introduce more mutations. [ 16 ]
Denaturants (such as dimethylsulfoxide ) can increase amplification specificity by destabilizing non-specific primer binding. Other chemicals, such as glycerol , are stabilizers for the activity of the polymerase during amplification. Detergents (such as Triton X-100 ) can prevent polymerase stick to itself or to the walls of the reaction tube.
DNA polymerases occasionally incorporate mismatch bases into the extending strand. High-fidelity PCR employs enzymes with 3'-5' exonuclease activity that decreases this rate of mis-incorporation. Examples of enzymes with proofreading activity include Pfu ; adjustments of the Mg ++ and dNTP concentrations may help maximize the number of products that exactly match the original target DNA. [ citation needed ]
Adjustments to the synthetic oligonucleotides used as primers in PCR are a rich source of modification:
Normally PCR primers are chosen from an invariant part of the genome, and might be used to amplify a polymorphic area between them. In allele-specific PCR the opposite is done. At least one of the primers is chosen from a polymorphic area, with the mutations located at (or near) its 3'-end. Under stringent conditions, a mismatched primer will not initiate replication, whereas a matched primer will. The appearance of an amplification product therefore indicates the genotype. (For more information, see SNP genotyping .)
InterSequence-Specific PCR (or ISSR-PCR ) is method for DNA fingerprinting that uses primers selected from segments repeated throughout a genome to produce a unique fingerprint of amplified product lengths. [ 17 ] The use of primers from a commonly repeated segment is called Alu-PCR , and can help amplify sequences adjacent (or between) these repeats.
Primers can also be designed to be 'degenerate' – able to initiate replication from a large number of target locations. Whole genome amplification (or WGA ) is a group of procedures that allow amplification to occur at many locations in an unknown genome, and which may only be available in small quantities. Other techniques use degenerate primers that are synthesized using multiple nucleotides at particular positions (the polymerase 'chooses' the correctly matched primers). Also, the primers can be synthesized with the nucleoside analog inosine , which hybridizes to three of the four normal bases. A similar technique can force PCR to perform Site-directed mutagenesis . (also see Overlap extension polymerase chain reaction )
Normally the primers used in PCR are designed to be fully complementary to the target. However, the polymerase is tolerant to mis-matches away from the 3' end. Tailed-primers include non-complementary sequences at their 5' ends. A common procedure is the use of linker-primers , which ultimately place restriction sites at the ends of the PCR products, facilitating their later insertion into cloning vectors.
An extension of the 'colony-PCR' method (above), is the use of vector primers . Target DNA fragments (or cDNA) are first inserted into a cloning vector , and a single set of primers are designed for the areas of the vector flanking the insertion site. Amplification occurs for whatever DNA has been inserted. [ 4 ]
PCR can easily be modified to produce a labeled product for subsequent use as a hybridization probe. One or both primers might be used in PCR with a radioactive or fluorescent label already attached, or labels might be added after amplification. These labeling methods can be combined with 'asymmetric-PCR' (above) to produce effective hybridization probes.
RNase H-dependent PCR (rhPCR) can reduce primer-dimer formation, and increase the number of assays in multiplex PCR. The method utilizes primers with a cleavable block on the 3’ end that is removed by the action of a thermostable RNase HII enzyme. [ 18 ]
There are several DNA polymerases that are used in PCR.
The Klenow fragment , derived from the original DNA Polymerase I from E. coli , was the first enzyme used in PCR. Because of its lack of stability at high temperature, it needs be replenished during each cycle, and therefore is not commonly used in PCR.
The bacteriophage T4 DNA polymerase (family A) was also initially used in PCR. It has a higher fidelity of replication than the Klenow fragment, but is also destroyed by heat. T7 DNA polymerase (family B) has similar properties and purposes. It has been applied to site-directed mutagenesis [ 19 ] and Sanger sequencing . [ 20 ]
Taq polymerase, the DNA Polymerase I from Thermus aquaticus , was the first thermostable polymerase used in PCR, and is still the one most commonly used. The enzyme can be isolated from its native source, or from its cloned gene expressed in E. coli . [ 4 ] A 61kDa truncated from lacking 5'-3' exonuclease activity is known as the Stoffel fragment , and is expressed in E. coli . [ 21 ] The lack of exonuclease activity may allow it to amplify longer targets than the native enzyme. It has been commercialized as AmpliTaq and Klentaq . [ 22 ] A variant designed for hot-start PCR called the "Faststart polymerase" has also been produced. It requires strong heat activation, thereby avoiding non-specific amplification due to polymerase activity at low temperature. Many other variants have been created. [ 23 ]
Other Thermus polymerases, such as Tth polymerase I ( P52028 ) from Thermus thermophilus , has seen some use. Tth has reverse transcriptase activity in the presence of Mn 2+ ions, allowing PCR amplification from RNA targets. [ 24 ]
The archean genus Pyrococcus has proven a rich source of thermostable polymerases with proofreading activity. Pfu DNA polymerase , isolated from the P. furiosus shows a 5-fold decrease in the error rate of replication compared to Taq. [ 25 ] Since errors increase as PCR progresses, Pfu is the preferred polymerase when products are to be individually cloned for sequencing or expression. Other lesser used polymerases from this genus include Pwo ( P61876 ) from Pyrococcus woesei , Pfx from an unnamed species, "Deep Vent" polymerase ( Q51334 ) from strain GB-D. [ 26 ]
Vent or Tli polymerase is an extremely thermostable DNA polymerase isolated from Thermococcus litoralis . The polymerase from Thermococcus fumicolans ( Tfu ) has also been commercialized. [ 26 ]
Sometimes even the basic mechanism of PCR can be modified.
Unlike normal PCR, Inverse PCR allows amplification and sequencing of DNA that surrounds a known sequence. It involves initially subjecting the target DNA to a series of restriction enzyme digestions , and then circularizing the resulting fragments by self ligation . Primers are designed to be extended outward from the known segment, resulting in amplification of the rest of the circle. This is especially useful in identifying sequences to either side of various genomic inserts. [ 27 ]
Similarly, thermal asymmetric interlaced PCR (or TAIL-PCR ) is used to isolate unknown sequences flanking a known area of the genome. Within the known sequence, TAIL-PCR uses a nested pair of primers with differing annealing temperatures. A 'degenerate' primer is used to amplify in the other direction from the unknown sequence. [ 28 ]
Some DNA amplification protocols have been developed that may be used alternatively to PCR. They are isothermal, meaning that they are run at a constant temperature. [ 29 ]
Helicase-dependent amplification (HDA) is similar to traditional PCR, but uses a constant temperature rather than cycling through denaturation and annealing/extension steps. DNA Helicase , an enzyme that unwinds DNA, is used in place of thermal denaturation. [ 30 ] Loop-mediated isothermal amplification is a similar idea, but done with a strand-displacing polymerase. [ 31 ]
Nicking enzyme amplification reaction (NEAR) and its cousin strand displacement amplification (SDA) are isothermal, replicating DNA at a constant temperature using a polymerase and nicking enzyme. [ 29 ]
Recombinase Polymerase Amplification (RPA) [ 32 ] uses a recombinase to specifically pair primers with double-stranded DNA on the basis of homology, thus directing DNA synthesis from defined DNA sequences present in the sample. Presence of the target sequence initiates DNA amplification, and no thermal or chemical melting of DNA is required. The reaction progresses rapidly and results in specific DNA amplification from just a few target copies to detectable levels typically within 5–10 minutes. The entire reaction system is stable as a dried formulation and does not need refrigeration. RPA can be used to replace PCR in a variety of laboratory applications and users can design their own assays. [ 33 ]
Other types of isothermal amplification include whole genome amplification (WGA), Nucleic acid sequence-based amplification (NASBA), and transcription-mediated amplification (TMA). [ 29 ]
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In computational physics , variational Monte Carlo (VMC) is a quantum Monte Carlo method that applies the variational method to approximate the ground state of a quantum system. [ 1 ]
The basic building block is a generic wave function | Ψ ( a ) ⟩ {\displaystyle |\Psi (a)\rangle } depending on some parameters a {\displaystyle a} . The optimal values of the parameters a {\displaystyle a} is then found upon minimizing the total energy of the system.
In particular, given the Hamiltonian H {\displaystyle {\mathcal {H}}} , and denoting with X {\displaystyle X} a many-body configuration, the expectation value of the energy can be written as: [ 2 ]
E ( a ) = ⟨ Ψ ( a ) | H | Ψ ( a ) ⟩ ⟨ Ψ ( a ) | Ψ ( a ) ⟩ = ∫ | Ψ ( X , a ) | 2 H Ψ ( X , a ) Ψ ( X , a ) d X ∫ | Ψ ( X , a ) | 2 d X . {\displaystyle E(a)={\frac {\langle \Psi (a)|{\mathcal {H}}|\Psi (a)\rangle }{\langle \Psi (a)|\Psi (a)\rangle }}={\frac {\int |\Psi (X,a)|^{2}{\frac {{\mathcal {H}}\Psi (X,a)}{\Psi (X,a)}}\,dX}{\int |\Psi (X,a)|^{2}\,dX}}.}
Following the Monte Carlo method for evaluating integrals , we can interpret | Ψ ( X , a ) | 2 ∫ | Ψ ( X , a ) | 2 d X {\displaystyle {\frac {|\Psi (X,a)|^{2}}{\int |\Psi (X,a)|^{2}\,dX}}} as a probability distribution function, sample it, and evaluate the energy expectation value E ( a ) {\displaystyle E(a)} as the average of the so-called local energy E loc ( X ) = H Ψ ( X , a ) Ψ ( X , a ) {\displaystyle E_{\textrm {loc}}(X)={\frac {{\mathcal {H}}\Psi (X,a)}{\Psi (X,a)}}} . Once E ( a ) {\displaystyle E(a)} is known for a given set of variational parameters a {\displaystyle a} , then optimization is performed in order to minimize the energy and obtain the best possible representation of the ground-state wave-function.
VMC is no different from any other variational method, except that the many-dimensional integrals are evaluated numerically. Monte Carlo integration is particularly crucial in this problem since the dimension of the many-body Hilbert space, comprising all the possible values of the configurations X {\displaystyle X} , typically grows exponentially with the size of the physical system. Other approaches to the numerical evaluation of the energy expectation values would therefore, in general, limit applications to much smaller systems than those analyzable thanks to the Monte Carlo approach.
The accuracy of the method then largely depends on the choice of the variational state. The simplest choice typically corresponds to a mean-field form, where the state Ψ {\displaystyle \Psi } is written as a factorization over the Hilbert space. This particularly simple form is typically not very accurate since it neglects many-body effects. One of the largest gains in accuracy over writing the wave function separably comes from the introduction of the so-called Jastrow factor. In this case the wave function is written as Ψ ( X ) = exp ( ∑ u ( r i j ) ) {\textstyle \Psi (X)=\exp(\sum {u(r_{ij})})} , where r i j {\displaystyle r_{ij}} is the distance between a pair of quantum particles and u ( r ) {\displaystyle u(r)} is a variational function to be determined. With this factor, we can explicitly account for particle-particle correlation, but the many-body integral becomes unseparable, so Monte Carlo is the only way to evaluate it efficiently. In chemical systems, slightly more sophisticated versions of this factor can obtain 80–90% of the correlation energy (see electronic correlation ) with less than 30 parameters. In comparison, a configuration interaction calculation may require around 50,000 parameters to reach that accuracy, although it depends greatly on the particular case being considered. In addition, VMC usually scales as a small power of the number of particles in the simulation, usually something like N 2−4 for calculation of the energy expectation value, depending on the form of the wave function.
QMC calculations crucially depend on the quality of the trial-function, and so it is essential to have an optimized wave-function as close as possible to the ground state.
The problem of function optimization is a very important research topic in numerical simulation. In QMC, in addition to the usual difficulties to find the minimum of multidimensional parametric function, the statistical noise is present in the estimate of the cost function (usually the energy), and its derivatives, required for an efficient optimization.
Different cost functions and different strategies were used to optimize a many-body trial-function. Usually three cost functions were used in QMC optimization energy, variance or a linear combination of them. The variance optimization method has the advantage that the exact wavefunction's variance is known. (Because the exact wavefunction is an eigenfunction of the Hamiltonian, the variance of the local energy is zero). This means that variance optimization is ideal in that it is bounded from below, it is positive defined and its minimum is known. Energy minimization may ultimately prove more effective, however, as different authors recently showed that the energy optimization is more effective than the variance one.
There are different motivations for this: first, usually one is interested in the lowest energy rather than in the lowest variance in both variational and diffusion Monte Carlo; second, variance optimization takes many iterations to optimize determinant parameters and often the optimization can get stuck in multiple local minimum and it suffers of the "false convergence" problem; third energy-minimized wave functions on average yield more accurate values of other expectation values than variance minimized wave functions do.
The optimization strategies can be divided into three categories. The first strategy is based on correlated sampling together with deterministic optimization methods. Even if this idea yielded very accurate results for the first-row atoms, this procedure can have problems if parameters affect the nodes, and moreover density ratio of the current and initial trial-function increases exponentially with the size of the system. In the second strategy one use a large bin to evaluate the cost function and its derivatives in such way that the noise can be neglected and deterministic methods can be used.
The third approach, is based on an iterative technique to handle directly with noise functions. The first example of these methods is the so-called Stochastic Gradient Approximation (SGA), that was used also for structure optimization. Recently an improved and faster approach of this kind was proposed the so-called Stochastic Reconfiguration (SR) method.
In 2017, Giuseppe Carleo and Matthias Troyer [ 3 ] used a VMC objective function to train an artificial neural network to find the ground state of a quantum mechanical system. More generally, artificial neural networks are being used as a wave function ansatz (known as neural network quantum states ) in VMC frameworks for finding ground states of quantum mechanical systems. The use of neural network ansatzes for VMC has been extended to fermions , enabling electronic structure calculations that are significantly more accurate than VMC calculations which do not use neural networks. [ 4 ] [ 5 ] [ 6 ]
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In quantum mechanics , the variational method is one way of finding approximations to the lowest energy eigenstate or ground state , and some excited states. This allows calculating approximate wavefunctions such as molecular orbitals . [ 1 ] The basis for this method is the variational principle . [ 2 ] [ 3 ]
The method consists of choosing a "trial wavefunction " depending on one or more parameters , and finding the values of these parameters for which the expectation value of the energy is the lowest possible. The wavefunction obtained by fixing the parameters to such values is then an approximation to the ground state wavefunction, and the expectation value of the energy in that state is an upper bound to the ground state energy. The Hartree–Fock method , density matrix renormalization group , and Ritz method apply the variational method.
Suppose we are given a Hilbert space and a Hermitian operator over it called the Hamiltonian H {\displaystyle H} . Ignoring complications about continuous spectra , we consider the discrete spectrum of H {\displaystyle H} and a basis of eigenvectors { | ψ λ ⟩ } {\displaystyle \{|\psi _{\lambda }\rangle \}} (see spectral theorem for Hermitian operators for the mathematical background): ⟨ ψ λ 1 | ψ λ 2 ⟩ = δ λ 1 λ 2 , {\displaystyle \left\langle \psi _{\lambda _{1}}|\psi _{\lambda _{2}}\right\rangle =\delta _{\lambda _{1}\lambda _{2}},} where δ i j {\displaystyle \delta _{ij}} is the Kronecker delta δ i j = { 0 if i ≠ j , 1 if i = j , {\displaystyle \delta _{ij}={\begin{cases}0&{\text{if }}i\neq j,\\1&{\text{if }}i=j,\end{cases}}} and the { | ψ λ ⟩ } {\displaystyle \{|\psi _{\lambda }\rangle \}} satisfy the eigenvalue equation H | ψ λ ⟩ = λ | ψ λ ⟩ . {\displaystyle H\left|\psi _{\lambda }\right\rangle =\lambda \left|\psi _{\lambda }\right\rangle .}
Once again ignoring complications involved with a continuous spectrum of H {\displaystyle H} , suppose the spectrum of H {\displaystyle H} is bounded from below and that its greatest lower bound is E 0 . The expectation value of H {\displaystyle H} in a state | ψ ⟩ {\displaystyle |\psi \rangle } is then ⟨ ψ | H | ψ ⟩ = ∑ λ 1 , λ 2 ∈ S p e c ( H ) ⟨ ψ | ψ λ 1 ⟩ ⟨ ψ λ 1 | H | ψ λ 2 ⟩ ⟨ ψ λ 2 | ψ ⟩ = ∑ λ ∈ S p e c ( H ) λ | ⟨ ψ λ | ψ ⟩ | 2 ≥ ∑ λ ∈ S p e c ( H ) E 0 | ⟨ ψ λ | ψ ⟩ | 2 = E 0 ⟨ ψ | ψ ⟩ . {\displaystyle {\begin{aligned}\left\langle \psi \right|H\left|\psi \right\rangle &=\sum _{\lambda _{1},\lambda _{2}\in \mathrm {Spec} (H)}\left\langle \psi |\psi _{\lambda _{1}}\right\rangle \left\langle \psi _{\lambda _{1}}\right|H\left|\psi _{\lambda _{2}}\right\rangle \left\langle \psi _{\lambda _{2}}|\psi \right\rangle \\&=\sum _{\lambda \in \mathrm {Spec} (H)}\lambda \left|\left\langle \psi _{\lambda }|\psi \right\rangle \right|^{2}\geq \sum _{\lambda \in \mathrm {Spec} (H)}E_{0}\left|\left\langle \psi _{\lambda }|\psi \right\rangle \right|^{2}=E_{0}\langle \psi |\psi \rangle .\end{aligned}}}
If we were to vary over all possible states with norm 1 trying to minimize the expectation value of H {\displaystyle H} , the lowest value would be E 0 {\displaystyle E_{0}} and the corresponding state would be the ground state, as well as an eigenstate of H {\displaystyle H} . Varying over the entire Hilbert space is usually too complicated for physical calculations, and a subspace of the entire Hilbert space is chosen, parametrized by some (real) differentiable parameters α i ( i = 1, 2, ..., N ) . The choice of the subspace is called the ansatz . Some choices of ansatzes lead to better approximations than others, therefore the choice of ansatz is important.
Let's assume there is some overlap between the ansatz and the ground state (otherwise, it's a bad ansatz). We wish to normalize the ansatz, so we have the constraints ⟨ ψ ( α ) | ψ ( α ) ⟩ = 1 {\displaystyle \left\langle \psi (\mathbf {\alpha } )|\psi (\mathbf {\alpha } )\right\rangle =1} and we wish to minimize ε ( α ) = ⟨ ψ ( α ) | H | ψ ( α ) ⟩ . {\displaystyle \varepsilon (\mathbf {\alpha } )=\left\langle \psi (\mathbf {\alpha } )\right|H\left|\psi (\mathbf {\alpha } )\right\rangle .}
This, in general, is not an easy task, since we are looking for a global minimum and finding the zeroes of the partial derivatives of ε over all α i is not sufficient. If ψ ( α ) is expressed as a linear combination of other functions ( α i being the coefficients), as in the Ritz method , there is only one minimum and the problem is straightforward. There are other, non-linear methods, however, such as the Hartree–Fock method , that are also not characterized by a multitude of minima and are therefore comfortable in calculations.
There is an additional complication in the calculations described. As ε tends toward E 0 in minimization calculations, there is no guarantee that the corresponding trial wavefunctions will tend to the actual wavefunction. This has been demonstrated by calculations using a modified harmonic oscillator as a model system, in which an exactly solvable system is approached using the variational method. A wavefunction different from the exact one is obtained by use of the method described above. [ citation needed ]
Although usually limited to calculations of the ground state energy, this method can be applied in certain cases to calculations of excited states as well. If the ground state wavefunction is known, either by the method of variation or by direct calculation, a subset of the Hilbert space can be chosen which is orthogonal to the ground state wavefunction.
| ψ ⟩ = | ψ test ⟩ − ⟨ ψ g r | ψ test ⟩ | ψ gr ⟩ {\displaystyle \left|\psi \right\rangle =\left|\psi _{\text{test}}\right\rangle -\left\langle \psi _{\mathrm {gr} }|\psi _{\text{test}}\right\rangle \left|\psi _{\text{gr}}\right\rangle }
The resulting minimum is usually not as accurate as for the ground state, as any difference between the true ground state and ψ gr {\displaystyle \psi _{\text{gr}}} results in a lower excited energy. This defect is worsened with each higher excited state.
In another formulation: E ground ≤ ⟨ ϕ | H | ϕ ⟩ . {\displaystyle E_{\text{ground}}\leq \left\langle \phi \right|H\left|\phi \right\rangle .}
This holds for any trial φ since, by definition, the ground state wavefunction has the lowest energy, and any trial wavefunction will have energy greater than or equal to it.
Proof: φ can be expanded as a linear combination of the actual eigenfunctions of the Hamiltonian (which we assume to be normalized and orthogonal): ϕ = ∑ n c n ψ n . {\displaystyle \phi =\sum _{n}c_{n}\psi _{n}.}
Then, to find the expectation value of the Hamiltonian: ⟨ H ⟩ = ⟨ ϕ | H | ϕ ⟩ = ⟨ ∑ n c n ψ n | H | ∑ m c m ψ m ⟩ = ∑ n ∑ m ⟨ c n ∗ ψ n | E m | c m ψ m ⟩ = ∑ n ∑ m c n ∗ c m E m ⟨ ψ n | ψ m ⟩ = ∑ n | c n | 2 E n . {\displaystyle {\begin{aligned}\left\langle H\right\rangle =\left\langle \phi \right|H\left|\phi \right\rangle ={}&\left\langle \sum _{n}c_{n}\psi _{n}\right|H\left|\sum _{m}c_{m}\psi _{m}\right\rangle \\={}&\sum _{n}\sum _{m}\left\langle c_{n}^{*}\psi _{n}\right|E_{m}\left|c_{m}\psi _{m}\right\rangle \\={}&\sum _{n}\sum _{m}c_{n}^{*}c_{m}E_{m}\left\langle \psi _{n}|\psi _{m}\right\rangle \\={}&\sum _{n}|c_{n}|^{2}E_{n}.\end{aligned}}}
Now, the ground state energy is the lowest energy possible, i.e., E n ≥ E ground {\displaystyle E_{n}\geq E_{\text{ground}}} . Therefore, if the guessed wave function φ is normalized: ⟨ ϕ | H | ϕ ⟩ ≥ E ground ∑ n | c n | 2 = E ground . {\displaystyle \left\langle \phi \right|H\left|\phi \right\rangle \geq E_{\text{ground}}\sum _{n}|c_{n}|^{2}=E_{\text{ground}}.}
For a hamiltonian H that describes the studied system and any normalizable function Ψ with arguments appropriate for the unknown wave function of the system, we define the functional ε [ Ψ ] = ⟨ Ψ | H ^ | Ψ ⟩ ⟨ Ψ | Ψ ⟩ . {\displaystyle \varepsilon \left[\Psi \right]={\frac {\left\langle \Psi \right|{\hat {H}}\left|\Psi \right\rangle }{\left\langle \Psi |\Psi \right\rangle }}.}
The variational principle states that
The variational principle formulated above is the basis of the variational method used in quantum mechanics and quantum chemistry to find approximations to the ground state .
Another facet in variational principles in quantum mechanics is that since Ψ {\displaystyle \Psi } and Ψ † {\displaystyle \Psi ^{\dagger }} can be varied separately (a fact arising due to the complex nature of the wave function), the quantities can be varied in principle just one at a time. [ 4 ]
The helium atom consists of two electrons with mass m and electric charge − e , around an essentially fixed nucleus of mass M ≫ m and charge +2 e . The Hamiltonian for it, neglecting the fine structure , is: H = − ℏ 2 2 m ( ∇ 1 2 + ∇ 2 2 ) − e 2 4 π ε 0 ( 2 r 1 + 2 r 2 − 1 | r 1 − r 2 | ) {\displaystyle H=-{\frac {\hbar ^{2}}{2m}}\left(\nabla _{1}^{2}+\nabla _{2}^{2}\right)-{\frac {e^{2}}{4\pi \varepsilon _{0}}}\left({\frac {2}{r_{1}}}+{\frac {2}{r_{2}}}-{\frac {1}{|\mathbf {r} _{1}-\mathbf {r} _{2}|}}\right)} where ħ is the reduced Planck constant , ε 0 is the vacuum permittivity , r i (for i = 1, 2 ) is the distance of the i -th electron from the nucleus, and | r 1 − r 2 | is the distance between the two electrons.
If the term V ee = e 2 /(4 πε 0 | r 1 − r 2 |) , representing the repulsion between the two electrons, were excluded, the Hamiltonian would become the sum of two hydrogen-like atom Hamiltonians with nuclear charge +2 e . The ground state energy would then be 8 E 1 = −109 eV , where E 1 is the Rydberg constant , and its ground state wavefunction would be the product of two wavefunctions for the ground state of hydrogen-like atoms: [ 2 ] : 262 ψ ( r 1 , r 2 ) = Z 3 π a 0 3 e − Z ( r 1 + r 2 ) / a 0 . {\displaystyle \psi (\mathbf {r} _{1},\mathbf {r} _{2})={\frac {Z^{3}}{\pi a_{0}^{3}}}e^{-Z\left(r_{1}+r_{2}\right)/a_{0}}.} where a 0 is the Bohr radius and Z = 2 , helium's nuclear charge. The expectation value of the total Hamiltonian H (including the term V ee ) in the state described by ψ 0 will be an upper bound for its ground state energy. ⟨ V ee ⟩ is −5 E 1 /2 = 34 eV , so ⟨ H ⟩ is 8 E 1 − 5 E 1 /2 = −75 eV .
A tighter upper bound can be found by using a better trial wavefunction with 'tunable' parameters. Each electron can be thought to see the nuclear charge partially "shielded" by the other electron, so we can use a trial wavefunction equal with an "effective" nuclear charge Z < 2 : The expectation value of H in this state is: ⟨ H ⟩ = [ − 2 Z 2 + 27 4 Z ] E 1 {\displaystyle \left\langle H\right\rangle =\left[-2Z^{2}+{\frac {27}{4}}Z\right]E_{1}}
This is minimal for Z = 27/16 implying shielding reduces the effective charge to ~1.69. Substituting this value of Z into the expression for H yields 729 E 1 /128 = −77.5 eV , within 2% of the experimental value, −78.975 eV. [ 5 ]
Even closer estimations of this energy have been found using more complicated trial wave functions with more parameters. This is done in physical chemistry via variational Monte Carlo .
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In science and especially in mathematical studies, a variational principle is one that enables a problem to be solved using calculus of variations , which concerns finding functions that optimize the values of quantities that depend on those functions. For example, the problem of determining the shape of a hanging chain suspended at both ends—a catenary —can be solved using variational calculus , and in this case, the variational principle is the following: The solution is a function that minimizes the gravitational potential energy of the chain.
The history of the variational principle in classical mechanics started with Maupertuis's principle in the 18th century.
Felix Klein 's 1872 Erlangen program attempted to identify invariants under a group of transformations.
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In evolutionary biology, the variational properties of an organism are those properties relating to the production of variation among its offspring. In a broader sense variational properties include phenotypic plasticity. [ 1 ] [ 2 ] [ 3 ] Variational properties contrast with functional properties. While the functional properties of an organism determine is level of adaptedness to its environment, it is the variational properties of the organisms in a species that chiefly determine its evolvability and genetic robustness .
Variational properties group together many classical and more recent concepts of evolutionary biology. It includes the classical concepts of pleiotropy , canalization , developmental constraints , developmental bias , morphological integration , developmental homeostasis and later concepts such as robustness , neutral networks , modularity , the G-matrix and distribution of fitness effects .
Variational properties also include the production of DNA sequence variation, epigenetic variation, and phenotypic variation. While the genome is typically thought of as the storehouse of information that generates the organism, it can also be seen as the set of heritable degrees of freedom for varying the organism. DNA thus has both a generative role in the organism, and variational role in the lineage.
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Variational transition-state theory [ 1 ] is a refinement of transition-state theory . When using transition-state theory to estimate a chemical reaction rate , the dividing surface is taken to be a surface that intersects a first-order saddle point and is also perpendicular to the reaction coordinate in all other dimensions . When using variational transition-state theory, the position of the dividing surface between reactant and product regions is variationally optimized to minimize the reaction rate. This minimizes the effects of recrossing, and gives a much more accurate result.
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In the mathematical fields of the calculus of variations and differential geometry , the variational vector field is a certain type of vector field defined on the tangent bundle of a differentiable manifold which gives rise to variations along a vector field in the manifold itself.
Specifically, let X be a vector field on M . Then X generates a one-parameter group of local diffeomorphisms Fl X t , the flow along X . The differential of Fl X t gives, for each t , a mapping
where TM denotes the tangent bundle of M . This is a one-parameter group of local diffeomorphisms of the tangent bundle. The variational vector field of X , denoted by T ( X ) is the tangent to the flow of d Fl X t .
This geometry-related article is a stub . You can help Wikipedia by expanding it .
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Varicode is a self-synchronizing code for use in PSK31 . It supports all ASCII characters, but the characters used most frequently in English have shorter codes. The space between characters is indicated by a 00 sequence, an implementation of Fibonacci coding . Originally created for speeding up real-time keyboard-to-keyboard exchanges over low bandwidth links, Varicode is freely available. [ 1 ] [ 2 ] [ 3 ]
Beginning with the single-bit code "1", valid varicode values may be formed by prefixing a "1" or "10" to a shorter code. Thus, the number of codes of length n is equal to the Fibonacci number F n . Varicode uses the 88 values of lengths up to 9 bits, and 40 of the 55 codes of length 10.
As transmitted, the codes are two bits longer due to the trailing delimiter 00.
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Variegation is the appearance of differently coloured zones in the foliage , flowers , and sometimes the stems and fruit of plants , granting a speckled, striped, or patchy appearance. The colors of the patches themselves vary from a slightly lighter shade of the natural coloration to yellow, to white, or other colors entirely such as red and pink. [ 1 ] This is caused by varying levels and types of pigment , such as chlorophyll in leaves. [ 2 ] Variegation can be caused by genetic mutations affecting pigment production, or by viral infections such as those resulting from mosaic viruses . [ 3 ] Many plants are also naturally variegated, such as Goeppertia insignis . Most of these are herbaceous or climbing plants, and are most often species native to tropical rainforests. [ 4 ]
Many species which are normally non-variegated are known to display variegation. Their appearance is desirable to enthusiasts, and many such plants are propagated and sold as unique cultivars . [ 1 ] However, in individuals where the variegation occurs in normally- photosynthetic cells, the lack of functioning chloroplasts can slow growth rate. [ 2 ] Conversely, naturally-variegated plants derive benefits from their appearance, such as improved photosynthetic efficiency in low-light conditions and herbivore deterrence. [ 5 ] [ 6 ]
The term is also sometimes used to refer to colour zonation in minerals and the integument of animals .
Chimeric plants contain tissues with more than one genotype . [ further explanation needed ] A variegated chimera contains some tissues that produce chlorophyll and other tissues which do not. [ 7 ] Because the variegation is due to the presence of two kinds of plant tissue, propagating the plant must be by a vegetative method of propagation that preserves both types of tissue in relation to each other. Typically, stem cuttings , bud and stem grafting , and other propagation methods that results in growth from leaf axil buds will preserve variegation. [ citation needed ] Cuttings with complete variegation may be difficult, if not impossible, to propagate. [ why? ] Root cuttings will not usually preserve variegation, since the new stem tissue is derived from a particular [ which? ] tissue type within the root. [ citation needed ]
Some variegation is caused by structural color , not pigment; the microscopic structure of the plant itself reflects light to produce varying colors. This can happen when an air layer is located just under the epidermis resulting in a white or silvery reflection. [ 8 ] It is sometimes called blister variegation. [ citation needed ] Pilea cadierei (aluminum plant) shows this effect. Leaves of most Cyclamen species show such patterned variegation, varying between plants, but consistent within each plant.
The presence of hairs on leaves, which may be coloured differently from the leaf itself, can also produce variable coloration. This is found in various Begonia species and their hybrids.
Sometimes venal variegation occurs – the veins of the leaf are picked out in white or yellow. This is due to lack of green tissue above the veins. It can be seen in some aroids . The blessed milk thistle , Silybum marianum , is a plant in which another type of venal variegation occurs, but in this case it is due to a blister variegation occurring along the veins.
A common cause of variegation is the masking of green pigment by other pigments, such as anthocyanins . This often extends to the whole leaf, causing it to be reddish or purplish. On some plants however, consistent zonal markings occur; such as on some clovers, bromeliads , certain Pelargonium and Oxalis species. On others, such as the commonly grown forms of Coleus , the variegation can vary widely within a population. In Nymphaea lotus , the tiger lotus, leaf variegations appear under intense illumination.
Virus infections may cause patterning to appear on the leaf surface. The patterning is often characteristic of the infection. Examples are the mosaic viruses , which produce a mosaic-type effect on the leaf surface or the citrus variegation virus (CVV). Recently [ when? ] a virus disease, Hosta virus X (HVX) has been identified that causes mottled leaf coloring in hostas . At first, diseased plants were propagated and grown for their mottled foliage, at the risk of infecting other healthy hostas. [ 9 ] While these diseases are usually serious enough that the gardener would not grow affected plants, there are a few affected plants that can survive indefinitely, and are attractive enough to be grown for ornament; e.g. some variegated Abutilon varieties. Nutrient deficiency symptoms may cause a temporary or variable yellowing in specific zones on the leaf. Iron and magnesium deficiencies are common causes of this. Transposable elements can cause colour variegation. [ 10 ]
It has been suggested that some patterns of leaf variegation may be part of a "defensive masquerade strategy." [ 11 ] In this, leaf variegation may appear to a leaf mining insect that the leaf is already infested, and this may reduce parasitization of the leaf by leaf miners. [ 12 ]
By convention, the italicised term 'variegata' as the second part of the Latin binomial name, indicates a species found in the wild with variegation ( Aloe variegata ). The much more common, non-italicised, inclusion of 'Variegata' as the third element of a name indicates a variegated cultivar of an unvariegated parent ( Aucuba japonica 'Variegata'). However, not all variegated plants have this Latin tag, for instance many cultivars of Pelargonium have some zonal variegation in their leaves. Other types of variegation may be indicated, e.g. Daphne odora 'Aureomarginata' has yellow edging on its leaves.
Variegated plants have long been valued by gardeners, as the usually lighter-coloured variegation can 'lift' what would otherwise be blocks of solid green foliage. Many gardening societies have specialist variegated plants groups, such as the Hardy Plant Society 's Variegated Plant Special Interest Group in the UK.
In 2020, a variegated Rhaphidophora tetrasperma plant sold at auction for US$5,300. [ 13 ] In June 2021, another variegated Rhaphidophora tetrasperma plant sold at auction for US$19,297. [ 14 ]
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In mathematics , and more precisely in semigroup theory, a variety of finite semigroups is a class of semigroups satisfying specific algebraic properties. Those classes can be defined in two distinct ways, using either algebraic notions or topological notions. Varieties of finite monoids , varieties of finite ordered semigroups and varieties of finite ordered monoids are defined similarly.
This notion is very similar to the general notion of varieties and pseudovarieties in universal algebra .
There are two standard equivalent definitions for a variety of finite semigroups.
A variety V {\displaystyle V} of finite (ordered) semigroups is a class of finite (ordered) semigroups that:
The first condition is equivalent to stating that V {\displaystyle V} is closed under taking subsemigroups and under taking quotients . The second property implies that the empty product—that is, the trivial semigroup of one element—belongs to each variety. Hence a variety is necessarily non-empty.
A variety of finite (ordered) monoids is a variety of finite (ordered) semigroups whose elements are monoids. That is, it is a class of (ordered) monoids satisfying the two conditions stated above.
In order to give the topological definition of a variety of finite semigroups, some other definitions related to profinite words are needed.
Let A {\displaystyle A} be an arbitrary finite alphabet . Let A + {\displaystyle A^{+}} be its free semigroup . Then let A ^ {\displaystyle {\hat {A}}} be the set of profinite words over A {\displaystyle A} . Given a semigroup morphism ϕ : A + → S {\displaystyle \phi :A^{+}\to S} , let ϕ ^ : A ^ → S {\displaystyle {\hat {\phi }}:{\hat {A}}\to S} be the unique continuous extension of ϕ {\displaystyle \phi } to A ^ {\displaystyle {\hat {A}}} .
A profinite identity is a pair u {\displaystyle u} and v {\displaystyle v} of profinite words. A semigroup S is said to satisfy the profinite identity u = v {\displaystyle u=v} if, for each semigroup morphism ϕ : A + → S {\displaystyle \phi :A^{+}\to S} , the equality ϕ ^ ( u ) = ϕ ^ ( v ) {\displaystyle {\hat {\phi }}(u)={\hat {\phi }}(v)} holds.
A variety of finite semigroups is the class of finite semigroups satisfying a set of profinite identities P {\displaystyle P} .
A variety of finite monoids is defined like a variety of finite semigroups, with the difference that one should consider monoid morphisms ϕ : A ∗ → M {\displaystyle \phi :A^{*}\to M} instead of semigroup morphisms ϕ : A + → M {\displaystyle \phi :A^{+}\to M} .
A variety of finite ordered semigroups/monoids is also given by a similar definition, with the difference that one should consider morphisms of ordered semigroups/monoids.
A few examples of classes of semigroups are given. The first examples uses finite identities—that is, profinite identities whose two words are finite words. The next example uses profinite identities. The last one is an example of a class that is not a variety.
More examples are given in the article Special classes of semigroups .
More generally, given a profinite word u and a letter x , the profinite equality ux = xu states that the set of possible images of u contains only elements of the centralizer. Similarly, ux = x states that the set of possible images of u contains only left identities. Finally ux = u states that the set of possible images of u is composed of left zeros.
Examples using profinite words that are not finite are now given.
Given a profinite word, x , let x ω {\displaystyle x^{\omega }} denote lim n → ∞ x n ! {\displaystyle \lim _{n\to \infty }x^{n!}} . Hence, given a semigroup morphism ϕ : A + → S {\displaystyle \phi :A^{+}\to S} , ϕ ^ ( x ω ) {\displaystyle {\hat {\phi }}(x^{\omega })} is the only idempotent power of ϕ ( x ) {\displaystyle \phi (x)} . Thus, in profinite equalities, x ω {\displaystyle x^{\omega }} represents an arbitrary idempotent.
The class G of finite groups is a variety of finite semigroups. Note that a finite group can be defined as a finite semigroup, with a unique idempotent, which in addition is a left and right identity. Once those two properties are translated in terms of profinite equality, one can see that the variety G is defined by the set of profinite equalities { x ω = y ω and x ω y = y x ω = y } . {\displaystyle \{x^{\omega }=y^{\omega }{\text{ and }}x^{\omega }y=yx^{\omega }=y\}.}
Note that the class of finite monoids is not a variety of finite semigroups. Indeed, this class is not closed under subsemigroups. To see this, take any finite semigroup S that is not a monoid. It is a subsemigroup of the monoid S 1 formed by adjoining an identity element.
Reiterman's theorem states that the two definitions above are equivalent. A scheme of the proof is now given.
Given a variety V of semigroups as in the algebraic definition , one can choose the set P of profinite identities to be the set of profinite identities satisfied by every semigroup of V .
Reciprocally, given a profinite identity u = v , one can remark that the class of semigroups satisfying this profinite identity is closed under subsemigroups, quotients, and finite products. Thus this class is a variety of finite semigroups. Furthermore, varieties are closed under arbitrary intersection, thus, given an arbitrary set P of profinite identities u i = v i , the class of semigroups satisfying P is the intersection of the class of semigroups satisfying all of those profinite identities. That is, it is an intersection of varieties of finite semigroups, and this a variety of finite semigroups.
The definition of a variety of finite semigroups is inspired by the notion of a variety of universal algebras . We recall the definition of a variety in universal algebra. Such a variety is, equivalently:
The main differences between the two notions of variety are now given. In this section "variety of (arbitrary) semigroups" means "the class of semigroups as a variety of universal algebra over the vocabulary of one binary operator". It follows from the definitions of those two kind of varieties that, for any variety V of (arbitrary) semigroups, the class of finite semigroups of V is a variety of finite semigroups.
We first give an example of a variety of finite semigroups that is not similar to any subvariety of the variety of (arbitrary) semigroups. We then give the difference between the two definition using identities. Finally, we give the difference between the algebraic definitions.
As shown above, the class of finite groups is a variety of finite semigroups. However, the class of groups is not a subvariety of the variety of (arbitrary) semigroups. Indeed, ⟨ Z , + ⟩ {\displaystyle \langle \mathbb {Z} ,+\rangle } is a monoid that is an infinite group . However, its submonoid ⟨ N , + ⟩ {\displaystyle \langle \mathbb {N} ,+\rangle } is not a group. Since the class of (arbitrary) groups contains a semigroup and does not contain one of its subsemigroups, it is not a variety. The main difference between the finite case and the infinite case, when groups are considered, is that a submonoid of a finite group is a finite group. While infinite groups are not closed under taking submonoids.
The class of finite groups is a variety of finite semigroups, while it is not a subvariety of the variety of (arbitrary) semigroups. Thus, Reiterman's theorem shows that this class can be defined using profinite identities. And Birkhoff's HSP theorem shows that this class can not be defined using identities (of finite words). This illustrates why the definition of a variety of finite semigroups uses the notion of profinite words and not the notion of identities.
We now consider the algebraic definitions of varieties. Requiring that varieties are closed under arbitrary direct products implies that a variety is either trivial or contains infinite structures. In order to restrict varieties to contain only finite structures, the definition of variety of finite semigroups uses the notion of finite product instead of notion of arbitrary direct product.
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In Euclidean geometry , Varignon's theorem holds that the midpoints of the sides of an arbitrary quadrilateral form a parallelogram , called the Varignon parallelogram . It is named after Pierre Varignon , whose proof was published posthumously in 1731. [ 1 ]
The midpoints of the sides of an arbitrary quadrilateral form a parallelogram. If the quadrilateral is convex or concave (not complex ), then the area of the parallelogram is half the area of the quadrilateral.
If one introduces the concept of oriented areas for n -gons , then this area equality also holds for complex quadrilaterals. [ 2 ]
The Varignon parallelogram exists even for a skew quadrilateral , and is planar whether the quadrilateral is planar or not. The theorem can be generalized to the midpoint polygon of an arbitrary polygon.
Referring to the diagram above, triangles ADC and HDG are similar by the side-angle-side criterion, so angles DAC and DHG are equal, making HG parallel to AC . In the same way EF is parallel to AC , so HG and EF are parallel to each other; the same holds for HE and GF .
Varignon's theorem can also be proved as a theorem of affine geometry organized as linear algebra with the linear combinations restricted to coefficients summing to 1, also called affine or barycentric coordinates . The proof applies even to skew quadrilaterals in spaces of any dimension.
Any three points E , F , G are completed to a parallelogram (lying in the plane containing E , F , and G ) by taking its fourth vertex to be E − F + G . In the construction of the Varignon parallelogram this is the point ( A + B )/2 − ( B + C )/2 + ( C + D )/2 = ( A + D )/2. But this is the point H in the figure, whence EFGH forms a parallelogram.
In short, the centroid of the four points A , B , C , D is the midpoint of each of the two diagonals EG and FH of EFGH , showing that the midpoints coincide.
From the first proof, one can see that the sum of the diagonals is equal to the perimeter of the parallelogram formed. Also, we can use vectors 1/2 the length of each side to first determine the area of the quadrilateral, and then to find areas of the four triangles divided by each side of the inner parallelogram.
A planar Varignon parallelogram also has the following properties:
In a convex quadrilateral with sides a , b , c and d , the length of the bimedian that connects the midpoints of the sides a and c is
where p and q are the length of the diagonals. [ 4 ] The length of the bimedian that connects the midpoints of the sides b and d is
Hence [ 3 ] : p.126
This is also a corollary to the parallelogram law applied in the Varignon parallelogram.
The lengths of the bimedians can also be expressed in terms of two opposite sides and the distance x between the midpoints of the diagonals. This is possible when using Euler's quadrilateral theorem in the above formulas. Whence [ 5 ]
and
The two opposite sides in these formulas are not the two that the bimedian connects.
In a convex quadrilateral, there is the following dual connection between the bimedians and the diagonals: [ 6 ]
The Varignon parallelogram is a rhombus if and only if the two diagonals of the quadrilateral have equal length, that is, if the quadrilateral is an equidiagonal quadrilateral . [ 7 ]
The Varignon parallelogram is a rectangle if and only if the diagonals of the quadrilateral are perpendicular , that is, if the quadrilateral is an orthodiagonal quadrilateral . [ 6 ] : p. 14 [ 7 ] : p. 169
For a self-crossing quadrilateral, the Varignon parallelogram can degenerate to four collinear points, forming a line segment traversed twice. This happens whenever the polygon is formed by replacing two parallel sides of a trapezoid by the two diagonals of the trapezoid, such as in the antiparallelogram . [ 8 ]
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Varignon's theorem is a theorem of French mathematician Pierre Varignon (1654–1722), published in 1687 in his book Projet d'une nouvelle mécanique . The theorem states that the torque of a resultant of two concurrent forces about any point is equal to the algebraic sum of the torques of its components about the same point.
In other words, "If many concurrent forces are acting on a body, then the algebraic sum of torques of all the forces about a point in the plane of the forces is equal to the torque of their resultant about the same point." [ 1 ]
Consider a set of N {\displaystyle N} force vectors f 1 , f 2 , . . . , f N {\displaystyle \mathbf {f} _{1},\mathbf {f} _{2},...,\mathbf {f} _{N}} that concur at a point O {\displaystyle \mathbf {O} } in space. Their resultant is:
The torque of each vector with respect to some other point O 1 {\displaystyle \mathbf {O} _{1}} is
Adding up the torques and pulling out the common factor ( O − O 1 ) {\displaystyle (\mathbf {O} -\mathbf {O_{1}} )} , one sees that the result may be expressed solely in terms of F {\displaystyle \mathbf {F} } , and is in fact the torque of F {\displaystyle \mathbf {F} } with respect to the point O 1 {\displaystyle \mathbf {O} _{1}} :
Proving the theorem, i.e. that the sum of torques about O 1 {\displaystyle \mathbf {O} _{1}} is the same as the torque of the sum of the forces about the same point.
This classical mechanics –related article is a stub . You can help Wikipedia by expanding it .
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The variome is the whole set of genetic variations found in populations of species that have gone through a relatively short evolution change. For example, among humans, about 1 in every 1,200 [ citation needed ] nucleotide bases differ. The size of human variome in terms of effective population size is claimed to be about 10,000 individuals. This variation rate is comparatively small compared to other species. For example, the effective population size of tigers which perhaps has the whole population size less than 10,000 in the wild is not much smaller than the human species indicating a much higher level of genetic diversity although they are close to extinction in the wild. In practice, the variome can be the sum of the single nucleotide polymorphisms (SNPs), indels, and structural variation (SV) of a population or species. The Human Variome Project seeks to compile this genetic variation data worldwide. Variomics is the study of variome and a branch of bioinformatics .
The human variome can be subdivided into smaller ethnicity specific variomes. Each variome can have a utility in terms of filtering out common ethnic specific variants in the analyses of cancer normal variants filtering for more efficient detection of somatic mutations that can be relevant to certain anti-cancer drugs. KoVariome is one such ethnic specific variome where the project uses the term variome to denote their identity and connection to the concept of the broader human variome. KoVariome founders have been affiliated with HVP since early days of HVP where Prof. Richard Cotton initiated various efforts to compile the human variome resources.
Many curated databases has been established to document the impact of clinically significant sequence variations, such as dbSNP [ 1 ] or ClinVar. [ 2 ] Similarly, many services have been developed by the bioinformatics community to search the literature for variants. [ 3 ]
The blend word 'variome' is from genetic variant (“a version of a gene that differs from other versions of the same gene which may or may not have an effect on human health”) and the suffix –ome (“the complete whole of a class of substances for a species or an individual”). ‘Variomics’ (which appeared in the literature before ‘variome’) was first coined by Professor Richard ‘Dick’ Cotton in 2002 to describe (“The systematic study of the effect of genetic variation on human health”). ‘Variome’ has since been most commonly used in reference to the ‘ Human Variome Project ’ founded four years in 2006 after ‘variomics’ was first coined.
There are a number of international project studying the human variome, including the International HapMap Project and the Human Variome Project (HVP). [ 4 ] The HapMap Project aims to identify and catalog genetic similarities and differences among humans. The HVP aims to collect data on all human genetic variation. The Korean Genome Project, which aims to collect all the East Asian ethnic Korean genetic variations has produced KoVariome that contains currently over 1,000 Korean whole genome variation information. [ 5 ] Turkish Genome Project has plans to analyze genomes of 100,000 people in Turkey . [ 6 ]
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In aviation , a variometer – also known as a rate of climb and descent indicator ( RCDI ), rate-of-climb indicator , vertical speed indicator ( VSI ), or vertical velocity indicator ( VVI ) – is one of the flight instruments in an aircraft used to inform the pilot of the rate of descent or climb . [ 1 ] It can be calibrated in metres per second , feet per minute (1 ft/min = 0.00508 m/s) or knots (1 kn ≈ 0.514 m/s), depending on country and type of aircraft. It is typically connected to the aircraft's external static pressure source.
In powered flight , the pilot makes frequent use of the VSI to ascertain that level flight is being maintained, especially during turning maneuvers. In gliding , the instrument is used almost continuously during normal flight, often with an audible output, to inform the pilot of rising or sinking air . It is usual for gliders to be equipped with more than one type of variometer. The simpler type does not need an external source of power and can therefore be relied upon to function regardless of whether a battery or power source has been fitted. The electronic type with audio needs a power source to be operative during the flight. The instrument is of little interest during launching and landing, with the exception of aerotow , where the pilot will usually want to avoid releasing in sink.
In 1930, according to Ann Welch , " Kronfeld ...was one of the first to use a variometer, a device suggested by Alexander Lippisch ." Welch goes on to state that the "first real thermal soaring" occurred in 1930 by A. Haller and Wolf Hirth , with Hirth using a variometer in his Musterle . Frank Irving states that Arthur Kantrowitz first mentioned total energy in 1940. However, as early as 1901, Wilbur Wright wrote about thermals, "when gliding operators have attained greater skill, they can, with comparative safety, maintain themselves in the air for hours at a time in this way, and thus by constant practice so increase their knowledge and skill that they can rise into the higher air and search out the currents which enable the soaring birds to transport themselves to any desired point, by first rising in a circle, and then sailing off at a descending angle." [ 2 ] [ 3 ]
According to Paul MacCready , "A variometer is essentially a pressure altimeter with a leak which tends to make it read the altitude of a moment earlier. It consists of a container vented to the outside air in such a way that the pressure inside the flask lags slightly behind the outside static pressure. The rate of climb measurement comes from the rate-of-air inflow or outflow from the container." [ 4 ]
Variometers measure the rate of change of altitude by detecting the change in air pressure (static pressure) as altitude changes. Common types of variometers include those based on a diaphragm, a vane (horn), a taut band, or are electric based. The vane variometer consists of a rotating vane, centered by a coil spring, dividing a chamber into two parts, one connected to a static port, and the other to an expansion chamber. Electric variometers use thermistors sensitive to airflow, or circuit boards consisting of variable resistors connected to the membrane of a tiny vacuum cavity. [ 5 ] [ 6 ] [ 7 ] [ 8 ]
A simple variometer can be constructed by adding a large reservoir (a vacuum flask ) to augment the storage capacity of a common aircraft rate-of-climb instrument. In its simplest electronic form, the instrument consists of an air bottle connected to the external atmosphere through a sensitive air flow meter. As the aircraft changes altitude, the atmospheric pressure outside the aircraft changes and air flows into or out of the air bottle to equalise the pressure inside the bottle and outside the aircraft. The rate and direction of flowing air is measured by the cooling of one of two self-heating thermistors and the difference between the thermistor resistances will cause a voltage difference; this is amplified and displayed to the pilot. The faster the aircraft is ascending (or descending), the faster the air flows. Air flowing out of the bottle indicates that the altitude of the aircraft is increasing. Air flowing into the bottle indicates that the aircraft is descending.
Newer variometer designs directly measure the static pressure of the atmosphere using a pressure sensor and detect changes in altitude directly from the change in air pressure instead of by measuring air flow. These designs tend to be smaller as they do not need the air bottle. They are more reliable as there is no bottle to be affected by changes in temperature and fewer chances for leaks to occur in the connecting tubes.
The designs described above, which measure the rate of change of altitude by automatically detecting the change in static pressure as the aircraft changes altitude are referred to as "uncompensated" variometers. The term "vertical speed indicator" or "VSI" is most often used for the instrument when it is installed in a powered aircraft. The term "variometer" is most often used when the instrument is installed in a glider or sailplane.
An "Inertial-lead" or "Instantaneous" VSI (IVSI) uses accelerometers to provide a quicker response to changes in vertical speed. [ 9 ]
Human beings, unlike birds and other flying animals, are not able directly to sense climb and sink rates. [ citation needed ] Before the invention of the variometer, sailplane pilots found it very hard to soar . Although they could readily detect abrupt changes in vertical speed ("in the seat of the pants"), their senses did not allow them to distinguish lift from sink, or strong lift from weak lift. The actual climb/sink rate could not even be guessed at, unless there was some clear fixed visual reference nearby. Being near a fixed reference means being near to a hillside, or to the ground. Except when hill-soaring (exploiting the lift close to the up-wind side of a hill), these are generally very unprofitable positions for glider pilots to be in. The most useful forms of lift ( thermal and wave lift) are found at higher altitudes and it is very hard for a pilot to detect or exploit them without the use of a variometer. After the variometer was invented in 1929 by Alexander Lippisch and Robert Kronfeld , [ 10 ] the sport of gliding moved into a new realm.
Variometers also became important in foot-launch hang gliding, where the open-to-air pilot hears the wind but needs the variometer to help him or her to detect regions of rising or sinking air. In early hang gliding, variometers were not needed for the short flights or flights close to ridge lift. But the variometer became key as pilots began making longer flights. The first portable variometer for use in hang gliders was the Colver Variometer, introduced in the 1970s by Colver Soaring Instruments, [ 11 ] which served to extend the sport into cross-country thermal flying. [ 12 ] [ 13 ] In the 1980s, Ball Variometers Inc., founded in 1971 by Richard Harding Ball (1921–2011), produced a wrist variometer powered by a 9-volt battery. [ 14 ] [ 15 ]
As the sport of gliding developed, however, it was found that these very simple "uncompensated" instruments had their limitations. The information that glider pilots really need to soar is the total change in energy experienced by the glider, including both altitude and speed. An uncompensated variometer will simply indicate vertical speed of the glider, giving rise to the possibility of a " stick thermal ," i.e., a change in altitude caused by stick input only. If a pilot pulls back on the stick, the glider will rise, but also slow down as well. But if a glider is rising without the speed changing, this is an indication of real lift, not "stick lift."
Compensated variometers also include information about the speed of the aircraft, so the total energy ( potential and kinetic ) is used, not just the change in altitude. For example, if a pilot pushes forward on the stick, speeding up as the plane dives, an uncompensated variometer only indicates that altitude is being lost. But the pilot could pull back on the stick, trading the extra speed for altitude again. A compensated variometer uses both speed and altitude to indicate the change in total energy. So the pilot that pushes the stick forward, diving to gain speed, and then pulls back again to regain altitude will notice no change in total energy on a compensated variometer (neglecting energy loss due to drag).
According to Helmut Reichmann , "The word 'variometer' means literally 'change meter,' and this is how it should be understood. Without further information it remains unclear what changes are being measured. The simple variometers...are rate of climb indicators. Since the actual sailplane climb and sink displayed on these instruments depends not only on airmass movement and sailplane performance, but also in large part on angle-of-attack changes ( elevator movements )...This makes it virtually impossible to extract useful information, such as - for instance - the location of thermals . While rate of climb indicators show altitude changes and hence changes in the potential energy of the sailplane, total-energy variometers indicate changes in the total energy of the sailplane, that is, both its potential energy (due to altitude) and its kinetic energy (due to airspeed)." [ 5 ]
Most modern sailplanes are equipped with Total Energy compensated variometers.
The total energy of the aircraft is:
1. E tot = E pot + E kin {\displaystyle E_{\text{tot}}=E_{\text{pot}}+E_{\text{kin}}}
where E pot {\displaystyle E_{\text{pot}}} is the potential energy, and E kin {\displaystyle E_{\text{kin}}} is the kinetic energy. So the change in total energy is:
2. Δ E tot = Δ E pot + Δ E kin {\displaystyle \Delta E_{\text{tot}}=\Delta E_{\text{pot}}+\Delta E_{\text{kin}}}
Since
3. Potential energy is proportional to height
E pot = m g h {\displaystyle E_{\text{pot}}=mgh}
where m {\displaystyle m} is the glider mass and g {\displaystyle g} the acceleration of gravity
and
4. Kinetic energy is proportional to velocity squared,
E kin = 1 2 m V 2 {\displaystyle E_{\text{kin}}={1 \over 2}mV^{2}}
then from 2:
5. Δ E tot = m g Δ h + 1 2 m Δ V 2 {\displaystyle \Delta E_{\text{tot}}=mg\Delta h+{1 \over 2}m{\Delta V}^{2}}
6. Typically, this is converted to an effective altitude change by dividing by the acceleration of gravity, and the mass of the aircraft, so:
Δ E tot m g = Δ h + Δ V 2 2 g {\displaystyle {\Delta E_{\text{tot}} \over mg}=\Delta h+{{\Delta V}^{2} \over 2g}}
Total-Energy Variometers use a membrane compensator, compensation by venturi , or are electronically compensated. The membrane compensator is an elastic membrane, which flexes according to the total pressure (pitot plus static) from airspeed. Thus, airspeed effects cancel out an increase in sink, due to acceleration, or a decrease in sink, due to deceleration. The venturi compensator supplies a speed-dependent negative pressure, so that the pressure reduces as speed increases, compensating for the increased static pressure due to sink. According to Helmut Reichmann , "...the least sensitive venturi mounting point would appear to be on the upper quarter of the vertical fin, some 60 cm (2 feet) forward of the leading edge." Venturi compensator types include the Irving Venturi (1948), the Althaus Venturi, the Hüttner Venturi, the Brunswick Tube, the Nicks Venturi, and the Double-Slotted Tube, developed by Bardowicks of Akaflieg Hannover, also known as the Braunschweig Tube. [ 5 ] [ 8 ] [ 16 ] [ 17 ]
Very few powered aircraft have total energy variometers. Pilots of powered aircraft are more interested in the true rate of change of altitude, as they often want to hold a constant altitude or maintain a steady climb or descent.
A second type of compensated variometer is the Netto or airmass variometer. In addition to TE compensation, the Netto variometer adjusts for the intrinsic sink rate of the glider at a given speed (the polar curve ) adjusted for the wing loading due to water ballast. The Netto variometer will always read zero in still air. This provides the pilot with the accurate measurement of air mass vertical movement critical for final glides (the last glide to the ultimate destination location).
In 1954, Paul MacCready wrote about a sinking speed correction for a total energy venturi. MacCready stated, "In still air...a glider has a different sinking speed at each airspeed...it would be nicer if the variometer automatically added the sink rate, and thus showed the vertical air motion instead of the vertical glider motion. The correction can be made by a variety of methods. Probably the nicest is to utilize the total energy venturi and the dynamic pressure from the pitot tube." [ 4 ] As Reichmann explained, a "Netto variometer shows the climb and sink of airmass (not of the sailplane!)...In order to achieve a 'net' indication, the always present polar sink of the sailplane must be 'compensated out' of the indication. To do this, one makes use of the fact that above the speed for best glide the polar sink speed of the sailplane increases roughly with the square of the airspeed. Since the pitot pressure also increases with the square of the speed, one can use it to 'compensate away' the effect of sailplane polar sink over virtually the entire speed range." [ 5 ] Tom Brandes states, "Netto is simply the German way of saying 'net,' and a Netto Variometer System (or polar compensator) is simply one that tells you the net vertical air movement with the sailplane movement or sink taken out of the usual variometer reading." [ 18 ]
The Relative Netto Variometer indicates the vertical speed the glider would achieve IF it flies at thermalling speed - independent of current air speed and attitude. This reading is calculated as the Netto reading minus the glider's minimum sink. When the glider circles to thermal, the pilot needs to know the glider's vertical speed instead of that of the air mass. The Relative Netto Variometer (or sometimes the super Netto ) includes a g-sensor to detect thermalling. When thermalling, the sensor will detect acceleration (gravity plus centrifugal) above 1 g and tell the relative netto variometer to stop subtracting the sailplane's wing load-adjusted polar sink rate for the duration. Some earlier nettos used a manual switch instead of the g sensor.
In 1954, MacCready pointed out the advantages of an Audio Variometer, "There is much to be gained if the variometer indication is presented to the pilot by sound. More than any other instrument except during blind flying, the variometer must be watched continuously. if the pilot can get the reading by ear, he can improve his thermal flying by watching nearby gliders, and he can materially improve the overall flight by studying the cloud formations to be used next." [ 4 ]
In modern gliders, most electronic variometers generate a sound whose pitch and rhythm depends on the instrument reading. Typically the audio tone increases in frequency as the variometer shows a higher rate of climb and decreases in frequency towards a deep groan as the variometer shows a faster rate of descent. When the variometer is showing a climb, the tone is often chopped and the rate of chopping may be increased as the climb rate increases, while during a descent the tone is not chopped. The vario is typically silent in still air or in lift which is weaker than the typical sink rate of the glider at minimum sink . This audio signal allows the pilot to concentrate on the external view instead of having to watch the instruments, thus improving safety and also giving the pilot more opportunity to search for promising looking clouds and other signs of lift. A variometer that produces this type of audible tone is known as an "audio variometer".
Advanced electronic variometers in gliders can present other information to the pilot from GPS receivers. The display can thus show the bearing, distance and height required to reach an objective. In cruise mode (used in straight flight), the vario can also give an audible indication of the correct speed to fly depending on whether the air is rising or sinking. The pilot merely has to input the estimated MacCready setting, which is the expected rate of climb in the next acceptable thermal.
There is an increasing trend for advanced variometers in gliders towards flight computers (with variometer indications) which can also present information such as controlled airspace, lists of turnpoints and even collision warnings. Some will also store positional GPS data during the flight for later analysis.
Variometers are also used in radio controlled gliders. Each variometer system consists of a radio transmitter in the glider, and a receiver on the ground for use by the pilot. Depending on the design, the receiver may give the pilot the current altitude of the glider, and a display that indicates if the glider is gaining or losing altitude—often via an audio tone. Other forms of telemetry may also be provided by the system, displaying parameters such as airspeed and battery voltage. Variometers used in radio controlled gliders may or may not feature total energy compensation.
Variometers are not essential in radio controlled gliders; a skilled pilot can usually determine if the glider is going up or down via visual cues alone. The use of variometers is prohibited in some soaring contests for radio controlled gliders.
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A variscale is variable length mechanical scale (ruler) designed to directly measure latitude and longitude on USGS maps. [ 1 ]
This cartography or mapping term article is a stub . You can help Wikipedia by expanding it .
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The Varrentrapp reaction , also named Varrentrapp degradation , is a name reaction in the organic chemistry . It is named after Franz Varrentrapp, who described this reaction in 1840. [ 1 ] The reaction entails the degradation of an unsaturated carboxylic acid into a saturated acid with two fewer carbon atoms and acetic acid . The fragmentation is induced by action of molten alkali . [ 2 ]
Below, the reaction mechanism is shown for the degradation of ( E )-4-hexenoic acid. [ 3 ]
First, the carboxylic acid reacts with the caustic potash 1 to form its carboxylate. A series of base-catalyzed isomerizations leads to migration of the alkene into conjugation with the carboxylate (from ( 2 to 5 ). Reaction of this α,β-unsaturated carbonyl compound with hydroxide leads to a retro- aldol-condensation reaction, fragmenting the molecule into a shortened carbon chain with an aldehyde ( 7 ) and a separate acetate ( 9 ). Hydroxide then causes dehydrogenation of the aldehyde to form an acid ( 10 ).
Further insight is obtained by study of the Varrentrapp reaction for the conversion of oleic acid to palmitic acid. If the reaction is quenched before formation of the acetate, the recovered C18 acid consists of numerous isomers of octadecenoic acid (but not α,β-octadecenoic acid). [ 2 ] This observation suggests that the base (KOH) isomerizes the double bond. It is speculated that this occurs via deprotonation of the allylic C-H's. [ 4 ]
Likewise cinnamic acid is converted to benzoic acid . [ 5 ]
The reaction conditions are harsh: medium molten potassium hydroxide at temperatures in the range of 250 to 300 °C. The reaction has been of some importance in structure elucidation of certain fatty acids , but has little practical synthetic value. [ 6 ] [ 4 ] The original 1840 Varrentrapp reaction concerned the conversion of oleate to palmitate and acetate . [ 1 ]
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Vartkess Ara Apkarian (born 1955) is an Armenian-American physical chemist and a Professor of Chemistry at The University of California, Irvine . [ 1 ]
He is the Director of Center for Chemistry at the Space-Time Limit , a National Science Foundation Center for Chemical Innovation . He graduated from University of Southern California with B.S. degrees in Chemistry followed by Ph.D. degree in chemistry from Northwestern University . Following a postdoctoral fellowship at Cornell University, he joined the University of California as Chemistry faculty in 1983. He served as the Chair of the Chemistry Department (2004-2007) at UC Irvine.
He is a Foreign Member of the National Academy of Sciences of Armenia , and a Fellow of American Physical Society , American Association for the Advancement of Sciences . His teaching and research has been recognized with awards including the Humboldt Prize (1996), USC Distinguished Alumnus (2007), Charles Bennett Service Through Chemistry Award of ACS (2008) ACS Award in Experimental Physical Chemistry (2014), Honorary Doctorate from the University of Jyväskylä , Finland (2016).
His recent scientific contributions include creating a single-molecule sensor [ 2 ] and developing tools to confine the light to atomic dimensions. [ 3 ] His team visualized the internal structure of single molecules and imaging the normal vibrational modes of single molecules. [ 2 ] [ 4 ]
This biographical article about an American chemist is a stub . You can help Wikipedia by expanding it .
This biography of an American academic is a stub . You can help Wikipedia by expanding it .
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Vascular endothelial growth factor ( VEGF , / v ɛ dʒ ˈ ɛ f / ), originally known as vascular permeability factor ( VPF ), [ 1 ] is a signal protein produced by many cells that stimulates the formation of blood vessels. To be specific, VEGF is a sub-family of growth factors , the platelet-derived growth factor family of cystine-knot growth factors. They are important signaling proteins involved in both vasculogenesis (the de novo formation of the embryonic circulatory system ) and angiogenesis (the growth of blood vessels from pre-existing vasculature).
It is part of the system that restores the oxygen supply to tissues when blood circulation is inadequate such as in hypoxic conditions. [ 2 ] Serum concentration of VEGF is high in bronchial asthma and diabetes mellitus . [ 3 ] VEGF's normal function is to create new blood vessels during embryonic development , new blood vessels after injury, muscle following exercise, and new vessels ( collateral circulation ) to bypass blocked vessels.
It can contribute to disease. Solid cancers cannot grow beyond a limited size without an adequate blood supply; cancers that can express VEGF are able to grow and metastasize. Overexpression of VEGF can cause vascular disease in the retina of the eye and other parts of the body. Drugs such as aflibercept , bevacizumab , ranibizumab , and pegaptanib can inhibit VEGF and control or slow those diseases.
In 1970, Judah Folkman et al . described a factor secreted by tumors causing angiogenesis and called it tumor angiogenesis factor . [ 4 ] In 1983 Senger et al. identified a vascular permeability factor secreted by tumors in guinea pigs and hamsters. [ 1 ] In 1989 Ferrara and Henzel described an identical factor in bovine pituitary follicular cells which they purified, cloned and named VEGF. [ 5 ] A similar VEGF alternative splicing was discovered by Tischer et al. in 1991. [ 6 ] Between 1996 and 1997, Christinger and De Vos obtained the crystal structure of VEGF, first at 2.5 Å resolution and later at 1.9 Å. [ 7 ] [ 8 ] [ 9 ]
Fms-like tyrosine kinase-1 (flt-1) was shown to be a VEGF receptor by Ferrara et al. in 1992. [ 10 ] The kinase insert domain receptor (KDR) was shown to be a VEGF receptor by Terman et al. in 1992 as well. [ 11 ] In 1998, neuropilin 1 and neuropilin 2 were shown to act as VEGF receptors. [ 12 ]
In mammals, the VEGF family comprises five members: VEGF-A , placenta growth factor ( PGF ), VEGF-B , VEGF-C and VEGF-D . The latter members were discovered after VEGF-A; before their discovery, VEGF-A was known as VEGF. A number of VEGF-related proteins encoded by viruses ( VEGF-E ) and in the venom of some snakes ( VEGF-F ) have also been discovered.
Activity of VEGF-A, as its name implies, has been studied mostly on cells of the vascular endothelium , although it does have effects on a number of other cell types (e.g., stimulation monocyte / macrophage migration, neurons, cancer cells, kidney epithelial cells). In vitro, VEGF-A has been shown to stimulate endothelial cell mitogenesis and cell migration . VEGF-A is also a vasodilator and increases microvascular permeability and was originally referred to as vascular permeability factor.
There are multiple isoforms of VEGF-A that result from alternative splicing of mRNA from a single, 8- exon VEGFA gene. These are classified into two groups which are referred to according to their terminal exon (exon 8) splice site: the proximal splice site (denoted VEGF xxx ) or distal splice site (VEGF xxx b). In addition, alternate splicing of exon 6 and 7 alters their heparin -binding affinity and amino acid number (in humans: VEGF 121 , VEGF 121 b, VEGF 145 , VEGF 165 , VEGF 165 b, VEGF 189 , VEGF 206 ; the rodent orthologs of these proteins contain one fewer amino acids). These domains have important functional consequences for the VEGF splice variants, as the terminal (exon 8) splice site determines whether the proteins are pro-angiogenic (proximal splice site, expressed during angiogenesis) or anti-angiogenic (distal splice site, expressed in normal tissues). In addition, inclusion or exclusion of exons 6 and 7 mediate interactions with heparan sulfate proteoglycans (HSPGs) and neuropilin co-receptors on the cell surface, enhancing their ability to bind and activate the VEGF receptors (VEGFRs). [ 18 ] Recently, VEGF-C has been shown to be an important inducer of neurogenesis in the murine subventricular zone, without exerting angiogenic effects. [ 19 ]
All members of the VEGF family stimulate cellular responses by binding to tyrosine kinase receptors (the VEGFRs ) on the cell surface, causing them to dimerize and become activated through transphosphorylation , although to different sites, times, and extents. The VEGF receptors have an extracellular portion consisting of 7 immunoglobulin-like domains, a single transmembrane spanning region, and an intracellular portion containing a split tyrosine-kinase domain. VEGF-A binds to VEGFR-1 ( Flt-1 ) and VEGFR-2 ( KDR/Flk-1 ). [ 21 ] VEGFR-2 appears to mediate almost all of the known cellular responses to VEGF. The function of VEGFR-1 is less well-defined, although it is thought to modulate VEGFR-2 signaling. [ 22 ] Another function of VEGFR-1 may be to act as a dummy/decoy receptor, sequestering VEGF from VEGFR-2 binding (this appears to be particularly important during vasculogenesis in the embryo). VEGF-C and VEGF-D, but not VEGF-A, are ligands for a third receptor ( VEGFR-3/Flt4 ), which mediates lymphangiogenesis . The receptor (VEGFR3) is the site of binding of main ligands (VEGFC and VEGFD), which mediates perpetual action and function of ligands on
target cells. Vascular endothelial growth factor-C can stimulate lymphangiogenesis (via VEGFR3) and angiogenesis via VEGFR2. Vascular endothelial growth factor-R3 has been detected in lymphatic endothelial cells in CL of many species, cattle, buffalo and primate. [ 23 ]
In addition to binding to VEGFRs , VEGF binds to receptor complexes consisting of both neuropilins and VEGFRs. This receptor complex has increased VEGF signalling activity in endothelial cells ( blood vessels ). [ 12 ] [ 24 ] Neuropilins (NRP) are pleiotropic receptors and therefore other molecules may interfere with the signalling of the NRP/VEGFR receptor complexes. For example, Class 3 semaphorins compete with VEGF 165 for NRP binding and could therefore regulate VEGF-mediated angiogenesis . [ 25 ]
VEGF-A production can be induced in a cell that is not receiving enough oxygen . [ 21 ] When a cell is deficient in oxygen, it produces HIF, hypoxia-inducible factor , a transcription factor. HIF stimulates the release of VEGF-A, among other functions (including modulation of erythropoiesis). Circulating VEGF-A then binds to VEGF receptors on endothelial cells, triggering a tyrosine kinase pathway leading to angiogenesis. [ clarification needed ] The expression of angiopoietin-2 in the absence of VEGF leads to endothelial cell death and vascular regression. [ 26 ] Conversely, a German study done in vivo found that VEGF concentrations actually decreased after a 25% reduction in oxygen intake for 30 minutes. [ 27 ] HIF1 alpha and HIF1 beta are constantly being produced but HIF1 alpha is highly O 2 labile, so, in aerobic conditions, it is degraded. When the cell becomes hypoxic, HIF1 alpha persists and the HIF1alpha/beta complex stimulates VEGF release. the combined use of microvesicles and 5-FU resulted in enhanced chemosensitivity of squamous cell carcinoma cells more than the use of either 5-FU or microvesicle alone. In addition, down regulation of VEGF gene expression was associated with decreased CD1 gene expression. [ 28 ]
VEGF-A and the corresponding receptors are rapidly up-regulated after traumatic injury of the central nervous system (CNS). VEGF-A is highly expressed in the acute and sub-acute stages of CNS injury, but the protein expression declines over time. This time-span of VEGF-A expression corresponds with the endogenous re-vascularization capacity after injury. [ 25 ] This would suggest that VEGF-A / VEGF 165 could be used as target to promote angiogenesis after traumatic CNS injuries. However, there are contradicting scientific reports about the effects of VEGF-A treatments in CNS injury models. [ 25 ]
Although it has not been associated as a biomarker for the diagnosis of acute ischemic stroke , [ 29 ] high levels of serum VEGF in the first 48 hours after an cerebral infarct have been associated with a poor prognosis after 6 months [ 30 ] and 2 years. [ 31 ]
VEGF-A has been implicated with poor prognosis in breast cancer . Numerous studies show a decreased overall survival and disease-free survival in those tumors overexpressing VEGF. The overexpression of VEGF-A may be an early step in the process of metastasis , a step that is involved in the "angiogenic" switch. Although VEGF-A has been correlated with poor survival, its exact mechanism of action in the progression of tumors remains unclear. [ 32 ]
VEGF-A is also released in rheumatoid arthritis in response to TNF-α , increasing endothelial permeability and swelling and also stimulating angiogenesis (formation of capillaries). [ 33 ]
VEGF-A is also important in diabetic retinopathy (DR). The microcirculatory problems in the retina of people with diabetes can cause retinal ischaemia, which results in the release of VEGF-A, and a switch in the balance of pro-angiogenic VEGF xxx isoforms over the normally expressed VEGF xxx b isoforms. VEGF xxx may then cause the creation of new blood vessels in the retina and elsewhere in the eye, heralding changes that may threaten the sight.
VEGF-A plays a role in the disease pathology of the wet form age-related macular degeneration (AMD), which is the leading cause of blindness for the elderly of the industrialized world. The vascular pathology of AMD shares certain similarities with diabetic retinopathy, although the cause of disease and the typical source of neovascularization differs between the two diseases.
VEGF-D serum levels are significantly elevated in patients with angiosarcoma . [ 34 ]
Once released, VEGF-A may elicit several responses. It may cause a cell to survive, move, or further differentiate. Hence, VEGF is a potential target for the treatment of cancer . The first anti-VEGF drug, a monoclonal antibody named bevacizumab , was approved in 2004. Approximately 10–15% of patients benefit from bevacizumab therapy; however, biomarkers for bevacizumab efficacy are not yet known.
Current studies show that VEGFs are not the only promoters of angiogenesis. In particular, FGF2 and HGF are potent angiogenic factors.
Patients suffering from pulmonary emphysema have been found to have decreased levels of VEGF in the pulmonary arteries.
VEGF-D has also been shown to be over expressed in lymphangioleiomyomatosis and is currently used as a diagnostic biomarker in the treatment of this rare disease. [ 35 ]
In the kidney , increased expression of VEGF-A in glomeruli directly causes the glomerular hypertrophy that is associated with proteinuria. [ 36 ]
VEGF alterations can be predictive of early-onset pre-eclampsia . [ 37 ]
Gene therapies for refractory angina establish expression of VEGF in epicardial cells to promote angiogenesis. [ 38 ]
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A vascular malformation is a type of vascular anomaly . [ 2 ] They may cause aesthetic problems as they have a growth cycle, and can continue to grow throughout life.
Vascular malformations of the brain include those involving capillaries , and those involving the veins and arteries . Capillary malformations in the brain are known as cerebral cavernous malformations or capillary cavernous malformations . Those involving the mix of vessels are known as cerebral arteriovenous malformations (AVMs or cAVMs). The arteriovenous type is the most common in the brain. [ 3 ]
The International Society for the Study of Vascular Anomalies (ISSVA) classification has 5 types of Vascular Malformation.
* denotes high-flow malformation
Vascular malformations can also be divided into low-flow and high-flow types. [ 2 ] Low-flow malformations involve a single type of blood or lymph vessel, and are known as simple vascular malformations ; high-flow malformations involve an artery. There are also malformations that are of mixed-flow involving more than one type of vessel, such as an arteriovenous malformation . [ 2 ] Low-flow vascular malformations include capillary malformations , venous malformations , and lymphatic malformations . [ 4 ]
Capillary malformations involve the capillaries , and are the most common type. They used to refer only to port-wine stains but now include others. [ 2 ] Capillary malformations are limited to the superficial layers of the skin but they can thicken, become nodular, and sometimes become disfiguring. [ 5 ] It has been proposed that the category of capillary malformations, also called vascular stains , be classified into seven major clinical types including nevus flammeus nuchae also known as nevus simplex, commonly known as stork bite or salmon patch . [ 6 ]
A capillary malformation is also a feature of the disorder macrocephaly-capillary malformation . [ 7 ] An example of capillary malformation is cerebral cavernous malformations. This disease is linked to the central nervous system (brain, eye, spinal cord). They are abnormal clusters of closely packed, thin-walled blood vessels that usually form caverns. The lesions contain slow-moving or clotted blood. Lesions in the brain and spinal cord are particularly fragile and likely to bleed. [ 8 ]
Lymphatic malformations are congenital, developing from badly-formed lymphatic vessels in early embryonic development . [ 9 ] Abnormal development of the lymph vessels results in their failure to connect and drain into the venous system. [ 9 ]
These lymph vessels can become blocked due to the collection of lymph which forms a cyst as a mass, and are known as lymphatic malformationss . They can be macrocystic, microcystic, or a combination of the two. [ 9 ] Macrocystic have cysts greater than 2 cubic centimetres (0.12 cu in), and microcystic lymphatic malformation have cysts that are smaller than 2 cubic centimetres (0.12 cu in). [ 10 ]
A severe venous malformation is known as a lymphaticovenous malformation that also involves the lymph vessels. [ 11 ]
Venous malformations are the type of vascular malformation that involves the veins. They can often extend deeper from their surface appearance, reaching underlying muscle or bone. [ 12 ] In the neck they may extend into the lining of the mouth cavity or into the salivary glands . [ 11 ] They are the most common of the vascular malformations . [ 13 ] A severe venous malformation can involve the lymph vessels as a lymphaticovenous malformation . [ 11 ]
Arteriovenous malformations occur between an artery and a vein.
In the brain a cerebral arteriovenous malformation causes arterial blood to be directly shunted into the veins as there is an absence of a capillary bed. This carries a high risk of an intracranial hemorrhage . [ 14 ]
Combined types are defined as two or more vascular malformations found in one lesion. Examples of combined types include lymphatic-venous malformation (LVM) or capillary-venous-arteriovenous malformation (VAVM).
The International Society for the Study of Vascular Anomalies (ISSVA) classification is a basic and systematic classification of vascular anomalies with international acceptance.
As such terms such as "Lymphangioma" and "Cystic Hygroma", which were used widely in the past, are outdated. Newer research may only reference ISSVA terminology and, as a consequence, sources of information can be missed by doctors and patients unaware of the ISSVA convention.
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Vascular plants (from Latin vasculum ' duct ' ), also called tracheophytes ( UK : / ˈ t r æ k iː ə ˌ f aɪ t s / , [ 5 ] US : / ˈ t r eɪ k iː ə ˌ f aɪ t s / ) [ 6 ] or collectively tracheophyta ( / ˌ t r eɪ k iː ˈ ɒ f ɪ t ə / ; [ 7 ] [ 8 ] [ 9 ] from Ancient Greek τραχεῖα ἀρτηρία ( trakheîa artēría ) ' windpipe ' and φυτά ( phutá ) ' plants ' ), [ 9 ] are plants that have lignified tissues (the xylem ) for conducting water and minerals throughout the plant. They also have a specialized non-lignified tissue (the phloem ) to conduct products of photosynthesis . The group includes most land plants ( c. 300,000 accepted known species) [ 10 ] excluding mosses .
Vascular plants include the clubmosses , horsetails , ferns , gymnosperms (including conifers ), and angiosperms ( flowering plants ). They are contrasted with nonvascular plants such as mosses and green algae . Scientific names for the vascular plants group include Tracheophyta, [ 11 ] [ 4 ] : 251 Tracheobionta [ 12 ] and Equisetopsida sensu lato . Some early land plants (the rhyniophytes ) had less developed vascular tissue; the term eutracheophyte has been used for all other vascular plants, including all living ones.
Historically, vascular plants were known as " higher plants ", as it was believed that they were further evolved than other plants due to being more complex organisms. However, this is an antiquated remnant of the obsolete scala naturae , and the term is generally considered to be unscientific. [ 13 ]
Botanists define vascular plants by three primary characteristics:
Cavalier-Smith (1998) treated the Tracheophyta as a phylum or botanical division encompassing two of these characteristics defined by the Latin phrase "facies diploida xylem et phloem instructa" (diploid phase with xylem and phloem). [ 4 ] : 251
One possible mechanism for the presumed evolution from emphasis on haploid generation to emphasis on diploid generation is the greater efficiency in spore dispersal with more complex diploid structures. Elaboration of the spore stalk enabled the production of more spores and the development of the ability to release them higher and to broadcast them further. Such developments may include more photosynthetic area for the spore-bearing structure, the ability to grow independent roots, woody structure for support, and more branching. [ citation needed ]
Sexual reproduction in vascular land plants involves the process of meiosis. Meiosis provides a direct DNA repair capability for dealing with DNA damages , including oxidative DNA damages, in germline reproductive tissues. [ 15 ]
A proposed phylogeny of the vascular plants after Kenrick and Crane 1997 [ 16 ] is as follows, with modification to the gymnosperms from Christenhusz et al. (2011a), [ 17 ] Pteridophyta from Smith et al. [ 18 ] and lycophytes and ferns by Christenhusz et al. (2011b) [ 19 ] The cladogram distinguishes the rhyniophytes from the "true" tracheophytes, the eutracheophytes. [ 16 ]
† Aglaophyton
† Horneophytopsida
† Rhyniophyta
Lycopodiophyta
† Zosterophyllophyta
† Cladoxylopsida
Equisetopsida (horsetails)
Marattiopsida
Psilotopsida (whisk ferns and adders'-tongues)
Pteridopsida (true ferns)
† Progymnospermophyta
Cycadophyta (cycads)
Ginkgophyta (ginkgo)
Gnetophyta
Pinophyta (conifers)
Magnoliophyta (flowering plants)
† Pteridospermatophyta (seed ferns)
This phylogeny is supported by several molecular studies. [ 18 ] [ 20 ] [ 21 ] Other researchers state that taking fossils into account leads to different conclusions, for example that the ferns (Pteridophyta) are not monophyletic. [ 22 ]
Hao and Xue presented an alternative phylogeny in 2013 for pre- euphyllophyte plants. [ 23 ]
† Horneophytaceae
† Cooksoniaceae
† Aglaophyton
† Rhyniopsida
† Catenalis
† Aberlemnia
† Hsuaceae
† Renaliaceae
† Adoketophyton
†? Barinophytopsida
† Zosterophyllopsida
† Hicklingia
† Gumuia
† Nothia
Lycopodiopsida
† Zosterophyllum deciduum
† Yunia
† Eophyllophyton
† Trimerophytopsida
† Ibyka
† Pauthecophyton
† Cladoxylopsida
Polypodiopsida
† Celatheca
† Pertica
† Progymnosperms (paraphyletic)
Spermatophytes
Water and nutrients in the form of inorganic solutes are drawn up from the soil by the roots and transported throughout the plant by the xylem . Organic compounds such as sucrose produced by photosynthesis in leaves are distributed by the phloem sieve-tube elements . [ citation needed ]
The xylem consists of vessels in flowering plants and of tracheids in other vascular plants. Xylem cells are dead, hard-walled hollow cells arranged to form files of tubes that function in water transport. A tracheid cell wall usually contains the polymer lignin . [ citation needed ]
The phloem , on the other hand, consists of living cells called sieve-tube members . Between the sieve-tube members are sieve plates, which have pores to allow molecules to pass through. Sieve-tube members lack such organs as nuclei or ribosomes , but cells next to them, the companion cells , function to keep the sieve-tube members alive. [ citation needed ]
The most abundant compound in all plants, as in all cellular organisms, is water , which has an important structural role and a vital role in plant metabolism . Transpiration is the main process of water movement within plant tissues. Plants constantly transpire water through their stomata to the atmosphere and replace that water with soil moisture taken up by their roots. When the stomata are closed at night, water pressure can build up in the plant. Excess water is excreted through pores known as hydathodes . [ 24 ] The movement of water out of the leaf stomata sets up transpiration pull or tension in the water column in the xylem vessels or tracheids. The pull is the result of water surface tension within the cell walls of the mesophyll cells, from the surfaces of which evaporation takes place when the stomata are open. Hydrogen bonds exist between water molecules , causing them to line up; as the molecules at the top of the plant evaporate, each pulls the next one up to replace it, which in turn pulls on the next one in line. The draw of water upwards may be entirely passive and can be assisted by the movement of water into the roots via osmosis . Consequently, transpiration requires the plant to expend very little energy on water movement. Transpiration assists the plant in absorbing nutrients from the soil as soluble salts . Transpiration plays an important role in the absorption of nutrients from the soil as soluble salts are transported along with the water from the soil to the leaves. Plants can adjust their transpiration rate to optimize the balance between water loss and nutrient absorption. [ 25 ]
Living root cells passively absorb water. Pressure within the root increases when transpiration demand via osmosis is low and decreases when water demand is high. No water movement towards the shoots and leaves occurs when evapotranspiration is absent. This condition is associated with high temperature, high humidity , darkness, and drought. [ citation needed ]
Xylem is the water-conducting tissue, and the secondary xylem provides the raw material for the forest products industry. [ 26 ] Xylem and phloem tissues each play a part in the conduction processes within plants. Sugars are conveyed throughout the plant in the phloem; water and other nutrients pass through the xylem. Conduction occurs from a source to a sink for each separate nutrient. Sugars are produced in the leaves (a source) by photosynthesis and transported to the growing shoots and roots (sinks) for use in growth, cellular respiration or storage. Minerals are absorbed in the roots (a source) and transported to the shoots to allow cell division and growth. [ 27 ] [ 28 ] [ 29 ]
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Vascularity , in bodybuilding , is the condition of having many highly visible, prominent, and often extensively-ramified superficial veins . [ 1 ] The skin appears "thin"—sometimes virtually transparent —due to an extreme reduction of subcutaneous fat , allowing for maximum muscle definition. [ citation needed ]
Vascularity is enhanced by extremely low body fat (usually below 10%) and low retained water, as well as the muscle engorgement ("pump") and venous distension accentuated by the vigorous flexing and potentially hazardous Valsalva effect which characterize competitive posing. Genetics and androgenic hormones [ 2 ] will affect vascularity, as will ambient temperature. Additionally, although some bodybuilders develop arterial hypertension from performance-enhancing substances and practices, "high" venous pressure—being an order of magnitude lower than that of arteries [ 3 ] — neither causes nor is caused by vascularity. Some bodybuilders use topical vasodilators to increase blood flow to the skin as well. Although historically controversial, [ 4 ] vascularity is a highly-sought-after aesthetic for many male bodybuilders, [ 5 ] but less so for female bodybuilders , [ 4 ] where the target aesthetic is relatively more towards aesthetic symmetry than extreme development. [ citation needed ]
Bodybuilders or athletes sometimes dehydrate themselves a few days before a competition or show to achieve this so-called "ripped," vascular look. Self-dehydration is not recommended by medical professionals, as the negative and sometimes-fatal effects of the resultant water-electrolyte imbalances are well documented. [ 6 ]
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Vasculogenesis is the process of blood vessel formation, occurring by a de novo production of endothelial cells . [ 1 ] It is the first stage of the formation of the vascular network, closely followed by angiogenesis . [ 2 ] [ 3 ]
In the sense distinguished from angiogenesis , vasculogenesis is different in one aspect: whereas angiogenesis is the formation of new blood vessels from pre-existing ones, vasculogenesis is the formation of new blood vessels, in blood islands , when there are no pre-existing ones. [ 4 ] For example, if a monolayer of endothelial cells begins sprouting to form capillaries , angiogenesis is occurring. Vasculogenesis, in contrast, is when endothelial precursor cells ( angioblasts ) migrate and differentiate in response to local cues (such as growth factors and extracellular matrices) to form new blood vessels. These vascular trees are then pruned and extended through angiogenesis.
Vasculogenesis occurs during embryonic development of the circulatory system . Specifically, around blood islands, which first arise in the mesoderm of the yolk sac at 3 weeks of development. [ 5 ]
Vasculogenesis can also arise in the adult organism from circulating endothelial progenitor cells (derivatives of stem cells). These cells are able to contribute, albeit to varying degrees, to neovascularization. Examples of where vasculogenesis can occur in adults are:
This cardiovascular system article is a stub . You can help Wikipedia by expanding it .
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A vasculum or a botanical box is a stiff container used by botanists to keep field samples viable for transportation. The main purpose of the vasculum is to transport plants without crushing them and by maintaining a cool, humid environment .
Vascula are cylinders typically made from tinned and sometimes lacquered iron, though wooden examples are known. The box was carried horizontally on a strap so that plant specimens lie flat and lined with moistened cloth. [ 1 ] Traditionally, British and American vascula were somewhat flat and valise -like with a single room, while continental examples were more cylindrical and often longer, sometimes with two separate compartments. [ 2 ] Access to the interior is through one (sometimes two) large lids in the side, allowing plants to be put in and taken out without bending or distorting them unnecessarily. This is particularly important with wildflowers , which are often fragile.
Some early 20th century specimen are made from sheet aluminium rather than tin, but otherwise follow the 19th century pattern. The exterior is usually left rough, or lacquered green.
The roots of the vasculum are lost in time, but may have evolved from the 17th century tin candle-box of similar construction. Linnaeus called it a vasculum dillenianum , from Latin vasculum – small container and dillenianum , referring to J.J. Dillenius , Linnaeus' friend and colleague at Oxford Botanic Garden . With rise of botany as a scientific field the mid 18th century, the vasculum became an indispensable part of the botanist's equipment. [ 3 ]
Together with the screw-down plant press, the vasculum was popularized in Britain by naturalist William Withering around 1770. [ 4 ] The shortened term "vasculum" appears to have become the common name applied to them around 1830. [ 2 ] Being a hallmark of field botany, vascula were in common use until World War II . With post-war emphasis on systematics rather than alpha taxonomy and new species often collected in far-away places, field botany and the use of vascula went into decline.
Aluminium vascula are still made and in use, though zipper bags and clear plastic folders are today cheaper and more common in use. [ 4 ]
The Vasculum was "An Illustrated Quarterly dealing primarily with the Natural History of Northumberland and Durham and the tracts immediately adjacent," from 1915 to 2015. [ 5 ]
The newsletter of the Society of Herbarium Curators is named "The Vasculum" since 2006. [ 6 ]
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A vase ( Swedish pronunciation: [ˈ̍vɑːsːɛ] ⓘ ) is a heraldic symbol that has been used by the Swedish and Polish-Lithuanian House of Vasa . It has been used as a symbol of the Swedish state even after the extinction of the Vasa lineage, and was reused in 1818 as part of the coat of arms of Sweden . The vase has been used by other families, both noble and common, and is still in use by the government-owned real estate enterprise Vasakronan .
The term originally referred to a bundle of twigs or branches and comes from the Old Swedish vasi and is related to vad (" seine ") . The term has been used in compounds such as risvase (a bundle of straw of twigs used to attract fish near shore) and stormvase ("fascine"). As a coat of arms the term vase has likely been used since the symbol has mostly resembled a fascine. [ 1 ]
The exact meaning of the heraldic symbol of the Vasa family has been debated among both historians and heralds . During the late 16th century, the vase was depicted to look more like a sheaf, while medieval depictions show a sort of bundle, and which possibly could have been a depiction of a type of anchor plate . In modern times, depictions of a vase as a sheaf are considered inaccurate. Due to its historical association with a sheaf, it has sometimes been called vasakärven , "the Vasa sheaf".
The vase was used as a nationalist symbol during the 1930s and 1940s, especially by fascist and nazist groups in Sweden, such as the Swedish National Socialist Workers' Party ( Svensk socialistisk samling ) under Sven Olov Lindholm (1903–1998). [ 2 ] The vase was also used as a party symbol by National Socialist Front which was active about 1984–2008. It was also used by democratic political organizations like Svenska Landsbygdens Kvinnoförbund, today Centerkvinnorna, the women's organization of the Centre Party .
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Vasilis M. Fthenakis is a Greek American chemical engineer , environmental scientist, author and academic. He is an adjunct professor, and founding director of the center for Life Cycle Analysis at Columbia University . [ 1 ] [ 2 ]
Fthenakis is most known for his research on the environmental sustainability of photovoltaic energy technologies and for demonstrating the feasibility of solar energy as a solution to meet US energy demands while addressing climate challenges. [ 3 ] His publications comprise journal articles and books including Electricity from Sunlight: Photovoltaics Systems Integration and Sustainability and Onshore and Offshore Wind Energy: Evolution, Grid Integration and Impact . He has received awards such as a Certificate of Appreciation from the US Department of Energy in 2006, [ 4 ] the Brookhaven National Laboratory 's Certificate of Recognition in 2015, the 2018 IEEE William Cherry Award, [ 5 ] [ 6 ] and the 2022 Karl Böer Solar Energy Medal of Merit from the International Solar Energy Society . [ 7 ]
Fthenakis is an elected Fellow of the American Institute of Chemical Engineers , the International Energy Foundation, and the Institute of Electrical and Electronics Engineers . [ 8 ] Additionally, he has served as Editor-in-Chief of Green Energy and Sustainability , [ 9 ] Section Editor-in-Chief of Energies , [ 10 ] and associate editor for Progress in Energy . [ 11 ]
Fthenakis earned a diploma in chemistry from the University of Athens in 1975 while working as a chemist at ChemiResearch in Greece from 1974 to 1976. He then completed an MS in Chemical Engineering at Columbia University in 1978 and held research roles at Columbia's Catalysis Laboratory and Fossil Energy Laboratory. In 1980, he joined Brookhaven National Laboratory, working as a research engineer and senior scientist across departments focused on sustainable energy and environmental sciences . He received a PhD in fluid dynamics and atmospheric science from New York University in 1991, with a focus on toxic gas release modeling and mitigation using water spray systems. [ 1 ]
Fthenakis continued his academic and research career at Columbia University, serving as an adjunct associate professor of earth and environmental engineering from 1995 to 2000, and has been an adjunct professor since 2006. In 2006, he became a senior research scientist and founded the Center for Life Cycle Analysis (CLCA), where he continues to serve as director. He also co-founded the Global Clean Water Desalination Alliance (GCWDA), where he served on the board of directors, leading efforts to integrate solar energy systems with desalination technologies. [ 12 ] Concurrently, at Brookhaven National Laboratory, he served in various roles from 1980 to 2016 and has been a distinguished scientist emeritus since 2017. [ 13 ] From 2002 to 2016, he led the National Photovoltaic Environmental Research Center and has coordinated international collaborations on life cycle assessment (LCA) under the direction of the US Department of Energy and the International Energy Agency . [ 14 ]
Fthenakis has led collaborations on silane safety and lead-free solder technologies, conducted foundational life-cycle studies on thin-film photovoltaics and PV recycling, and anticipated regulatory trends concerning lead and cadmium , supporting industry adaptation. [ 1 ] [ 3 ] In later years, the scope of his research expanded to topics at the energy-water-environment nexus and he led applied research on solar-powered water desalination with applications in the United States [ 15 ] and Chile . [ 16 ]
In 2004, Fthenakis began studying life cycle analysis (LCA) to address what he identified as an unbalanced portrayal of photovoltaics' environmental impacts and started an international collaboration to update LCA studies on photovoltaics. He established an ad-hoc committee and organized scoping meetings with researchers from institutions such as the University of Utrecht , the Energy Research Center of the Netherlands , Chalmers University , the University of Stuttgart , Siena University , and Ambient Italia to assess the LCA needs of the photovoltaic industry. [ 17 ] [ 18 ]
Through related research, Fthenakis has developed and advocated for a proactive, long-term environmental strategy for photovoltaics, including recycling processes for end-of-life photovoltaic modules. In 1999, he organized a workshop to promote lead-free solder technology. [ 19 ] From 2002 to 2005, he established a laboratory focused on recycling spent photovoltaic modules and manufacturing scrap, employing hydrometallurgical separation technologies and resulting in a patented method for separating copper , cadmium , and tellurium , with applications in cadmium telluride (CdTe) and copper indium gallium selenide (CIGS) technologies. [ 20 ] He also conducted studies on optimizing the collection of end-of-life photovoltaics to reduce recycling costs. [ 21 ] Recognizing the environmental concerns surrounding the growth of the CdTe and CIGS markets, he designed experiments simulating fire effects on photovoltaics using techniques such as NSLS-x ray diffraction analysis. [ 22 ] In 2005, he led a European Union workshop, organized with the Joint Research Center and the German Ministry of the Environment , facilitating a US company's establishment of a manufacturing facility in Germany . In 2007, he launched and led a five-year International Energy Agency (IEA) Photovoltaic Environmental Health and Safety Task (Task 12), serving as the US Operating Agent until 2012. [ 23 ]
Over the years, Fthenakis’ research has been highlighted by news outlets, including The New York Times , [ 24 ] Science News , [ 25 ] Environmental Science & Technology , IEEE Spectrum , [ 26 ] Scientific American , [ 27 ] Spiegel , and NRC Handelsblad .
Fthenakis' research on photovoltaics and the environment has led to approximately 300 journal and conference papers, contributing to over 450 publications on energy and environmental topics. As of November 15, 2024, his publications have been cited 18,464 times and his h-index is 66. [ 3 ] In 2007, he co-authored the Grand Plan for Solar Energy with Ken Zweibel and James Mason, a study demonstrating the feasibility of solar energy to meet most of the US electricity needs; this was the prelude of the detailed SunShot Solar Vision studies. [ 28 ] Earlier, in 1993, he published his first book, Prevention and Control of Accidental Releases of Hazardous Gases , which was used by the chemical and oil refinery industries as a primer on the prevention of industrial disasters. His publications have also focused on electricity generation through renewable energy sources such as wind and solar. [ 29 ]
Fthenakis is the son of Menelaos Fthenakis and Antonia Korkidis-Fthenakis, who died in the sinking of the SS Heraklion in Greece when he was 14 years old. [ 30 ] He is married to Christina Georgakopoulos and has two children and two grandchildren. [ 31 ]
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Vasilis Gregoriou (born 1965, Trikala, Greece) [ 1 ] is a researcher, inventor, technology entrepreneur and former Director and Chairman of the Board of Directors at National Hellenic Research Foundation (NHRF) in Athens, Greece. [ 2 ] [ 3 ] [ 4 ] During his career, he has achieved international recognition by serving in research and administrative positions both in Greece and the US. [ 5 ] [ 6 ] [ 7 ] His studies in Greece began at the University of Patras (BSc. Chemistry) while his studies in the United States took place at Duke University where he received a PhD degree in Physical Chemistry . [ 2 ] [ 8 ] He was also a National Research Service Award recipient at Princeton University . [ 9 ]
His academic teaching experience spans in both undergraduate level at the University of Massachusetts and postgraduate level at the University of Connecticut and the University of Patras. [ 10 ] [ 11 ] His published work as co-author includes three books, six chapters in other authors' books, 92 scientific papers and 146 research presentations. [ 6 ] [ 12 ] Vasilis Gregoriou is also co-inventor of 15 patents. [ 2 ] His research interests include flexible photovoltaic cells based on organic semiconductors , optically active materials based on conjugated oligomers, and nanostructured polymer materials. [ 7 ] [ 4 ] He has served as President of Society for Applied Spectroscopy (SAS) in 2001 and now he participates as National Representative of Greece in the Committee of the European Research Council (ERC) for the Horizon 2020 program, the Mari Sklodowska-Curie actions and the Future and Emerging Technologies (FET). [ 13 ] Vasilis Gregoriou has been the Director of the National Hellenic Research Foundation since 2013. [ 14 ] As a technology entrepreneur, Vasilis Gregoriou is the co-founder and the CEO of Advent Technologies which is based in Cambridge , Massachusetts. [ 15 ] Advent Technologies develops advanced technology and devices in the field of energy and defense and it has also developed research collaborations with Northeastern University in Boston , US, Patras University in Greece, and the Institute of Chemical Engineering Sciences (ICE-HT/FORTH). [ 16 ] [ 15 ] [ 17 ]
the Mari Sklodowska-Curie actions, and the Future and Emerging Technologies (FET) Brussels, Belgium (2014–present) [ 18 ]
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Vasily Leonidovich Omelianski ( Vasilij Leonidovič Omeljanskij , Russian: Василий Леонидович Омелянский ; 10 March 1867 – 21 April 1928) was a Russian microbiologist and author of the first original Russian text book on microbiology. He was the only student of Sergei Winogradsky and succeeded him as head of the department of General Microbiology at the Institute of Experimental Medicine in Saint Petersburg .
Omelianski was the youngest son of a college teacher in Zhytomyr . In 1885 or 1886, Omelianski enrolled in the natural history division of the physico-mathematical faculty of the University of Saint Petersburg . [ 1 ] [ 2 ] During his studies he visited the lectures of D. I. Mendeleev and N. A. Menshutkin . [ 2 ]
After finishing his studies with distinction in 1889 or 1890, he worked in the chemical laboratory of Menshutkin for further two years and published the first time. In 1891, financial difficulties forced Omelianski to work as laboratory chemist in a metallurgical factory in Southern Russia. [ 1 ] However, two years later he became the assistant of S. N. Winogradsky , who hired him on recommendation of Menshutkin, at the new-founded Imperial Institute of Experimental Medicine . Omelianski supported Winogradskys work on nitrification . Later on he studied the fermentation of cellulose and did research on nitrogen fixation on his own. [ 2 ]
In 1909, he published the textbook "Principles of Microbiology" [ 1 ] ( Основы микробиологии ) which was the first original Russian textbook on microbiology and remained a standard work at Soviet universities till the 1950s. Omelianski had conceived this text from his lectures he held on a women's college since 1906 or 1909. [ 2 ] In 1922, he published his second textbook "Practical Manual of Microbiology" [ 1 ] ( Практическое руководство по микробиологии ) in which he spread the methodology of Winogradsky (using enrichment cultures ) and the so-called "Delft school of microbiology" [ 3 ] (founded by M. Beijerinck ) in Russia. [ 2 ] Since 1912 till his death he led the department of General Microbiology at the Institute of Experimental Medicine succeeding Winogradsky. As head of the department he edited the "Archive of Biological Sciences" ( Архив биологических наук ), the first biology journal publishing in Russian. In 1924, Omelianski became editor of the popular journal “Progress of biological chemistry” ( Успехи биологической химии ). The last textbook he could finish in 1927 was “Short course in general and soil microbiology” ( Краткий курс общей и почвенной микробиологии ). [ 2 ]
In 1916, Omelianski became a corresponding member of the Russian Academy of Sciences and he was appointed to Doctor botanicus h. c. without examination in 1917. In 1923, he became of full member of the Russian Academy of Sciences. [ 2 ] In 1926, he affiliated with the Society of American Bacteriologists [ 1 ] and the Lombardic Academical Society.
Omelianski was married and had a daughter, Maria Vasilevna Stepanova (1901-1946, an ethnographer ). [ 2 ] During World War I , the Russian Revolution and Russian Civil War , Omelianski was able to stay in Saint Petersburg, while Winogradsky (as a rich landowner) had to escape. Possibly, he was saved by his poor bourgeois ancestry, his interest in the starving poor, his popular commitment by publishing Russian text books and journals and lecturing in a women’s college or thanks to the Bolsheviks scientific progress friendly stance. In springtime 1927, Omelianski travelled to the Pasteur Institute in Paris to visit his mentor Winogradsky. There he suffered a first heart attack . Omelianski had a second heart attack in December 1927 but could recover. [ 2 ] During a vacation in Gagra ( Abkhazia ) he died on April 21, 1928. [ 1 ] Omelianski was also a gifted chess player who entered into competitions as a student, a reportedly gifted portraitist and wrote several short-stories, four of which have since been stored in the archive at the Russian Academy of Sciences in St. Petersburg. [ 2 ]
Omelianski published only once in English on “aroma-producing microorganisms” in the American Journal of Bacteriology in 1923. [ 4 ] However, today his international reputation is connected to microbial methanogenesis in syntrophic co-cultures . [ 5 ] This is based on his French publication of 1916 on "methane fermentation of ethanol". [ 6 ] Following this research, microbiologist Horace Barker isolated an ethanol -degrading microbe called Methanobacterium omelianskii . [ 7 ] Barker had used the methodological approach of the "Delft school of microbiology" [ 3 ] developed by Barkers mentors Albert Kluyver and Cornelis van Niel . In 1967, the renamed Methanobacterium omelianskii was specified as a co-culture of the ethanol-oxidizing S -organism and a methanogen , which uses hydrogen produced by its bacterial partner to reduce carbon dioxide to methane . [ 8 ] Certainly, Omelianski was one of the founding fathers of methanogenesis research and the first scientist investigating methanogenic fermentation of cellulose and ethanol systematically. He even discovered hydrogen as a product of cellulose fermentation around 1900 but, of course, did not discover the concept of syntrophic electron transfer. [ 2 ]
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Vasishtha Siddhanta is one of the earliest astronomical systems in use in India, which is summarized in Varahamihira 's Pancha-siddhantika (6th century). It is attributed to sage Vasishtha and claims a date of composition of 1,299,101 BCE. [ 1 ] The original text probably dated to the 4th century, but it has been lost and our knowledge of it is restricted to Varahamira's account. Alberuni ascribes the work to Vishnuchandra .
There is a modern work bearing the title Vasishtha Siddhantika . [ 2 ]
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Vaterite is a mineral, a polymorph of calcium carbonate ( Ca C O 3 ). It was named after the German mineralogist Heinrich Vater . It is also known as mu- calcium carbonate (μ-CaCO 3 ). Vaterite belongs to the hexagonal crystal system , whereas calcite is trigonal and aragonite is orthorhombic .
Vaterite, like aragonite, is a metastable phase of calcium carbonate at ambient conditions at the surface of the Earth. As it is less stable than either calcite, the most stable polymorph, [ 5 ] or aragonite, vaterite has a higher solubility than either of these phases. Therefore, once vaterite is exposed to water , it converts to calcite (at low temperature) or aragonite (at high temperature: ~60 °C). At 37 °C for example a solution-mediated transition from vaterite to calcite occurs, where the vaterite dissolves and subsequently precipitates as calcite assisted by an Ostwald ripening process. [ 6 ]
However, vaterite does occur naturally in mineral springs , organic tissue, gallstones , urinary calculi and plants. In those circumstances, some impurities ( metal ions or organic matter) may stabilize the vaterite and prevent its transformation into calcite or aragonite. Vaterite is usually colorless.
Vaterite can be produced as the first mineral deposits repairing natural or experimentally-induced shell damage in some aragonite-shelled mollusks (e.g. gastropods). Subsequent shell deposition occurs as aragonite. In 2018, vaterite was identified as a constituent of a deposit formed on the leaves of Saxifraga at Cambridge University Botanic Garden . [ 7 ] [ 8 ]
Vaterite is tapped as an effective intermediate form of cement whose production consumes carbon dioxide rather than emitting it. Research into the vaterite production process was inspired by its discovery in the hard skeletons of coral . [ 9 ] [ unreliable source? ]
Vaterite has a JCPDS number of 13-192.
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In mathematics and analytic number theory , Vaughan's identity is an identity found by R. C. Vaughan ( 1977 ) that can be used to simplify Vinogradov 's work on trigonometric sums . It can be used to estimate summatory functions of the form
where f is some arithmetic function of the natural integers n , whose values in applications are often roots of unity, and Λ is the von Mangoldt function .
The motivation for Vaughan's construction of his identity is briefly discussed at the beginning of Chapter 24 in Davenport. For now, we will skip over most of the technical details motivating the identity and its usage in applications, and instead focus on the setup of its construction by parts. Following from the reference, we construct four distinct sums based on the expansion of the logarithmic derivative of the Riemann zeta function in terms of functions which are partial Dirichlet series respectively truncated at the upper bounds of U {\displaystyle U} and V {\displaystyle V} , respectively. More precisely, we define F ( s ) = ∑ m ≤ U Λ ( m ) m − s {\displaystyle F(s)=\sum _{m\leq U}\Lambda (m)m^{-s}} and G ( s ) = ∑ d ≤ V μ ( d ) d − s {\displaystyle G(s)=\sum _{d\leq V}\mu (d)d^{-s}} , which leads us to the exact identity that
This last expansion implies that we can write
where the component functions are defined to be
We then define the corresponding summatory functions for 1 ≤ i ≤ 4 {\displaystyle 1\leq i\leq 4} to be
so that we can write
Finally, at the conclusion of a multi-page argument of technical and at times delicate estimations of these sums, [ 1 ] we obtain the following form of Vaughan's identity when we assume that | f ( n ) | ≤ 1 , ∀ n {\displaystyle |f(n)|\leq 1,\ \forall n} , U , V ≥ 2 {\displaystyle U,V\geq 2} , and U V ≤ N {\displaystyle UV\leq N} :
It is remarked that in some instances sharper estimates can be obtained from Vaughan's identity by treating the component sum S 2 {\displaystyle S_{2}} more carefully by expanding it in the form of
The optimality of the upper bound obtained by applying Vaughan's identity appears to be application-dependent with respect to the best functions U = f U ( N ) {\displaystyle U=f_{U}(N)} and V = f V ( N ) {\displaystyle V=f_{V}(N)} we can choose to input into equation (V1). See the applications cited in the next section for specific examples that arise in the different contexts respectively considered by multiple authors.
In particular, we obtain an asymptotic upper bound for these sums (typically evaluated at irrational α ∈ R ∖ Q {\displaystyle \alpha \in \mathbb {R} \setminus \mathbb {Q} } ) whose rational approximations satisfy
of the form
The argument for this estimate follows from Vaughan's identity by proving by a somewhat intricate argument that
and then deducing the first formula above in the non-trivial cases when q ≤ N {\displaystyle q\leq N} and with U = V = N 2 / 5 {\displaystyle U=V=N^{2/5}} .
Vaughan's identity was generalized by Heath-Brown (1982) .
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In astronomy , the Vaughan–Preston gap is an observed absence of F-, G- and K-type stars with intermediate levels of magnetic activity . In 1980, Vaughan and Preston noted there were two populations of stars of these classifications, with either high or low levels of activity, separated by an apparent gap. [ 1 ] There remains no consensus on the cause of the gap. [ 2 ]
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The Vaught conjecture is a conjecture in the mathematical field of model theory originally proposed by Robert Lawson Vaught in 1961. It states that the number of countable models of a first-order complete theory in a countable language is finite or ℵ 0 or 2 ℵ 0 . Morley showed that the number of countable models is finite or ℵ 0 or ℵ 1 or 2 ℵ 0 , which solves the conjecture except for the case of ℵ 1 models when the continuum hypothesis fails. For this remaining case, Robin Knight ( 2002 , 2007 ) has announced a counterexample to the Vaught conjecture and the topological Vaught conjecture . As of 2021, the counterexample has not been verified.
Let T {\displaystyle T} be a first-order, countable, complete theory with infinite models. Let I ( T , α ) {\displaystyle I(T,\alpha )} denote the number of models of T of cardinality α {\displaystyle \alpha } up to isomorphism—the spectrum of the theory T {\displaystyle T} . Morley proved that if I ( T , ℵ 0 ) is infinite then it must be ℵ 0 or ℵ 1 or the cardinality of the continuum . The Vaught conjecture is the statement that it is not possible for ℵ 0 < I ( T , ℵ 0 ) < 2 ℵ 0 {\displaystyle \aleph _{0}<I(T,\aleph _{0})<2^{\aleph _{0}}} . The conjecture is a trivial consequence of the continuum hypothesis ; so this axiom is often excluded in work on the conjecture. Alternatively, there is a sharper form of the conjecture that states that any countable complete T with uncountably many countable models will have a perfect set of uncountable models (as pointed out by John Steel , in "On Vaught's conjecture". Cabal Seminar 76–77 (Proc. Caltech-UCLA Logic Sem., 1976–77), pp. 193–208, Lecture Notes in Math., 689, Springer, Berlin, 1978, this form of the Vaught conjecture is equiprovable with the original).
The original formulation by Vaught was not stated as a conjecture, but as a problem: Can it be proved, without the use of the continuum hypothesis, that there exists a complete theory having exactly ℵ 1 non-isomorphic denumerable models? By the result by Morley mentioned at the beginning, a positive solution to the conjecture essentially corresponds to a negative answer to Vaught's problem as originally stated.
Vaught proved that the number of countable models of a complete theory cannot be 2. It can be any finite number other than 2, for example:
The idea of the proof of Vaught's theorem is as follows. If there are at most countably many countable models, then there is a smallest one: the atomic model , and a largest one, the saturated model , which are different if there is more than one model. If they are different, the saturated model must realize some n -type omitted by the atomic model. Then one can show that an atomic model of the theory of structures realizing this n -type (in a language expanded by finitely many constants) is a third model, not isomorphic to either the atomic or the saturated model. In the example above with 3 models, the atomic model is the one where the sequence is unbounded, the saturated model is the one where the sequence converges, and an example of a type not realized by the atomic model is an element greater than all elements of the sequence.
The topological Vaught conjecture is the statement that whenever a Polish group acts continuously on a Polish space , there are either countably many orbits or continuum many orbits. The topological Vaught conjecture is more general than the original Vaught conjecture: Given a countable language we can form the space of all structures on the natural numbers for that language. If we equip this with the topology generated by first-order formulas, then it is known from A. Gregorczyk , A. Mostowski , C. Ryll-Nardzewski , "Definability of sets of models of axiomatic theories" ( Bulletin of the Polish Academy of Sciences (series Mathematics, Astronomy, Physics) , vol. 9 (1961), pp. 163–7) that the resulting space is Polish. There is a continuous action of the infinite symmetric group (the collection of all permutations of the natural numbers with the topology of point-wise convergence) that gives rise to the equivalence relation of isomorphism. Given a complete first-order theory T , the set of structures satisfying T is a minimal, closed invariant set, and hence Polish in its own right.
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Vault 7 is a series of documents that WikiLeaks began to publish on 7 March 2017, detailing the activities and capabilities of the United States Central Intelligence Agency (CIA) to perform electronic surveillance and cyber warfare . The files, dating from 2013 to 2016, include details on the agency's software capabilities, such as the ability to compromise cars , smart TVs , [ 1 ] web browsers including Google Chrome , Microsoft Edge , Mozilla Firefox , and Opera , [ 2 ] [ 3 ] the operating systems of most smartphones including Apple 's iOS and Google 's Android , and computer operating systems including Microsoft Windows , macOS , and Linux . [ 4 ] [ 5 ] A CIA internal audit identified 91 malware tools out of more than 500 tools in use in 2016 being compromised by the release. [ 6 ] The tools were developed by the Operations Support Branch of the CIA. [ 7 ]
The Vault 7 release led the CIA to redefine WikiLeaks as a "non-state hostile intelligence service." [ 8 ] In July 2022, former CIA software engineer Joshua Schulte was convicted of leaking the documents to WikiLeaks, [ 9 ] and in February 2024 sentenced to 40 years' imprisonment, on espionage counts and separately to 80 months for child pornography counts. [ 10 ]
In February 2017, WikiLeaks began teasing the release of "Vault 7" with a series of cryptic messages on Twitter, according to media reports. [ 11 ] Later on in February, WikiLeaks released classified documents describing how the CIA monitored the 2012 French presidential election . [ 12 ] The press release for the leak stated that it was published "as context for its forthcoming CIA Vault 7 series." [ 13 ]
In March 2017, US intelligence and law enforcement officials said to the international wire agency Reuters that they had been aware of the CIA security breach which led to Vault 7 since late 2016. Two officials said they were focusing on "contractors" as the possible source of the leaks. [ 14 ]
In 2017, federal law enforcement identified CIA software engineer Joshua Adam Schulte as a suspected source of Vault 7. [ 15 ] [ 16 ] Schulte plead not guilty and was convicted in July 2022 of leaking the documents to WikiLeaks.
On 13 April 2017, CIA director Mike Pompeo declared WikiLeaks to be a "hostile intelligence service." [ 17 ] In September 2021, Yahoo! News reported that in 2017 in the wake of the Vault 7 leaks, the CIA considered kidnapping or assassinating Julian Assange , the founder of WikiLeaks. The CIA also considered spying on associates of WikiLeaks, sowing discord among its members, and stealing their electronic devices. After many months of deliberation, all proposed plans had been scrapped due to a combination of legal and moral objections. Per the 2021 Yahoo News article, a former Trump national security official stated, "We should never act out of a desire for revenge". [ 18 ]
The Vault 7 release led the CIA to redefine WikiLeaks as a "non-state hostile intelligence service." [ 8 ] In July 2022, former CIA software engineer Joshua Schulte was convicted of leaking the documents to WikiLeaks, [ 9 ] and in February 2024 sentenced to 40 years' imprisonment. [ 10 ]
The first batch of documents named "Year Zero" was published by WikiLeaks on 7 March 2017, consisting of 7,818 web pages with 943 attachments, purportedly from the Center for Cyber Intelligence, [ 19 ] which contained more pages than former NSA contractor and leaker, Edward Snowden 's NSA release at the time. [ 20 ] WikiLeaks had released Year Zero online in a locked archive earlier that week, and revealing the passphrase on the 7th. The passphrase referred to a President Kennedy quote that he wanted “to splinter the CIA in a thousand pieces and scatter it to the winds”. [ 21 ]
WikiLeaks did not name the source, but said that the files had "circulated among former U.S. government hackers and contractors in an unauthorized manner, one of whom has provided WikiLeaks with portions of the archive." [ 1 ] According to WikiLeaks, the source "wishes to initiate a public debate about the security, creation, use, proliferation and democratic control of cyberweapons" since these tools raise questions that "urgently need to be debated in public, including whether the C.I.A.'s hacking capabilities exceed its mandated powers and the problem of public oversight of the agency." [ 1 ]
WikiLeaks attempted to redact names and other identifying information from the documents before their release, [ 1 ] but faced criticism for leaving some key details unredacted. [ 22 ] WikiLeaks also attempted to allow for connections between people to be drawn via unique identifiers generated by WikiLeaks. [ 23 ] [ 24 ] It also said that it would postpone releasing the source code for the cyber weapons, which is reportedly several hundred million lines long, "until a consensus emerges on the technical and political nature of the C.I.A.'s program and how such 'weapons' should be analyzed, disarmed and published." [ 1 ] WikiLeaks founder Julian Assange claimed this was only part of a larger series. [ 20 ]
The CIA released a statement saying, "The American public should be deeply troubled by any WikiLeaks disclosure designed to damage the Intelligence Community's ability to protect America against terrorists or other adversaries. Such disclosures not only jeopardize US personnel and operations, but also equip our adversaries with tools and information to do us harm." [ 25 ]
In a statement issued on 19 March 2017, Assange said the technology companies who had been contacted had not agreed to, disagreed with, or questioned what he termed as WikiLeaks' standard industry disclosure plan. The standard disclosure time for a vulnerability is 90 days after the company responsible for patching the software is given full details of the flaw. [ 26 ] According to WikiLeaks, only Mozilla had been provided with information on the vulnerabilities, while "Google and some other companies" only confirmed receiving the initial notification. WikiLeaks stated: "Most of these lagging companies have conflicts of interest due to their classified work with US government agencies. In practice such associations limit industry staff with US security clearances from fixing holes based on leaked information from the CIA. Should such companies choose to not secure their users against CIA or NSA attacks users may prefer organizations such as Mozilla or European companies that prioritize their users over government contracts". [ 27 ] [ 28 ]
On 23 March 2017 WikiLeaks published the second release of Vault 7 material, entitled "Dark Matter". The publication included documentation for several CIA efforts to hack Apple's iPhones and Macs. [ 29 ] [ 30 ] [ 31 ] These included the Sonic Screwdriver malware that could use the Thunderbolt interface to bypass Apple's password firmware protection. [ 32 ]
On 31 March 2017, WikiLeaks published the third part, "Marble". It contained 676 source code files for the CIA's Marble Framework. It is used to obfuscate, or scramble, malware code in an attempt to make it so that anti-virus firms or investigators cannot understand the code or attribute its source. According to WikiLeaks, the code also included a de-obfuscator to reverse the obfuscation effects. [ 33 ] [ 34 ]
On 7 April 2017, WikiLeaks published the fourth set, "Grasshopper". The publication contains 27 documents from the CIA's Grasshopper framework, which is used by the CIA to build customized and persistent malware payloads for the Microsoft Windows operating systems. Grasshopper focused on Personal Security Product (PSP) avoidance. PSPs are antivirus software such as MS Security Essentials , Symantec Endpoint or Kaspersky IS . [ 34 ] [ 35 ]
On 14 April 2017, WikiLeaks published the fifth part, "HIVE". Based on the CIA top-secret virus program created by its "Embedded Development Branch" (EDB). The six documents published by WikiLeaks are related to the HIVE multi-platform CIA malware suite. A CIA back-end infrastructure with a public-facing HTTPS interface used by CIA to transfer information from target desktop computers and smartphones to the CIA, and open those devices to receive further commands from CIA operators to execute specific tasks, all the while hiding its presence behind unsuspicious-looking public domains through a masking interface known as "Switchblade" (also known as Listening Post (LP) and Command and Control (C2)). [ 36 ]
On 21 April 2017, WikiLeaks published the sixth part, "Weeping Angel" (named for a monster in the TV show Doctor Who [ 37 ] [ 38 ] ), a hacking tool co-developed by the CIA and MI5 used to exploit a series of early smart TVs for the purpose of covert intelligence gathering. Once installed in suitable televisions with a USB stick, the hacking tool enables those televisions' built-in microphones and possibly video cameras to record their surroundings, while the televisions falsely appear to be turned off. The recorded data is then either stored locally into the television's memory or sent over the internet to the CIA. Allegedly both the CIA and MI5 agencies collaborated to develop that malware in Joint Development Workshops. Security expert Sarah Zatko said about the data "nothing in this suggests it would be used for mass surveillance," and Consumer Reports said that only some of the earliest smart TVs with built-in microphones and cameras were affected. [ 39 ] [ 40 ] [ 41 ]
On 28 April 2017, WikiLeaks published the seventh part, "Scribbles". The leak includes documentation and source code of a tool intended to track documents leaked to whistleblowers and journalists by embedding web beacon tags into classified documents to trace who leaked them. [ 42 ] The tool affects Microsoft Office documents, specifically "Microsoft Office 2013 (on Windows 8.1 x64), documents from Office versions 97-2016 (Office 95 documents will not work) and documents that are not locked, encrypted, or password-protected". When a CIA watermarked document is opened, an invisible image within the document that is hosted on the agency's server is loaded, generating a HTTP request . The request is then logged on the server, giving the intelligence agency information about who is opening it and where it is being opened. However, if a watermarked document is opened in an alternative word processor the image may be visible to the viewer. The documentation also states that if the document is viewed offline or in protected view, the watermarked image will not be able to contact its home server. This is overridden only when a user enables editing. [ 43 ]
On 5 May 2017, WikiLeaks published the eighth part, "Archimedes". According to U.S. SANS Institute instructor Jake Williams, who analyzed the published documents, Archimedes is a virus previously codenamed "Fulcrum". According to cyber security expert and ENISA member Pierluigi Paganini, the CIA operators use Archimedes to redirect local area network (LAN) web browser sessions from a targeted computer through a computer controlled by the CIA before the sessions are routed to the users. This type of attack is known as man-in-the-middle (MitM). With their publication WikiLeaks included a number of hashes that they claim can be used to potentially identify the Archimedes virus and guard against it in the future. Paganini stated that potential targeted computers can search for those hashes on their systems to check if their systems had been attacked by the CIA. [ 44 ]
On 12 May 2017, WikiLeaks published part nine, "AfterMidnight" and "Assassin". AfterMidnight is a piece of malware installed on a target personal computer and disguises as a DLL file, which is executed while the user's computer reboots. It then triggers a connection to the CIA's Command and Control (C2) computer, from which it downloads various modules to run. As for Assassin, it is very similar to its AfterMidnight counterpart, but deceptively runs inside a Windows service process. CIA operators reportedly use Assassin as a C2 to execute a series of tasks, collect, and then periodically send user data to the CIA Listening Post(s) (LP). Similar to backdoor Trojan behavior. Both AfterMidnight and Assassin run on Windows operating system, are persistent, and periodically beacon to their configured LP to either request tasks or send private information to the CIA, as well as automatically uninstall themselves on a set date and time. [ 45 ]
On 19 May 2017, WikiLeaks published the tenth part, "Athena". The published user guide, demo, and related documents were created between September 2015 and February 2016. They are about a malware allegedly developed for the CIA in August 2015, about a month after Microsoft released Windows 10 with their firm statements about how difficult it was to compromise. Both the primary "Athena" malware and its secondary malware named "Hera" are similar in theory to Grasshopper and AfterMidnight malware but with some significant differences. One of those differences is that Athena and Hera were developed by the CIA with a New Hampshire private corporation called Siege Technologies. During a Bloomberg 2014 interview the founder of Siege Technologies confirmed and justified their development of such malware. Athena malware completely hijacks Windows' Remote Access services , while Hera hijacks Windows Dnscache service . Both Athena and Hera also affect all then current versions of Windows including Windows Server 2012 and Windows 10. Another difference is in the types of encryption used between the infected computers and the CIA Listening Posts (LP). As for the similarities, they exploit persistent DLL files to create a backdoor to communicate with CIA's LP, steal private data, then send it to CIA servers, or delete private data on the target computer, as well as Command and Control (C2) for CIA operatives to send additional malicious software to further run specific tasks on the attacked computer. All of the above designed to deceive computer security software. Beside the published detailed documents, WikiLeaks did not provide any evidence suggesting the CIA used Athena or not. [ 46 ]
On 1 June 2017, WikiLeaks published part 11, "Pandemic". This tool is a persistent implant affecting Windows machines with shared folders. It functions as a file system filter driver on an infected computer, and listens for Server Message Block traffic while detecting download attempts from other computers on a local network. "Pandemic" will answer a download request on behalf of the infected computer. However, it will replace the legitimate file with malware. In order to obfuscate its activities, "Pandemic" only modifies or replaces the legitimate file in transit, leaving the original on the server unchanged. The implant allows 20 files to be modified at a time, with a maximum individual file size of 800MB. While not stated in the leaked documentation, it is possible that newly infected computers could themselves become "Pandemic" file servers, allowing the implant to reach new targets on a local network. [ 47 ]
On 15 June 2017, WikiLeaks published part 12, entitled "Cherry Blossom". Cherry Blossom used a command and control server called Cherry Tree and custom router firmware called FlyTrap to monitor internet activity of targets, scan for “email addresses, chat usernames, MAC addresses and VoIP numbers" and redirect traffic. [ 48 ]
On 22 June 2017, WikiLeaks published part 13, the manuals for "Brutal Kangaroo". Brutal Kangaroo was a project focused on CIA malware designed to compromise air-gapped computer networks with infected USB drives. Brutal Kangaroo included the tools Drifting Deadline, the main tool, Shattered Assurance, a server that automates thumb drive infection, Shadow, a tool to coordinate compromised machines, and Broken Promise, a tool for exfiltrating data from the air-gapped networks. [ 49 ]
On 28 June 2017, WikiLeaks published part 14, the manual for the project entitled "Elsa". Elsa was a tool used for tracking Windows devices on nearby WiFi networks. [ 50 ]
On 29 June 2017, WikiLeaks published part 15, the manual for project "OutlawCountry". OutlawCountry was a kernel module for Linux 2.6 that let CIA agents spy on Linux servers and redirect outgoing traffic from a Linux computer to a chosen site. [ 51 ]
On 6 July 2017, WikiLeaks published part 16, the manual for project "BothanSpy". BothanSpy was a CIA hacking tool made to steal SSH credentials from Windows computers. [ 52 ]
On 13 July 2017, WikiLeaks published part 17, the manual for project "Highrise". The Highrise hacking tool, also known as Tidecheck, was used to intercept and redirect SMS messages to Android phones using versions 4.0 through 4.3. Highrise could also be used as an encrypted communications channel between CIA agents and supervisors. [ 53 ]
On 19 July 2017, WikiLeaks published part 18, documents from Raytheon Blackbird Technologies for the "UMBRAGE Component Library" (UCL) project reports on malware and their attack vectors . According to WikiLeaks, it analysed malware attacks in the wild and gave "recommendations to the CIA development teams for further investigation and PoC development for their own malware projects." It mostly contained Proof-of-Concept ideas partly based on public documents. [ 54 ]
On 27 July 2017, WikiLeaks published part 19, manuals for project "Imperial". Imperial included three tools: Achilles, Aeris and SeaPea. Achilles turned MacOS DMG install files into trojan malware. Aeris was a malware implant for POSIX systems, and SeaPea was an OS X rootkit . [ 55 ]
On 3 August 2017, WikiLeaks published part 20, manuals for project "Dumbo". Dumbo was a tool that the Agency used to disable webcams, microphones, and other surveillance tools over WiFi and bluetooth to allow field agents to perform their missions. [ 56 ]
On 10 August 2017, WikiLeaks published part 21, the manual for project CouchPotato. CouchPotato was a tool for intercepting and saving remote video streams, which let the CIA tap into other people's surveillance systems. [ 57 ]
On 24 August 2017, WikiLeaks published part 22, the "ExpressLane" project. These documents highlighted one of the cyber operations the CIA conducts against other services it liaises with, including the National Security Agency (NSA), the Department of Homeland Security (DHS) and the Federal Bureau of Investigation (FBI).
ExpressLane, a covert information collection tool, was used by the CIA to exfiltrate the biometric data collection systems of services it liaises with. ExpressLane was installed and run under the cover of upgrading the biometric software of liaison services by the CIA's Office of Technical Services (OTS) agents without their knowledge. [ 58 ] [ unreliable source ]
On 31 August 2017, WikiLeaks published part 23, the manual for the project Angelfire. Angelfire was a malware framework made to infect computers running Windows XP and Windows 7, made of five parts. Solartime was the malware that modified the boot sector to load Wolfcreek, which was a self-loading driver that loaded other drivers. Keystone was responsible for loading other malware. BadMFS was a covert file system that hid the malware, and Windows Transitory File System was a newer alternative to BadMFS. The manual included a long list of problems with the tools. [ 59 ]
Protego, part 24 of the Vault 7 documents, was published on 7 September 2017. According to WikiLeaks, Protego "is a PIC -based missile control system that was developed by Raytheon." [ 60 ] [ unreliable source ]
On 9 November 2017, WikiLeaks began publishing Vault 8, which it described as "source code and analysis for CIA software projects including those described in the Vault7 series." The stated intention of the Vault 8 publication was to "enable investigative journalists, forensic experts and the general public to better identify and understand covert CIA infrastructure components." [ 61 ] The only Vault 8 release has been the source code and development logs for Hive, a covert communications platform for CIA malware. WikiLeaks published the Hive documentation as part of Vault 7 on 14 April 2017.
In October 2021, a new backdoor based on the Hive source code was discovered being used "to collect sensitive information and provide a foothold for subsequent intrusions." Researchers called it xdr33 and released a report on it in January 2022. [ 62 ] [ 63 ] [ 64 ] The malware targets an unspecified F5 appliance and allowed hackers to upload and download files. [ 65 ] It also allowed network traffic spying and execute commands on the appliance. [ 64 ] [ 66 ]
WikiLeaks said that the documents came from "an isolated, high-security network situated inside the CIA's Center for Cyber Intelligence (CCI) in Langley , Virginia." [ 67 ] The documents allowed WikiLeaks to partially determine the structure and organization of the CCI. The CCI reportedly has a unit devoted to compromising Apple products. [ 68 ]
The cybersecurity firm Symantec analyzed Vault 7 documents and found some of the described software closely matched cyberattacks by "Longhorn," which it had monitored since 2014. Symantec had previously suspected that "Longhorn" was government-sponsored and had tracked its usage against 40 targets in 16 countries. [ 69 ] [ 70 ]
The first portion of the documents made public on 7 March 2017, Vault 7 "Year Zero", revealed that a top secret CIA unit used the German city of Frankfurt as the starting point for hacking attacks on Europe , China and the Middle East . According to the documents, the U.S. government uses its Consulate General Office in Frankfurt as a hacker base for cyber operations . WikiLeaks documents reveal the Frankfurt hackers, part of the Center for Cyber Intelligence Europe (CCIE), were given cover identities and diplomatic passports to obfuscate customs officers to gain entry to Germany. [ 68 ] [ 71 ]
The chief Public Prosecutor General of the Federal Court of Justice in Karlsruhe Peter Frank announced on 8 March 2017 that the government was conducting a preliminary investigation to see if it will launch a major probe into the activities being conducted out of the consulate and also more broadly whether people in Germany were being attacked by the CIA. [ 72 ] Germany's foreign minister Sigmar Gabriel from the Social Democratic Party responded to the documents of Vault 7 "Year Zero" that the CIA used Frankfurt as a base for its digital espionage operations, saying that Germany did not have any information about the cyber attacks. [ 73 ]
The documents reportedly revealed that the agency had amassed a large collection of cyberattack techniques and malware produced by other hackers. This library was reportedly maintained by the CIA's Remote Devices Branch's UMBRAGE group, with examples of using these techniques and source code contained in the "Umbrage Component Library" git repository. [ citation needed ]
On the day the Vault 7 documents were first released, WikiLeaks described UMBRAGE as "a substantial library of attack techniques 'stolen' from malware produced in other states including the Russian Federation," and tweeted, "CIA steals other groups virus and malware facilitating false flag attacks." [ 74 ] According to WikiLeaks, by recycling the techniques of third parties through UMBRAGE, the CIA can not only increase its total number of attacks, [ 75 ] but can also mislead forensic investigators by disguising these attacks as the work of other groups and nations. [ 1 ] [ 68 ] Among the techniques borrowed by UMBRAGE was the file wiping implementation used by Shamoon . According to PC World , some of the techniques and code snippets have been used by CIA in its internal projects, whose result cannot be inferred from the leaks. PC World commented that the practice of planting " false flags " to deter attribution was not a new development in cyberattacks: Russian, North Korean and Israeli hacker groups are among those suspected of using false flags. [ 76 ]
A conspiracy theory soon emerged alleging that the CIA framed the Russian government for interfering in the 2016 U.S. elections . Conservative commentators such as Sean Hannity and Ann Coulter speculated about this possibility on Twitter, and Rush Limbaugh discussed it on his radio show. [ 77 ] Russian foreign minister Sergey Lavrov said that Vault 7 showed that "the CIA could get access to such 'fingerprints' and then use them." [ 74 ]
Cybersecurity writers and experts, such as Ben Buchanan and Kevin Poulsen , were skeptical of those theories. [ 12 ] [ 78 ] Poulsen said the theories were "disinformation" being taken advantage of by Russia and spread by bots. He also wrote, "The leaked catalog isn't organized by country of origin, and the specific malware used by the Russian DNC hackers is nowhere on the list." Robert M. Lee, who founded the cybersecurity firm Dragos, said the "narrative emerged far too quickly to have been organic." [ 12 ]
According to a study by Kim Zetter in The Intercept , UMBRAGE was probably much more focused on speeding up development by repurposing existing tools, rather than on planting false flags. [ 75 ] Robert Graham, CEO of Errata Security told The Intercept that the source code referenced in the UMBRAGE documents is "extremely public", and is likely used by a multitude of groups and state actors. Graham added: "What we can conclusively say from the evidence in the documents is that they're creating snippets of code for use in other projects and they're reusing methods in code that they find on the internet. ... Elsewhere they talk about obscuring attacks so you can't see where it's coming from, but there's no concrete plan to do a false flag operation. They're not trying to say 'We're going to make this look like Russia'." [ 79 ]
The documents describe the Marble framework, a string obfuscator used to hide text fragments in malware from visual inspection. Some outlets reported that foreign languages were used to cover up the source of CIA hacks, but technical analysis refuted the idea. [ 80 ] [ 81 ] [ 82 ] According to WikiLeaks, it reached 1.0 in 2015 and was used by the CIA throughout 2016. [ 82 ]
In its release, WikiLeaks said "Marble" was used to insert foreign language text into the malware to mask viruses, trojans and hacking attacks, making it more difficult for them to be tracked to the CIA and to cause forensic investigators to falsely attribute code to the wrong nation. The source code revealed that Marble had examples in Chinese, Russian, Korean, Arabic and Persian . [ 82 ]
Analysts called WikiLeaks' description of Marble's main purpose inaccurate, telling The Hill its main purpose was probably to avoid detection by antivirus programs. [ 83 ]
Marble also contained a deobfuscator tool with which the CIA could reverse text obfuscation. [ 82 ] [ 84 ]
Security researcher Nicholas Weaver from International Computer Science Institute in Berkeley told the Washington Post: "This appears to be one of the most technically damaging leaks ever done by WikiLeaks, as it seems designed to directly disrupt ongoing CIA operations." [ 85 ] [ 86 ]
HammerDrill is a CD/DVD collection tool that collects directory walks and files to a configured directory and filename pattern as well as logging CD/DVD insertion and removal events. [ 87 ]
After WikiLeaks released the first installment of Vault 7, "Year Zero", Apple stated that "many of the issues leaked today were already patched in the latest iOS," and that the company will "continue work to rapidly address any identified vulnerabilities." [ 88 ]
On 23 March 2017, WikiLeaks released "Dark Matter", the second batch of documents in its Vault 7 series, detailing the hacking techniques and tools all focusing on Apple products developed by the Embedded Development Branch (EDB) of the CIA. The leak also revealed the CIA had been targeting the iPhone since 2008, and that some projects attacked Apple's firmware. [ 89 ] The "Dark Matter" archive included documents from 2009 and 2013. Apple issued a second statement assuring that based on an "initial analysis, the alleged iPhone vulnerability affected iPhone 3G only and was fixed in 2009 when iPhone 3GS was released." Additionally, a preliminary assessment showed "the alleged Mac vulnerabilities were previously fixed in all Macs launched after 2013". [ 90 ] [ 91 ]
WikiLeaks said on 19 March 2017 on Twitter that the "CIA was secretly exploiting" a vulnerability in a huge range of Cisco router models discovered thanks to the Vault 7 documents. [ 92 ] [ 93 ] The CIA had learned more than a year ago how to exploit flaws in Cisco's widely used internet switches , which direct electronic traffic, to enable eavesdropping. Cisco quickly reassigned staff from other projects to turn their focus solely on analyzing the attack and to figure out how the CIA hacking worked, so they could help customers patch their systems and prevent criminal hackers or spies from using similar methods. [ 94 ]
On 20 March, Cisco researchers confirmed that their study of the Vault 7 documents showed the CIA had developed malware which could exploit a flaw found in 318 of Cisco's switch models and alter or take control of the network. [ 95 ] Cisco issued a warning on security risks, patches were not available, but Cisco provided mitigation advice. [ 93 ]
The electronic tools can reportedly compromise both Apple 's iOS and Google 's Android operating systems. By adding malware to the Android operating system, the tools could gain access to secure communications made on a device. [ 96 ]
According to WikiLeaks, once an Android smartphone is penetrated the agency can collect "audio and message traffic before encryption is applied". [ 1 ] Some of the agency's software is reportedly able to gain access to messages sent by instant messaging services. [ 1 ] This method of accessing messages differs from obtaining access by decrypting an already encrypted message. [ 96 ] While the encryption of messengers that offer end-to-end encryption , such as Telegram , WhatsApp and Signal , wasn't reported to be cracked, their encryption can be bypassed by capturing input before their encryption is applied, by methods such as keylogging and recording the touch input from the user. [ 96 ]
Commentators, including Snowden and cryptographer and security pundit Bruce Schneier , observed that Wikileaks incorrectly implied that the messaging apps themselves, and their underlying encryption, had been compromised - an implication which was in turn reported for a period by the New York Times and other mainstream outlets . [ 1 ] [ 97 ]
One document reportedly showed that the CIA was researching ways to infect vehicle control systems. WikiLeaks stated, "The purpose of such control is not specified, but it would permit the CIA to engage in nearly undetectable assassinations." [ 68 ] This statement brought renewed attention to conspiracy theories surrounding the death of Michael Hastings . [ 98 ]
The documents refer to a "Windows FAX DLL injection " exploit in Windows XP , Windows Vista and Windows 7 operating systems. [ 19 ] This would allow a user with malicious intent to hide malware under [ clarification needed ] the DLL of another application. However, a computer must have already been compromised through another method for the injection to take place. [ 99 ] [ better source needed ]
On 7 March 2017, Edward Snowden commented on the importance of the release, stating that it reveals the United States Government to be "developing vulnerabilities in US products" and "then intentionally keeping the holes open", which he considered highly reckless. [ 100 ] On 7 March 2017, Nathan White, Senior Legislative Manager at the Internet advocacy group Access Now , wrote: [ 101 ]
Today, our digital security has been compromised because the CIA has been stockpiling vulnerabilities rather than working with companies to patch them. The United States is supposed to have a process that helps secure our digital devices and services — the ' Vulnerabilities Equities Process .' Many of these vulnerabilities could have been responsibly disclosed and patched. This leak proves the inherent digital risk of stockpiling vulnerabilities rather than fixing them.
On 8 March 2017, Lee Mathews, a contributor to Forbes , wrote that most of the hacking techniques described in Vault 7 were already known to many cybersecurity experts. [ 102 ] On 8 March 2017, some noted that the revealed techniques and tools are most likely to be used for more targeted surveillance [ 103 ] [ 104 ] revealed by Edward Snowden. [ 105 ]
On 8 April 2017, Ashley Gorski, an American Civil Liberties Union staff attorney called it "critical" to understand that "these vulnerabilities can be exploited not just by our government but by foreign governments and cyber criminals around the world." Justin Cappos , professor in the Computer Science and Engineering department at New York University asks "if the government knows of a problem in your phone that bad guys could use to hack your phone and have the ability to spy on you, is that a weakness that they themselves should use for counterterrorism, or for their own spying capabilities, or is it a problem they should fix for everyone?" [ 106 ]
On 8 April 2017, Cindy Cohn , executive director of the San Francisco-based international non-profit digital rights group Electronic Frontier Foundation , said: "If the C.I.A. was walking past your front door and saw that your lock was broken, they should at least tell you and maybe even help you get it fixed." "And worse, they then lost track of the information they had kept from you so that now criminals and hostile foreign governments know about your broken lock." [ 107 ] Furthermore, she stated that the CIA had "failed to accurately assess the risk of not disclosing vulnerabilities. Even spy agencies like the CIA have a responsibility to protect the security and privacy of Americans." [ 108 ] "The freedom to have a private conversation – free from the worry that a hostile government, a rogue government agent or a competitor or a criminal are listening – is central to a free society". [ 109 ]
On 12 May 2017 Microsoft President, Brad Smith, wrote that both NSA and CIA had stockpiled vulnerabilities, which were stolen from them and published on Wikileaks, resulting in security breaches for Microsoft customers. Smith requested, for a second time, a "digital Geneva Convention" which would require governments to report vulnerabilities to vendors. [ 45 ] [ 110 ]
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A Vavilov center or center of origin is a geographical area where a group of organisms, either domesticated or wild, first developed its distinctive properties. [ 1 ] Centers of origin were first identified in 1924 by Nikolai Vavilov . Vavilov posited that the center of origin for a species or genus is the same as its center of diversity , the geographic area where it has the highest genetic diversity , but this equivalence has been disputed by later scholars. [ 2 ]
Locating the origin of crop plants is basic to plant breeding . This allows one to locate wild relatives, related species, and new genes (especially dominant genes , which may provide resistance to diseases).
Knowledge of the origins of crop plants is important in order to avoid genetic erosion , the loss of germplasm due to the loss of ecotypes and landraces , loss of habitat (such as rainforests), and increased urbanization. Germplasm preservation is accomplished through gene banks (largely seed collections but now frozen stem sections) and preservation of natural habitats (especially in centers of origin).
A Vavilov Center (of Diversity) is a region of the world first indicated by Nikolai Vavilov to be an original center for the domestication of plants. [ 4 ] For crop plants, Nikolai Vavilov identified differing numbers of centers: three in 1924, five in 1926, six in 1929, seven in 1931, eight in 1935 and reduced to seven again in 1940. [ 5 ] [ 6 ]
Vavilov argued that plants were not domesticated somewhere in the world at random, but that there were regions where domestication started. The center of origin is also considered the center of diversity.
Vavilov centers are regions where a high diversity of crop wild relatives can be found, representing the natural relatives of domesticated crop plants.
[ 10 ]
1) eastern North America
Chenopodium berlandieri , Iva annua , and Helianthus annuus
4,500–4,000 years
2) Mesoamerica
Cucurbita pepo
10,000
Zea mays
9,000–7,000
2a) northern lowland neotropics
Cucurbita moschata , Ipomoea batatas , Phaseolus vulgaris , tree crops
9,000–8,000
3) central mid-altitude Andes
Chenopodium quinoa , Amaranthus caudatus
5,000
3a) north and central Andes , mid-altitude and high altitude areas
Solanum tuberosum , Oxalis tuberosa , Chenopodium pallidicaule
8,000
3b) lowland southern Amazonia
Manihot esculenta and Arachis hypogaea
8,000
3c) Ecuador (part of 3, 3a, and/or 3b?) and northwest Peru
Phaseolus lunatus , Canavalia plagiosperma , and Cucurbita ecuadorensis
10,000
4) western sub-Saharan African
Pennisetum glaucum
4,500
4a) west African savanna and woodlands
Vigna unguiculata
3,700
Digitaria exilis and Oryza glaberrima
<3,000
4b) west African rainforests
Dioscorea rotundata and Elaeis guineensis
poorly documented
5) east Sudanic Africa
Sorghum bicolor
>4,000?
6) east African uplands
Eragrostis tef and Eleusine coracana
4,000?
east African lowlands
vegeculture of Dioscorea cayennensis and Ensete ventricosum
poorly documented
7) Near East
Hordeum vulgare , Triticum spp., Lens culinaris , Pisum sativum , Cicer arietinum , Vicia faba
13,000–10,000
7a) eastern Fertile Crescent
additional Hordeum vulgare
goats
9,000
8a) Gujarat , India
Panicum sumatrense and Vigna mungo
5,000?
8b) Upper Indus
Panicum sumatrense , Vigna radiata , and Vigna aconitifolia
5,000
8c) Ganges
Oryza sativa subsp. indica
8,500–4,500
8d) southern India
Brachiaria ramosa , Vigna radiata , and Macrotyloma uniflorum
5,000–4,000
9) eastern Himalayas and Yunnan uplands
Fagopyrum esculentum
5,000?
10) northern China
Setaria italica and Panicum miliaceum
8,000
Glycine max
4,500?
11) southern Hokkaido , Japan
Echinochloa crusgalli
4,500
12) Yangtze River Valley , China
Oryza sativa subsp. japonica
9,000–6,000
12a) southern China
Colocasia spp., Coix lacryma-jobi
poorly documented, 4,500?
13) New Guinea and Wallacea
Colocasia esculenta , Dioscorea esculenta , and Musa acuminata
7,000
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Vaxart, Inc. is an American biotechnology company focused on the discovery, development, and commercialization of oral recombinant vaccines administered using temperature-stable tablets that can be stored and shipped without refrigeration, eliminating the need for needle injection. Its development programs for oral vaccine delivery (Vector-Adjuvant-Antigen Standardized Technology known as VAAST ) include prophylactic , enteric-coated tablet vaccines for inhibiting norovirus , seasonal influenza , respiratory syncytial virus , and human papillomavirus . It was founded in 2004 by Sean Tucker. [ 2 ] Originally incorporated as West Coast Biologicals, Inc. in California in 2004, the company later changed its name to Vaxart, Inc. in July 2007, after reincorporating in Delaware. A significant development in the company's history was the reverse merger with Aviragen Therapeutics, Inc. on February 13, 2018, which led to Vaxart becoming a wholly-owned subsidiary of Aviragen. Post-merger, the company continued as Vaxart, Inc. [ 3 ]
Vaxart's development portfolio includes a range of oral vaccines targeting infectious diseases such as norovirus, COVID-19, and seasonal influenza, as well as therapeutic vaccines like those targeting HPV. The company has achieved significant milestones in its vaccine development, including completing Phase 1 trials for its COVID-19 vaccine candidate and embarking on Phase 2 studies. Vaxart’s vaccines are designed to elicit not only systemic immune responses but also mucosal and T cell responses, which may enhance protection against specific diseases and offer potential benefits for certain cancers and chronic viral infections. The tablet form of these vaccines represents a significant advancement in vaccine administration, potentially improving patient acceptance and resolving distribution challenges. [ 4 ]
The Vaxart technology is based on the potential to prevent or inhibit infectious diseases by using orally-delivered vaccines by tablets , eliminating intramuscular injection concerns which may involve pain, fear of needles , cross-contamination, dosing inconsistencies, and higher cost for large-scale immunizations . [ 5 ] [ 6 ] As a proof of concept for oral vaccination efficacy, an oral vaccine against polio was proved to be safe and effective, and is in common use in many countries. [ 7 ] [ 8 ] [ 9 ]
Vaxart uses enteric-coated tablets to protect the active vaccine from acidic degradation in the stomach , delivering the vaccine into the small intestine where it can engage the immune system to stimulate systemic and mucosal immune responses against a virus. [ 10 ] [ 11 ] [ 6 ] [ 12 ]
Vaxart uses a specific virus called adenovirus type 5 (Ad5) as a delivery biological "vector" to carry genes coding for the antigen to generate a protective immune response. [ 13 ] [ 6 ] [ 12 ] The Ad5 vector delivers the antigen to the epithelial cells lining the mucosa of the small intestine where it stimulates the immune system to respond against the vaccine antigen, creating a systemic immune response against a virus. [ 11 ] [ 6 ] [ 12 ]
In addition to its focus on infectious diseases, Vaxart has expanded its vaccine development efforts to include therapeutic vaccines for chronic viral infections and cancer. These efforts are anchored in the company's innovative technology that enables the creation of tablet-based vaccines, which can generate broad and durable immune responses, including systemic, mucosal, and T-cell responses. This comprehensive immune activation is particularly critical in the context of chronic viral infections and cancer, where both systemic and localized immune responses play a pivotal role in effective treatment. [ 14 ]
The lead vaccine candidate by Vaxart is an influenza oral tablet vaccine, which showed safety and neutralizing antibody responses to influenza virus in a 2015 Phase I clinical trial . [ 15 ] A 2016-17 Phase II trial of the Vaxart oral flu vaccine, VXA-A1.1, showed that the vaccine was well-tolerated and provided immunity against virus shedding , outperforming the effectiveness of an established intramuscular vaccine. [ 16 ] In 2018, Vaxart completed a Phase II challenge study, in which the Vaxart influenza tablet vaccine demonstrated a 39 percent reduction in clinical disease relative to placebo, compared to a 27 percent reduction by the injectable flu vaccine, Fluzone . [ 11 ]
In January 2020, Vaxart announced they were intending to develop a tablet vaccine intended to prevent COVID-19 . It faced competition from other vaccine makers, such as Novavax , Inovio Pharmaceuticals , and Moderna . [ 17 ] [ 13 ] [ 18 ] [ 11 ] By February 2020, Vaxart had begun the program. [ 17 ] [ 13 ] [ 18 ] In April 2020, the company reported positive immune responses in laboratory animals from tests conducted with its COVID-19 vaccine candidate, suggesting potential effectiveness in its approach to vaccination. [ 19 ]
In January 2024, Vaxart received a grant from the United States Biomedical Advanced Research and Development Authority (BARDA). The grant, amounting to $9.27 million, is allocated for the preparation of a large-scale Phase 2b clinical trial involving 10,000 participants. This trial is designed to evaluate the efficacy of Vaxart's innovative oral pill XBB COVID-19 vaccine candidate in comparison to an approved mRNA vaccine. The funding is part of Project NextGen, a substantial $5 billion initiative by the U.S. Department of Health and Human Services (HHS) focusing on the development of new, innovative vaccines and therapeutics that provide broader and more durable protection against COVID-19. The oral vaccine platform developed by Vaxart is noted for its potential advantages, including generating mucosal immunity and offering a cross-reactive response to various COVID-19 variants. [ 20 ] A favorable recommendation from an independent Data Safety Monitoring Board (DSMB) reviewing the sentinel cohort of 400 participants further affirmed the trial's safety and allowed it to proceed without modifications. [ 21 ]
Vaxart has been progressing in the development of an oral tablet vaccine for Norovirus , the leading cause of acute gastroenteritis globally. Norovirus significantly impacts all age groups, particularly young children and the elderly, causing millions of cases annually with substantial healthcare and economic burdens.
Its vaccine candidate is a bivalent oral tablet targeting major norovirus genogroups GI and GII. The tablet form is specifically crafted to generate antibodies in the intestine, the primary site of norovirus infection.
In December 2022, Vaxart received significant funding and support from the Bill & Melinda Gates Foundationfor a study focusing on breastfeeding mothers and their infants. This study aligns with Vaxart’s broader norovirus vaccine program goals, targeting critical and at-risk populations globally. [ 22 ]
In 2023, Vaxart initiated a Phase 1 trial (VXA-NVV-108) focused on lactating mothers, aiming to evaluate the vaccine's ability to induce breast milk antibodies and their transfer to infants. [ 23 ] Furthermore, in July 2023, Vaxart announced positive topline data from a dose-ranging Phase 2 clinical trial of its bivalent norovirus vaccine candidate. The study met all primary endpoints, showing the vaccine was well-tolerated with robust immunogenicity. [ 24 ]
Additionally, in September 2023, Vaxart announced top-line data from its Phase 2 challenge study of the monovalent norovirus vaccine candidate. [ 25 ] This study met 5 of its 6 primary endpoints, demonstrating a statistically significant reduction in norovirus infection rate and a substantial reduction in viral shedding, while maintaining a well-tolerated safety profile.
In 2019, several hedge funds invested in Vaxart, with the largest investment coming from Armistice Capital which acquired 25.2 million shares. [ 26 ] [ 27 ]
Vaxart’s investments and stock trading activities led to two major legal cases:
In 2021, a group of shareholders filed a lawsuit in the Delaware Court of Chancery (In re Vaxart, Inc. Stockholder Litigation), alleging that company executives engaged in insider trading and violated their fiduciary duties by profiting from misleading vaccine-related statements. [ 28 ] The case was dismissed in early 2024.
A separate securities fraud class action lawsuit (In re Vaxart, Inc. Securities Litigation) was filed in the U.S. District Court for the Northern District of California, accusing Vaxart of misleading investors about its involvement in Operation Warp Speed. Judge Vince Chhabria presides over the case, which was certified as a class action in December 2024. Vaxart was dismissed from the case in late 2024 after the court ruled there was insufficient evidence to support the claims against the company. [ 29 ]
Vaxart has denied all allegations and continues to focus on developing its oral vaccine platform.
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In mathematics , the Veblen functions are a hierarchy of normal functions ( continuous strictly increasing functions from ordinals to ordinals), introduced by Oswald Veblen in Veblen (1908) . If φ 0 is any normal function, then for any non-zero ordinal α , φ α is the function enumerating the common fixed points of φ β for β < α . These functions are all normal.
In the special case when φ 0 ( α )=ω α this family of functions is known as the Veblen hierarchy .
The function φ 1 is the same as the ε function : φ 1 ( α )= ε α . [ 1 ] If α < β , {\displaystyle \alpha <\beta \,,} then φ α ( φ β ( γ ) ) = φ β ( γ ) {\displaystyle \varphi _{\alpha }(\varphi _{\beta }(\gamma ))=\varphi _{\beta }(\gamma )} . [ 2 ] From this and the fact that φ β is strictly increasing we get the ordering: φ α ( β ) < φ γ ( δ ) {\displaystyle \varphi _{\alpha }(\beta )<\varphi _{\gamma }(\delta )} if and only if either ( α = γ {\displaystyle \alpha =\gamma } and β < δ {\displaystyle \beta <\delta } ) or ( α < γ {\displaystyle \alpha <\gamma } and β < φ γ ( δ ) {\displaystyle \beta <\varphi _{\gamma }(\delta )} ) or ( α > γ {\displaystyle \alpha >\gamma } and φ α ( β ) < δ {\displaystyle \varphi _{\alpha }(\beta )<\delta } ). [ 2 ]
The fundamental sequence for an ordinal with cofinality ω is a distinguished strictly increasing ω-sequence that has the ordinal as its limit. If one has fundamental sequences for α and all smaller limit ordinals, then one can create an explicit constructive bijection between ω and α , (i.e. one not using the axiom of choice ). Here we will describe fundamental sequences for the Veblen hierarchy of ordinals. The image of n under the fundamental sequence for α will be indicated by α [ n ].
A variation of Cantor normal form used in connection with the Veblen hierarchy is: every nonzero ordinal number α can be uniquely written as α = φ β 1 ( γ 1 ) + φ β 2 ( γ 2 ) + ⋯ + φ β k ( γ k ) {\displaystyle \alpha =\varphi _{\beta _{1}}(\gamma _{1})+\varphi _{\beta _{2}}(\gamma _{2})+\cdots +\varphi _{\beta _{k}}(\gamma _{k})} , where k >0 is a natural number and each term after the first is less than or equal to the previous term, φ β m ( γ m ) ≥ φ β m + 1 ( γ m + 1 ) , {\displaystyle \varphi _{\beta _{m}}(\gamma _{m})\geq \varphi _{\beta _{m+1}}(\gamma _{m+1})\,,} and each γ m < φ β m ( γ m ) . {\displaystyle \gamma _{m}<\varphi _{\beta _{m}}(\gamma _{m}).} If a fundamental sequence can be provided for the last term, then that term can be replaced by such a sequence to get α [ n ] = φ β 1 ( γ 1 ) + ⋯ + φ β k − 1 ( γ k − 1 ) + ( φ β k ( γ k ) [ n ] ) . {\displaystyle \alpha [n]=\varphi _{\beta _{1}}(\gamma _{1})+\cdots +\varphi _{\beta _{k-1}}(\gamma _{k-1})+(\varphi _{\beta _{k}}(\gamma _{k})[n])\,.}
For any β , if γ is a limit with γ < φ β ( γ ) , {\displaystyle \gamma <\varphi _{\beta }(\gamma )\,,} then let φ β ( γ ) [ n ] = φ β ( γ [ n ] ) . {\displaystyle \varphi _{\beta }(\gamma )[n]=\varphi _{\beta }(\gamma [n])\,.}
No such sequence can be provided for φ 0 ( 0 ) {\displaystyle \varphi _{0}(0)} = ω 0 = 1 because it does not have cofinality ω.
For φ 0 ( γ + 1 ) = ω γ + 1 = ω γ ⋅ ω , {\displaystyle \varphi _{0}(\gamma +1)=\omega ^{\gamma +1}=\omega ^{\gamma }\cdot \omega \,,} we choose φ 0 ( γ + 1 ) [ n ] = φ 0 ( γ ) ⋅ n = ω γ ⋅ n . {\displaystyle \varphi _{0}(\gamma +1)[n]=\varphi _{0}(\gamma )\cdot n=\omega ^{\gamma }\cdot n\,.}
For φ β + 1 ( 0 ) , {\displaystyle \varphi _{\beta +1}(0)\,,} we use φ β + 1 ( 0 ) [ 0 ] = 0 {\displaystyle \varphi _{\beta +1}(0)[0]=0} and φ β + 1 ( 0 ) [ n + 1 ] = φ β ( φ β + 1 ( 0 ) [ n ] ) , {\displaystyle \varphi _{\beta +1}(0)[n+1]=\varphi _{\beta }(\varphi _{\beta +1}(0)[n])\,,} i.e. 0, φ β ( 0 ) {\displaystyle \varphi _{\beta }(0)} , φ β ( φ β ( 0 ) ) {\displaystyle \varphi _{\beta }(\varphi _{\beta }(0))} , etc..
For φ β + 1 ( γ + 1 ) {\displaystyle \varphi _{\beta +1}(\gamma +1)} , we use φ β + 1 ( γ + 1 ) [ 0 ] = φ β + 1 ( γ ) + 1 {\displaystyle \varphi _{\beta +1}(\gamma +1)[0]=\varphi _{\beta +1}(\gamma )+1} and φ β + 1 ( γ + 1 ) [ n + 1 ] = φ β ( φ β + 1 ( γ + 1 ) [ n ] ) . {\displaystyle \varphi _{\beta +1}(\gamma +1)[n+1]=\varphi _{\beta }(\varphi _{\beta +1}(\gamma +1)[n])\,.}
Now suppose that β is a limit:
If β < φ β ( 0 ) {\displaystyle \beta <\varphi _{\beta }(0)} , then let φ β ( 0 ) [ n ] = φ β [ n ] ( 0 ) . {\displaystyle \varphi _{\beta }(0)[n]=\varphi _{\beta [n]}(0)\,.}
For φ β ( γ + 1 ) {\displaystyle \varphi _{\beta }(\gamma +1)} , use φ β ( γ + 1 ) [ n ] = φ β [ n ] ( φ β ( γ ) + 1 ) . {\displaystyle \varphi _{\beta }(\gamma +1)[n]=\varphi _{\beta [n]}(\varphi _{\beta }(\gamma )+1)\,.}
Otherwise, the ordinal cannot be described in terms of smaller ordinals using φ {\displaystyle \varphi } and this scheme does not apply to it.
The function Γ enumerates the ordinals α such that φ α (0) = α .
Γ 0 is the Feferman–Schütte ordinal , i.e. it is the smallest α such that φ α (0) = α .
For Γ 0 , a fundamental sequence could be chosen to be Γ 0 [ 0 ] = 0 {\displaystyle \Gamma _{0}[0]=0} and Γ 0 [ n + 1 ] = φ Γ 0 [ n ] ( 0 ) . {\displaystyle \Gamma _{0}[n+1]=\varphi _{\Gamma _{0}[n]}(0)\,.}
For Γ β+1 , let Γ β + 1 [ 0 ] = Γ β + 1 {\displaystyle \Gamma _{\beta +1}[0]=\Gamma _{\beta }+1} and Γ β + 1 [ n + 1 ] = φ Γ β + 1 [ n ] ( 0 ) . {\displaystyle \Gamma _{\beta +1}[n+1]=\varphi _{\Gamma _{\beta +1}[n]}(0)\,.}
For Γ β where β < Γ β {\displaystyle \beta <\Gamma _{\beta }} is a limit, let Γ β [ n ] = Γ β [ n ] . {\displaystyle \Gamma _{\beta }[n]=\Gamma _{\beta [n]}\,.}
To build the Veblen function of a finite number of arguments (finitary Veblen function), let the binary function φ ( α , γ ) {\displaystyle \varphi (\alpha ,\gamma )} be φ α ( γ ) {\displaystyle \varphi _{\alpha }(\gamma )} as defined above.
Let z {\displaystyle z} be an empty string or a string consisting of one or more comma-separated zeros 0 , 0 , . . . , 0 {\displaystyle 0,0,...,0} and s {\displaystyle s} be an empty string or a string consisting of one or more comma-separated ordinals α 1 , α 2 , . . . , α n {\displaystyle \alpha _{1},\alpha _{2},...,\alpha _{n}} with α 1 > 0 {\displaystyle \alpha _{1}>0} . The binary function φ ( β , γ ) {\displaystyle \varphi (\beta ,\gamma )} can be written as φ ( s , β , z , γ ) {\displaystyle \varphi (s,\beta ,z,\gamma )} where both s {\displaystyle s} and z {\displaystyle z} are empty strings.
The finitary Veblen functions are defined as follows:
For example, φ ( 1 , 0 , γ ) {\displaystyle \varphi (1,0,\gamma )} is the ( 1 + γ ) {\displaystyle (1+\gamma )} -th fixed point of the functions ξ ↦ φ ( ξ , 0 ) {\displaystyle \xi \mapsto \varphi (\xi ,0)} , namely Γ γ {\displaystyle \Gamma _{\gamma }} ; then φ ( 1 , 1 , γ ) {\displaystyle \varphi (1,1,\gamma )} enumerates the fixed points of that function, i.e., of the ξ ↦ Γ ξ {\displaystyle \xi \mapsto \Gamma _{\xi }} function; and φ ( 2 , 0 , γ ) {\displaystyle \varphi (2,0,\gamma )} enumerates the fixed points of all the ξ ↦ φ ( 1 , ξ , 0 ) {\displaystyle \xi \mapsto \varphi (1,\xi ,0)} . Each instance of the generalized Veblen functions is continuous in the last nonzero variable (i.e., if one variable is made to vary and all later variables are kept constantly equal to zero).
The ordinal φ ( 1 , 0 , 0 , 0 ) {\displaystyle \varphi (1,0,0,0)} is sometimes known as the Ackermann ordinal . The limit of the φ ( 1 , 0 , . . . , 0 ) {\displaystyle \varphi (1,0,...,0)} where the number of zeroes ranges over ω, is sometimes known as the "small" Veblen ordinal .
Every non-zero ordinal α {\displaystyle \alpha } less than the small Veblen ordinal (SVO) can be uniquely written in normal form for the finitary Veblen function:
α = φ ( s 1 ) + φ ( s 2 ) + ⋯ + φ ( s k ) {\displaystyle \alpha =\varphi (s_{1})+\varphi (s_{2})+\cdots +\varphi (s_{k})}
where
For limit ordinals α < S V O {\displaystyle \alpha <SVO} , written in normal form for the finitary Veblen function:
More generally, Veblen showed that φ can be defined even for a transfinite sequence of ordinals α β , provided that all but a finite number of them are zero. Notice that if such a sequence of ordinals is chosen from those less than an uncountable regular cardinal κ, then the sequence may be encoded as a single ordinal less than κ κ (ordinal exponentiation). So one is defining a function φ from κ κ into κ.
The definition can be given as follows: let α be a transfinite sequence of ordinals (i.e., an ordinal function with finite support) that ends in zero (i.e., such that α 0 =0), and let α [γ@0] denote the same function where the final 0 has been replaced by γ. Then γ↦φ( α [γ@0]) is defined as the function enumerating the common fixed points of all functions ξ↦φ( β ) where β ranges over all sequences that are obtained by decreasing the smallest-indexed nonzero value of α and replacing some smaller-indexed value with the indeterminate ξ (i.e., β = α [ζ@ι 0 ,ξ@ι] meaning that for the smallest index ι 0 such that α ι 0 is nonzero the latter has been replaced by some value ζ<α ι 0 and that for some smaller index ι<ι 0 , the value α ι =0 has been replaced with ξ).
For example, if α =(1@ω) denotes the transfinite sequence with value 1 at ω and 0 everywhere else, then φ(1@ω) is the smallest fixed point of all the functions ξ↦φ(ξ,0,...,0) with finitely many final zeroes (it is also the limit of the φ(1,0,...,0) with finitely many zeroes, the small Veblen ordinal).
The smallest ordinal α such that α is greater than φ applied to any function with support in α (i.e., that cannot be reached "from below" using the Veblen function of transfinitely many variables) is sometimes known as the "large" Veblen ordinal , or "great" Veblen number. [ 3 ]
In Massmann & Kwon (2023) , the Veblen function was extended further to a somewhat technical system known as dimensional Veblen . In this, one may take fixed points or row numbers, meaning expressions such as φ (1@(1,0)) are valid (representing the large Veblen ordinal), visualised as multi-dimensional arrays. It was proven that all ordinals below the Bachmann–Howard ordinal could be represented in this system, and that the representations for all ordinals below the large Veblen ordinal were aesthetically the same as in the original system.
The function takes on several prominent values:
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A vector-valued function , also referred to as a vector function , is a mathematical function of one or more variables whose range is a set of multidimensional vectors or infinite-dimensional vectors . The input of a vector-valued function could be a scalar or a vector (that is, the dimension of the domain could be 1 or greater than 1); the dimension of the function's domain has no relation to the dimension of its range.
A common example of a vector-valued function is one that depends on a single real parameter t , often representing time , producing a vector v ( t ) as the result. In terms of the standard unit vectors i , j , k of Cartesian 3-space , these specific types of vector-valued functions are given by expressions such as r ( t ) = f ( t ) i + g ( t ) j + h ( t ) k {\displaystyle \mathbf {r} (t)=f(t)\mathbf {i} +g(t)\mathbf {j} +h(t)\mathbf {k} } where f ( t ) , g ( t ) and h ( t ) are the coordinate functions of the parameter t , and the domain of this vector-valued function is the intersection of the domains of the functions f , g , and h . It can also be referred to in a different notation: r ( t ) = ⟨ f ( t ) , g ( t ) , h ( t ) ⟩ {\displaystyle \mathbf {r} (t)=\langle f(t),g(t),h(t)\rangle } The vector r ( t ) has its tail at the origin and its head at the coordinates evaluated by the function.
The vector shown in the graph to the right is the evaluation of the function ⟨ 2 cos t , 4 sin t , t ⟩ {\displaystyle \langle 2\cos t,\,4\sin t,\,t\rangle } near t = 19.5 (between 6π and 6.5π ; i.e., somewhat more than 3 rotations). The helix is the path traced by the tip of the vector as t increases from zero through 8 π .
In 2D, we can analogously speak about vector-valued functions as: r ( t ) = f ( t ) i + g ( t ) j {\displaystyle \mathbf {r} (t)=f(t)\mathbf {i} +g(t)\mathbf {j} } or r ( t ) = ⟨ f ( t ) , g ( t ) ⟩ {\displaystyle \mathbf {r} (t)=\langle f(t),g(t)\rangle }
In the linear case the function can be expressed in terms of matrices : y = A x , {\displaystyle \mathbf {y} =A\mathbf {x} ,} where y is an n × 1 output vector, x is a k × 1 vector of inputs, and A is an n × k matrix of parameters . Closely related is the affine case (linear up to a translation ) where the function takes the form y = A x + b , {\displaystyle \mathbf {y} =A\mathbf {x} +\mathbf {b} ,} where in addition b'' is an n × 1 vector of parameters.
The linear case arises often, for example in multiple regression , [ clarification needed ] where for instance the n × 1 vector y ^ {\displaystyle {\hat {y}}} of predicted values of a dependent variable is expressed linearly in terms of a k × 1 vector β ^ {\displaystyle {\hat {\boldsymbol {\beta }}}} ( k < n ) of estimated values of model parameters: y ^ = X β ^ , {\displaystyle {\hat {\mathbf {y} }}=X{\hat {\boldsymbol {\beta }}},} in which X (playing the role of A in the previous generic form) is an n × k matrix of fixed (empirically based) numbers.
A surface is a 2-dimensional set of points embedded in (most commonly) 3-dimensional space. One way to represent a surface is with parametric equations , in which two parameters s and t determine the three Cartesian coordinates of any point on the surface: ( x , y , z ) = ( f ( s , t ) , g ( s , t ) , h ( s , t ) ) ≡ F ( s , t ) . {\displaystyle (x,y,z)=(f(s,t),g(s,t),h(s,t))\equiv \mathbf {F} (s,t).} Here F is a vector-valued function. For a surface embedded in n -dimensional space, one similarly has the representation ( x 1 , x 2 , … , x n ) = ( f 1 ( s , t ) , f 2 ( s , t ) , … , f n ( s , t ) ) ≡ F ( s , t ) . {\displaystyle (x_{1},x_{2},\dots ,x_{n})=(f_{1}(s,t),f_{2}(s,t),\dots ,f_{n}(s,t))\equiv \mathbf {F} (s,t).}
Many vector-valued functions, like scalar-valued functions , can be differentiated by simply differentiating the components in the Cartesian coordinate system. Thus, if r ( t ) = f ( t ) i + g ( t ) j + h ( t ) k {\displaystyle \mathbf {r} (t)=f(t)\mathbf {i} +g(t)\mathbf {j} +h(t)\mathbf {k} } is a vector-valued function, then d r d t = f ′ ( t ) i + g ′ ( t ) j + h ′ ( t ) k . {\displaystyle {\frac {d\mathbf {r} }{dt}}=f'(t)\mathbf {i} +g'(t)\mathbf {j} +h'(t)\mathbf {k} .} The vector derivative admits the following physical interpretation: if r ( t ) represents the position of a particle, then the derivative is the velocity of the particle v ( t ) = d r d t . {\displaystyle \mathbf {v} (t)={\frac {d\mathbf {r} }{dt}}.} Likewise, the derivative of the velocity is the acceleration d v d t = a ( t ) . {\displaystyle {\frac {d\mathbf {v} }{dt}}=\mathbf {a} (t).}
The partial derivative of a vector function a with respect to a scalar variable q is defined as [ 1 ] ∂ a ∂ q = ∑ i = 1 n ∂ a i ∂ q e i {\displaystyle {\frac {\partial \mathbf {a} }{\partial q}}=\sum _{i=1}^{n}{\frac {\partial a_{i}}{\partial q}}\mathbf {e} _{i}} where a i is the scalar component of a in the direction of e i . It is also called the direction cosine of a and e i or their dot product . The vectors e 1 , e 2 , e 3 form an orthonormal basis fixed in the reference frame in which the derivative is being taken.
If a is regarded as a vector function of a single scalar variable, such as time t , then the equation above reduces to the first ordinary time derivative of a with respect to t , [ 1 ] d a d t = ∑ i = 1 n d a i d t e i . {\displaystyle {\frac {d\mathbf {a} }{dt}}=\sum _{i=1}^{n}{\frac {da_{i}}{dt}}\mathbf {e} _{i}.}
If the vector a is a function of a number n of scalar variables q r ( r = 1, ..., n ) , and each q r is only a function of time t , then the ordinary derivative of a with respect to t can be expressed, in a form known as the total derivative , as [ 1 ] d a d t = ∑ r = 1 n ∂ a ∂ q r d q r d t + ∂ a ∂ t . {\displaystyle {\frac {d\mathbf {a} }{dt}}=\sum _{r=1}^{n}{\frac {\partial \mathbf {a} }{\partial q_{r}}}{\frac {dq_{r}}{dt}}+{\frac {\partial \mathbf {a} }{\partial t}}.}
Some authors prefer to use capital D to indicate the total derivative operator, as in D / Dt . The total derivative differs from the partial time derivative in that the total derivative accounts for changes in a due to the time variance of the variables q r .
Whereas for scalar-valued functions there is only a single possible reference frame , to take the derivative of a vector-valued function requires the choice of a reference frame (at least when a fixed Cartesian coordinate system is not implied as such). Once a reference frame has been chosen, the derivative of a vector-valued function can be computed using techniques similar to those for computing derivatives of scalar-valued functions. A different choice of reference frame will, in general, produce a different derivative function. The derivative functions in different reference frames have a specific kinematical relationship .
The above formulas for the derivative of a vector function rely on the assumption that the basis vectors e 1 , e 2 , e 3 are constant, that is, fixed in the reference frame in which the derivative of a is being taken, and therefore the e 1 , e 2 , e 3 each has a derivative of identically zero. This often holds true for problems dealing with vector fields in a fixed coordinate system, or for simple problems in physics . However, many complex problems involve the derivative of a vector function in multiple moving reference frames, which means that the basis vectors will not necessarily be constant. In such a case where the basis vectors e 1 , e 2 , e 3 are fixed in reference frame E, but not in reference frame N, the more general formula for the ordinary time derivative of a vector in reference frame N is [ 1 ] N d a d t = ∑ i = 1 3 d a i d t e i + ∑ i = 1 3 a i N d e i d t {\displaystyle {\frac {{}^{\mathrm {N} }d\mathbf {a} }{dt}}=\sum _{i=1}^{3}{\frac {da_{i}}{dt}}\mathbf {e} _{i}+\sum _{i=1}^{3}a_{i}{\frac {{}^{\mathrm {N} }d\mathbf {e} _{i}}{dt}}} where the superscript N to the left of the derivative operator indicates the reference frame in which the derivative is taken. As shown previously , the first term on the right hand side is equal to the derivative of a in the reference frame where e 1 , e 2 , e 3 are constant, reference frame E. It also can be shown that the second term on the right hand side is equal to the relative angular velocity of the two reference frames cross multiplied with the vector a itself. [ 1 ] Thus, after substitution, the formula relating the derivative of a vector function in two reference frames is [ 1 ] N d a d t = E d a d t + N ω E × a {\displaystyle {\frac {{}^{\mathrm {N} }d\mathbf {a} }{dt}}={\frac {{}^{\mathrm {E} }d\mathbf {a} }{dt}}+{}^{\mathrm {N} }\mathbf {\omega } ^{\mathrm {E} }\times \mathbf {a} } where N ω E is the angular velocity of the reference frame E relative to the reference frame N.
One common example where this formula is used is to find the velocity of a space-borne object, such as a rocket , in the inertial reference frame using measurements of the rocket's velocity relative to the ground. The velocity N v R in inertial reference frame N of a rocket R located at position r R can be found using the formula N d d t ( r R ) = E d d t ( r R ) + N ω E × r R . {\displaystyle {\frac {{}^{\mathrm {N} }d}{dt}}(\mathbf {r} ^{\mathrm {R} })={\frac {{}^{\mathrm {E} }d}{dt}}(\mathbf {r} ^{\mathrm {R} })+{}^{\mathrm {N} }\mathbf {\omega } ^{\mathrm {E} }\times \mathbf {r} ^{\mathrm {R} }.} where N ω E is the angular velocity of the Earth relative to the inertial frame N. Since velocity is the derivative of position, N v R and E v R are the derivatives of r R in reference frames N and E, respectively. By substitution, N v R = E v R + N ω E × r R {\displaystyle {}^{\mathrm {N} }\mathbf {v} ^{\mathrm {R} }={}^{\mathrm {E} }\mathbf {v} ^{\mathrm {R} }+{}^{\mathrm {N} }\mathbf {\omega } ^{\mathrm {E} }\times \mathbf {r} ^{\mathrm {R} }} where E v R is the velocity vector of the rocket as measured from a reference frame E that is fixed to the Earth.
The derivative of a product of vector functions behaves similarly to the derivative of a product of scalar functions. [ a ] Specifically, in the case of scalar multiplication of a vector, if p is a scalar variable function of q , [ 1 ] ∂ ∂ q ( p a ) = ∂ p ∂ q a + p ∂ a ∂ q . {\displaystyle {\frac {\partial }{\partial q}}(p\mathbf {a} )={\frac {\partial p}{\partial q}}\mathbf {a} +p{\frac {\partial \mathbf {a} }{\partial q}}.}
In the case of dot multiplication , for two vectors a and b that are both functions of q , [ 1 ] ∂ ∂ q ( a ⋅ b ) = ∂ a ∂ q ⋅ b + a ⋅ ∂ b ∂ q . {\displaystyle {\frac {\partial }{\partial q}}(\mathbf {a} \cdot \mathbf {b} )={\frac {\partial \mathbf {a} }{\partial q}}\cdot \mathbf {b} +\mathbf {a} \cdot {\frac {\partial \mathbf {b} }{\partial q}}.}
Similarly, the derivative of the cross product of two vector functions is [ 1 ] ∂ ∂ q ( a × b ) = ∂ a ∂ q × b + a × ∂ b ∂ q . {\displaystyle {\frac {\partial }{\partial q}}(\mathbf {a} \times \mathbf {b} )={\frac {\partial \mathbf {a} }{\partial q}}\times \mathbf {b} +\mathbf {a} \times {\frac {\partial \mathbf {b} }{\partial q}}.}
A function f of a real number t with values in the space R n {\displaystyle \mathbb {R} ^{n}} can be written as f ( t ) = ( f 1 ( t ) , f 2 ( t ) , … , f n ( t ) ) {\displaystyle \mathbf {f} (t)=(f_{1}(t),f_{2}(t),\ldots ,f_{n}(t))} . Its derivative equals f ′ ( t ) = ( f 1 ′ ( t ) , f 2 ′ ( t ) , … , f n ′ ( t ) ) . {\displaystyle \mathbf {f} '(t)=(f_{1}'(t),f_{2}'(t),\ldots ,f_{n}'(t)).} If f is a function of several variables, say of t ∈ R m {\displaystyle t\in \mathbb {R} ^{m}} , then the partial derivatives of the components of f form a n × m {\displaystyle n\times m} matrix called the Jacobian matrix of f .
If the values of a function f lie in an infinite-dimensional vector space X , such as a Hilbert space , then f may be called an infinite-dimensional vector function .
If the argument of f is a real number and X is a Hilbert space, then the derivative of f at a point t can be defined as in the finite-dimensional case: f ′ ( t ) = lim h → 0 f ( t + h ) − f ( t ) h . {\displaystyle \mathbf {f} '(t)=\lim _{h\to 0}{\frac {\mathbf {f} (t+h)-\mathbf {f} (t)}{h}}.} Most results of the finite-dimensional case also hold in the infinite-dimensional case too, mutatis mutandis . Differentiation can also be defined to functions of several variables (e.g., t ∈ R n {\displaystyle t\in \mathbb {R} ^{n}} or even t ∈ Y {\displaystyle t\in Y} , where Y is an infinite-dimensional vector space).
N.B. If X is a Hilbert space, then one can easily show that any derivative (and any other limit ) can be computed componentwise: if f = ( f 1 , f 2 , f 3 , … ) {\displaystyle \mathbf {f} =(f_{1},f_{2},f_{3},\ldots )} (i.e., f = f 1 e 1 + f 2 e 2 + f 3 e 3 + ⋯ {\displaystyle \mathbf {f} =f_{1}\mathbf {e} _{1}+f_{2}\mathbf {e} _{2}+f_{3}\mathbf {e} _{3}+\cdots } , where e 1 , e 2 , e 3 , … {\displaystyle \mathbf {e} _{1},\mathbf {e} _{2},\mathbf {e} _{3},\ldots } is an orthonormal basis of the space X ), and f ′ ( t ) {\displaystyle f'(t)} exists, then f ′ ( t ) = ( f 1 ′ ( t ) , f 2 ′ ( t ) , f 3 ′ ( t ) , … ) . {\displaystyle \mathbf {f} '(t)=(f_{1}'(t),f_{2}'(t),f_{3}'(t),\ldots ).} However, the existence of a componentwise derivative does not guarantee the existence of a derivative, as componentwise convergence in a Hilbert space does not guarantee convergence with respect to the actual topology of the Hilbert space.
Most of the above hold for other topological vector spaces X too. However, not as many classical results hold in the Banach space setting, e.g., an absolutely continuous function with values in a suitable Banach space need not have a derivative anywhere. Moreover, in most Banach spaces setting there are no orthonormal bases.
In vector calculus and physics , a vector field is an assignment of a vector to each point in a space , most commonly Euclidean space R n {\displaystyle \mathbb {R} ^{n}} . [ 2 ] A vector field on a plane can be visualized as a collection of arrows with given magnitudes and directions, each attached to a point on the plane. Vector fields are often used to model, for example, the speed and direction of a moving fluid throughout three dimensional space , such as the wind , or the strength and direction of some force , such as the magnetic or gravitational force, as it changes from one point to another point.
The elements of differential and integral calculus extend naturally to vector fields. When a vector field represents force , the line integral of a vector field represents the work done by a force moving along a path, and under this interpretation conservation of energy is exhibited as a special case of the fundamental theorem of calculus . Vector fields can usefully be thought of as representing the velocity of a moving flow in space, and this physical intuition leads to notions such as the divergence (which represents the rate of change of volume of a flow) and curl (which represents the rotation of a flow).
A vector field is a special case of a vector-valued function , whose domain's dimension has no relation to the dimension of its range; for example, the position vector of a space curve is defined only for smaller subset of the ambient space.
Likewise, n coordinates , a vector field on a domain in n -dimensional Euclidean space R n {\displaystyle \mathbb {R} ^{n}} can be represented as a vector-valued function that associates an n -tuple of real numbers to each point of the domain. This representation of a vector field depends on the coordinate system, and there is a well-defined transformation law ( covariance and contravariance of vectors ) in passing from one coordinate system to the other.
Vector fields are often discussed on open subsets of Euclidean space, but also make sense on other subsets such as surfaces , where they associate an arrow tangent to the surface at each point (a tangent vector ).
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VectorDB was a database of sequence information for common vectors used in molecular biology [ 1 ]
This Biological database -related article is a stub . You can help Wikipedia by expanding it .
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In molecular cloning , a vector is any particle (e.g., plasmids , cosmids , Lambda phages ) used as a vehicle to artificially carry a foreign nucleic sequence – usually DNA – into another cell , where it can be replicated and/or expressed . [ 1 ] A vector containing foreign DNA is termed recombinant DNA . The four major types of vectors are plasmids , viral vectors , cosmids , and artificial chromosomes . Of these, the most commonly used vectors are plasmids. [ 2 ] Common to all engineered vectors are an origin of replication , a multicloning site , and a selectable marker .
The vector itself generally carries a DNA sequence that consists of an insert (in this case the transgene ) and a larger sequence that serves as the "backbone" of the vector. The purpose of a vector which transfers genetic information to another cell is typically to isolate, multiply, or express the insert in the target cell. All vectors may be used for cloning and are therefore cloning vectors , but there are also vectors designed specially for cloning, while others may be designed specifically for other purposes, such as transcription and protein expression. Vectors designed specifically for the expression of the transgene in the target cell are called expression vectors , and generally have a promoter sequence that drives expression of the transgene. Simpler vectors called transcription vectors are only capable of being transcribed but not translated: they can be replicated in a target cell but not expressed, unlike expression vectors. Transcription vectors are used to amplify their insert.
The manipulation of DNA is normally conducted on E. coli vectors, which contain elements necessary for their maintenance in E. coli . However, vectors may also have elements that allow them to be maintained in another organism such as yeast, plant or mammalian cells, and these vectors are called shuttle vectors . Such vectors have bacterial or viral elements which may be transferred to the non-bacterial host organism, however other vectors termed intragenic vectors have also been developed to avoid the transfer of any genetic material from an alien species. [ 3 ]
Insertion of a vector into the target cell is usually called transformation for bacterial cells, [ 4 ] transfection for eukaryotic cells, [ 5 ] although insertion of a viral vector is often called transduction. [ 6 ]
Plasmids are double-stranded extra chromosomal and generally circular DNA sequences that are capable of replication using the host cell's replication machinery. [ 7 ] Plasmid vectors minimalistically consist of an origin of replication that allows for semi-independent replication of the plasmid in the host. Plasmids are found widely in many bacteria, for example in Escherichia coli , but may also be found in a few eukaryotes, for example in yeast such as Saccharomyces cerevisiae . [ 8 ] Bacterial plasmids may be conjugative/transmissible and non-conjugative:
Plasmids with specially-constructed features are commonly used in laboratory for cloning purposes . These plasmid are generally non-conjugative but may have many more features, notably a " multiple cloning site " where multiple restriction enzyme cleavage sites allow for the insertion of a transgene insert. The bacteria containing the plasmids can generate millions of copies of the vector within the bacteria in hours, and the amplified vectors can be extracted from the bacteria for further manipulation. Plasmids may be used specifically as transcription vectors and such plasmids may lack crucial sequences for protein expression. Plasmids used for protein expression, called expression vectors , would include elements for translation of protein, such as a ribosome binding site , start and stop codons .
Viral vectors are genetically engineered viruses carrying modified viral DNA or RNA that has been rendered noninfectious, but still contain viral promoters and the transgene, thus allowing for translation of the transgene through a viral promoter. However, because viral vectors frequently lack infectious sequences, they require helper viruses or packaging lines for large-scale transfection. Viral vectors are often designed to permanently incorporate the insert into the host genome, and thus leave distinct genetic markers in the host genome after incorporating the transgene. For example, retroviruses leaves a characteristic retroviral integration pattern after insertion that is detectable and indicates that the viral vector has incorporated into the host genome.
Artificial chromosomes are manufactured chromosomes in the context of yeast artificial chromosomes (YACs), bacterial artificial chromosomes (BACs), or human artificial chromosomes (HACs). An artificial chromosome can carry a much larger DNA fragment than other vectors. [ 9 ] YACs and BACs can carry a DNA fragment up to 300,000 nucleotides long. Three structural necessities of an artificial chromosome include an origin of replication, a centromere, and telomeric end sequences. [ 10 ]
Transcription of the cloned gene is a necessary component of the vector when expression of the gene is required: one gene may be amplified through transcription to generate multiple copies of mRNAs , the template on which protein may be produced through translation. [ 11 ] A larger number of mRNAs would express a greater amount of protein, and how many copies of mRNA are generated depends on the promoter used in the vector. [ 12 ] The expression may be constitutive, meaning that the protein is produced constantly in the background, or it may be inducible whereby the protein is expressed only under certain condition, for example when a chemical inducer is added. These two different types of expression depend on the types of promoter and operator used.
Viral promoters are often used for constitutive expression in plasmids and in viral vectors because they normally force constant transcription in many cell lines and types reliably. [ 13 ] Inducible expression depends on promoters that respond to the induction conditions: for example, the murine mammary tumor virus promoter only initiates transcription after dexamethasone application and the Drosophila heat shock promoter only initiates after high temperatures.
Some vectors are designed for transcription only, for example for in vitro mRNA production. These vectors are called transcription vectors. They may lack the sequences necessary for polyadenylation and termination, therefore may not be used for protein production.
Expression vectors produce proteins through the transcription of the vector's insert followed by translation of the mRNA produced, they therefore require more components than the simpler transcription-only vectors. Expression in different host organism would require different elements, although they share similar requirements, for example a promoter for initiation of transcription, a ribosomal binding site for translation initiation, and termination signals.
Eukaryote expression vectors require sequences that encode for:
Modern artificially-constructed vectors contain essential components found in all vectors, and may contain other additional features found only in some vectors:
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https://en.wikipedia.org/wiki/Vector_(molecular_biology)
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In mathematics , the Laplace operator or Laplacian is a differential operator given by the divergence of the gradient of a scalar function on Euclidean space . It is usually denoted by the symbols ∇ ⋅ ∇ {\displaystyle \nabla \cdot \nabla } , ∇ 2 {\displaystyle \nabla ^{2}} (where ∇ {\displaystyle \nabla } is the nabla operator ), or Δ {\displaystyle \Delta } . In a Cartesian coordinate system , the Laplacian is given by the sum of second partial derivatives of the function with respect to each independent variable . In other coordinate systems , such as cylindrical and spherical coordinates , the Laplacian also has a useful form. Informally, the Laplacian Δ f ( p ) of a function f at a point p measures by how much the average value of f over small spheres or balls centered at p deviates from f ( p ) .
The Laplace operator is named after the French mathematician Pierre-Simon de Laplace (1749–1827), who first applied the operator to the study of celestial mechanics : the Laplacian of the gravitational potential due to a given mass density distribution is a constant multiple of that density distribution. Solutions of Laplace's equation Δ f = 0 are called harmonic functions and represent the possible gravitational potentials in regions of vacuum .
The Laplacian occurs in many differential equations describing physical phenomena. Poisson's equation describes electric and gravitational potentials ; the diffusion equation describes heat and fluid flow ; the wave equation describes wave propagation ; and the Schrödinger equation describes the wave function in quantum mechanics . In image processing and computer vision , the Laplacian operator has been used for various tasks, such as blob and edge detection . The Laplacian is the simplest elliptic operator and is at the core of Hodge theory as well as the results of de Rham cohomology .
The Laplace operator is a second-order differential operator in the n -dimensional Euclidean space , defined as the divergence ( ∇ ⋅ {\displaystyle \nabla \cdot } ) of the gradient ( ∇ f {\displaystyle \nabla f} ). Thus if f {\displaystyle f} is a twice-differentiable real-valued function , then the Laplacian of f {\displaystyle f} is the real-valued function defined by:
where the latter notations derive from formally writing: ∇ = ( ∂ ∂ x 1 , … , ∂ ∂ x n ) . {\displaystyle \nabla =\left({\frac {\partial }{\partial x_{1}}},\ldots ,{\frac {\partial }{\partial x_{n}}}\right).} Explicitly, the Laplacian of f is thus the sum of all the unmixed second partial derivatives in the Cartesian coordinates x i :
As a second-order differential operator, the Laplace operator maps C k functions to C k −2 functions for k ≥ 2 . It is a linear operator Δ : C k ( R n ) → C k −2 ( R n ) , or more generally, an operator Δ : C k (Ω) → C k −2 (Ω) for any open set Ω ⊆ R n .
Alternatively, the Laplace operator can be defined as:
∇ 2 f ( x → ) = lim R → 0 2 n R 2 ( f s h e l l R − f ( x → ) ) = lim R → 0 2 n A n − 1 R 1 + n ∫ s h e l l R f ( r → ) − f ( x → ) d r n − 1 {\displaystyle \nabla ^{2}f({\vec {x}})=\lim _{R\rightarrow 0}{\frac {2n}{R^{2}}}(f_{shell_{R}}-f({\vec {x}}))=\lim _{R\rightarrow 0}{\frac {2n}{A_{n-1}R^{1+n}}}\int _{shell_{R}}f({\vec {r}})-f({\vec {x}})dr^{n-1}}
where n {\displaystyle n} is the dimension of the space, f s h e l l R {\displaystyle f_{shell_{R}}} is the average value of f {\displaystyle f} on the surface of an n-sphere of radius R {\displaystyle R} , ∫ s h e l l R f ( r → ) d r n − 1 {\displaystyle \int _{shell_{R}}f({\vec {r}})dr^{n-1}} is the surface integral over an n-sphere of radius R {\displaystyle R} , and A n − 1 {\displaystyle A_{n-1}} is the hypervolume of the boundary of a unit n-sphere . [ 1 ]
There are two conflicting conventions as to how the Laplace operator is defined:
Δ = ∇ 2 = ∑ j = 1 n ( ∂ ∂ x j ) 2 , {\displaystyle \Delta =\nabla ^{2}=\sum _{j=1}^{n}{\Big (}{\frac {\partial }{\partial x_{j}}}{\Big )}^{2},} which is negative-definite in the sense that ∫ R n φ ( x ) ¯ Δ φ ( x ) d x = − ∫ R n | ∇ φ ( x ) | 2 d x < 0 {\displaystyle \int _{\mathbb {R} ^{n}}{\overline {\varphi (x)}}\Delta \varphi (x)\,dx=-\int _{\mathbb {R} ^{n}}|\nabla \varphi (x)|^{2}\,dx<0} for any smooth compactly supported function φ ∈ C c ∞ ( R n ) {\displaystyle \varphi \in C_{c}^{\infty }(\mathbb {R} ^{n})} which is not identically zero);
Δ = − ∇ 2 = − ∑ j = 1 n ( ∂ ∂ x j ) 2 . {\displaystyle \Delta =-\nabla ^{2}=-\sum _{j=1}^{n}{\Big (}{\frac {\partial }{\partial x_{j}}}{\Big )}^{2}.}
In the physical theory of diffusion , the Laplace operator arises naturally in the mathematical description of equilibrium . [ 2 ] Specifically, if u is the density at equilibrium of some quantity such as a chemical concentration, then the net flux of u through the boundary ∂ V (also called S ) of any smooth region V is zero, provided there is no source or sink within V : ∫ S ∇ u ⋅ n d S = 0 , {\displaystyle \int _{S}\nabla u\cdot \mathbf {n} \,dS=0,} where n is the outward unit normal to the boundary of V . By the divergence theorem , ∫ V div ∇ u d V = ∫ S ∇ u ⋅ n d S = 0. {\displaystyle \int _{V}\operatorname {div} \nabla u\,dV=\int _{S}\nabla u\cdot \mathbf {n} \,dS=0.}
Since this holds for all smooth regions V , one can show that it implies: div ∇ u = Δ u = 0. {\displaystyle \operatorname {div} \nabla u=\Delta u=0.} The left-hand side of this equation is the Laplace operator, and the entire equation Δ u = 0 is known as Laplace's equation . Solutions of the Laplace equation, i.e. functions whose Laplacian is identically zero, thus represent possible equilibrium densities under diffusion.
The Laplace operator itself has a physical interpretation for non-equilibrium diffusion as the extent to which a point represents a source or sink of chemical concentration, in a sense made precise by the diffusion equation . This interpretation of the Laplacian is also explained by the following fact about averages.
Given a twice continuously differentiable function f : R n → R {\displaystyle f:\mathbb {R} ^{n}\to \mathbb {R} } and a point p ∈ R n {\displaystyle p\in \mathbb {R} ^{n}} , the average value of f {\displaystyle f} over the ball with radius h {\displaystyle h} centered at p {\displaystyle p} is: [ 3 ] f ¯ B ( p , h ) = f ( p ) + Δ f ( p ) 2 ( n + 2 ) h 2 + o ( h 2 ) for h → 0 {\displaystyle {\overline {f}}_{B}(p,h)=f(p)+{\frac {\Delta f(p)}{2(n+2)}}h^{2}+o(h^{2})\quad {\text{for}}\;\;h\to 0}
Similarly, the average value of f {\displaystyle f} over the sphere (the boundary of a ball) with radius h {\displaystyle h} centered at p {\displaystyle p} is: f ¯ S ( p , h ) = f ( p ) + Δ f ( p ) 2 n h 2 + o ( h 2 ) for h → 0. {\displaystyle {\overline {f}}_{S}(p,h)=f(p)+{\frac {\Delta f(p)}{2n}}h^{2}+o(h^{2})\quad {\text{for}}\;\;h\to 0.}
If φ denotes the electrostatic potential associated to a charge distribution q , then the charge distribution itself is given by the negative of the Laplacian of φ : q = − ε 0 Δ φ , {\displaystyle q=-\varepsilon _{0}\Delta \varphi ,} where ε 0 is the electric constant .
This is a consequence of Gauss's law . Indeed, if V is any smooth region with boundary ∂ V , then by Gauss's law the flux of the electrostatic field E across the boundary is proportional to the charge enclosed: ∫ ∂ V E ⋅ n d S = ∫ V div E d V = 1 ε 0 ∫ V q d V . {\displaystyle \int _{\partial V}\mathbf {E} \cdot \mathbf {n} \,dS=\int _{V}\operatorname {div} \mathbf {E} \,dV={\frac {1}{\varepsilon _{0}}}\int _{V}q\,dV.} where the first equality is due to the divergence theorem . Since the electrostatic field is the (negative) gradient of the potential, this gives: − ∫ V div ( grad φ ) d V = 1 ε 0 ∫ V q d V . {\displaystyle -\int _{V}\operatorname {div} (\operatorname {grad} \varphi )\,dV={\frac {1}{\varepsilon _{0}}}\int _{V}q\,dV.}
Since this holds for all regions V , we must have div ( grad φ ) = − 1 ε 0 q {\displaystyle \operatorname {div} (\operatorname {grad} \varphi )=-{\frac {1}{\varepsilon _{0}}}q}
The same approach implies that the negative of the Laplacian of the gravitational potential is the mass distribution . Often the charge (or mass) distribution are given, and the associated potential is unknown. Finding the potential function subject to suitable boundary conditions is equivalent to solving Poisson's equation .
Another motivation for the Laplacian appearing in physics is that solutions to Δ f = 0 in a region U are functions that make the Dirichlet energy functional stationary : E ( f ) = 1 2 ∫ U ‖ ∇ f ‖ 2 d x . {\displaystyle E(f)={\frac {1}{2}}\int _{U}\lVert \nabla f\rVert ^{2}\,dx.}
To see this, suppose f : U → R is a function, and u : U → R is a function that vanishes on the boundary of U . Then: d d ε | ε = 0 E ( f + ε u ) = ∫ U ∇ f ⋅ ∇ u d x = − ∫ U u Δ f d x {\displaystyle \left.{\frac {d}{d\varepsilon }}\right|_{\varepsilon =0}E(f+\varepsilon u)=\int _{U}\nabla f\cdot \nabla u\,dx=-\int _{U}u\,\Delta f\,dx}
where the last equality follows using Green's first identity . This calculation shows that if Δ f = 0 , then E is stationary around f . Conversely, if E is stationary around f , then Δ f = 0 by the fundamental lemma of calculus of variations .
The Laplace operator in two dimensions is given by:
In Cartesian coordinates , Δ f = ∂ 2 f ∂ x 2 + ∂ 2 f ∂ y 2 {\displaystyle \Delta f={\frac {\partial ^{2}f}{\partial x^{2}}}+{\frac {\partial ^{2}f}{\partial y^{2}}}} where x and y are the standard Cartesian coordinates of the xy -plane.
In polar coordinates , Δ f = 1 r ∂ ∂ r ( r ∂ f ∂ r ) + 1 r 2 ∂ 2 f ∂ θ 2 = ∂ 2 f ∂ r 2 + 1 r ∂ f ∂ r + 1 r 2 ∂ 2 f ∂ θ 2 , {\displaystyle {\begin{aligned}\Delta f&={\frac {1}{r}}{\frac {\partial }{\partial r}}\left(r{\frac {\partial f}{\partial r}}\right)+{\frac {1}{r^{2}}}{\frac {\partial ^{2}f}{\partial \theta ^{2}}}\\&={\frac {\partial ^{2}f}{\partial r^{2}}}+{\frac {1}{r}}{\frac {\partial f}{\partial r}}+{\frac {1}{r^{2}}}{\frac {\partial ^{2}f}{\partial \theta ^{2}}},\end{aligned}}} where r represents the radial distance and θ the angle.
In three dimensions, it is common to work with the Laplacian in a variety of different coordinate systems.
In Cartesian coordinates , Δ f = ∂ 2 f ∂ x 2 + ∂ 2 f ∂ y 2 + ∂ 2 f ∂ z 2 . {\displaystyle \Delta f={\frac {\partial ^{2}f}{\partial x^{2}}}+{\frac {\partial ^{2}f}{\partial y^{2}}}+{\frac {\partial ^{2}f}{\partial z^{2}}}.}
In cylindrical coordinates , Δ f = 1 ρ ∂ ∂ ρ ( ρ ∂ f ∂ ρ ) + 1 ρ 2 ∂ 2 f ∂ φ 2 + ∂ 2 f ∂ z 2 , {\displaystyle \Delta f={\frac {1}{\rho }}{\frac {\partial }{\partial \rho }}\left(\rho {\frac {\partial f}{\partial \rho }}\right)+{\frac {1}{\rho ^{2}}}{\frac {\partial ^{2}f}{\partial \varphi ^{2}}}+{\frac {\partial ^{2}f}{\partial z^{2}}},} where ρ {\displaystyle \rho } represents the radial distance, φ the azimuth angle and z the height.
In spherical coordinates : Δ f = 1 r 2 ∂ ∂ r ( r 2 ∂ f ∂ r ) + 1 r 2 sin θ ∂ ∂ θ ( sin θ ∂ f ∂ θ ) + 1 r 2 sin 2 θ ∂ 2 f ∂ φ 2 , {\displaystyle \Delta f={\frac {1}{r^{2}}}{\frac {\partial }{\partial r}}\left(r^{2}{\frac {\partial f}{\partial r}}\right)+{\frac {1}{r^{2}\sin \theta }}{\frac {\partial }{\partial \theta }}\left(\sin \theta {\frac {\partial f}{\partial \theta }}\right)+{\frac {1}{r^{2}\sin ^{2}\theta }}{\frac {\partial ^{2}f}{\partial \varphi ^{2}}},} or Δ f = 1 r ∂ 2 ∂ r 2 ( r f ) + 1 r 2 sin θ ∂ ∂ θ ( sin θ ∂ f ∂ θ ) + 1 r 2 sin 2 θ ∂ 2 f ∂ φ 2 , {\displaystyle \Delta f={\frac {1}{r}}{\frac {\partial ^{2}}{\partial r^{2}}}(rf)+{\frac {1}{r^{2}\sin \theta }}{\frac {\partial }{\partial \theta }}\left(\sin \theta {\frac {\partial f}{\partial \theta }}\right)+{\frac {1}{r^{2}\sin ^{2}\theta }}{\frac {\partial ^{2}f}{\partial \varphi ^{2}}},} by expanding the first and second term, these expressions read Δ f = ∂ 2 f ∂ r 2 + 2 r ∂ f ∂ r + 1 r 2 sin θ ( cos θ ∂ f ∂ θ + sin θ ∂ 2 f ∂ θ 2 ) + 1 r 2 sin 2 θ ∂ 2 f ∂ φ 2 , {\displaystyle \Delta f={\frac {\partial ^{2}f}{\partial r^{2}}}+{\frac {2}{r}}{\frac {\partial f}{\partial r}}+{\frac {1}{r^{2}\sin \theta }}\left(\cos \theta {\frac {\partial f}{\partial \theta }}+\sin \theta {\frac {\partial ^{2}f}{\partial \theta ^{2}}}\right)+{\frac {1}{r^{2}\sin ^{2}\theta }}{\frac {\partial ^{2}f}{\partial \varphi ^{2}}},} where φ represents the azimuthal angle and θ the zenith angle or co-latitude . In particular, the above is equivalent to
Δ f = ∂ 2 f ∂ r 2 + 2 r ∂ f ∂ r + 1 r 2 Δ S 2 f , {\displaystyle \Delta f={\frac {\partial ^{2}f}{\partial r^{2}}}+{\frac {2}{r}}{\frac {\partial f}{\partial r}}+{\frac {1}{r^{2}}}\Delta _{S^{2}}f,}
where Δ S 2 f {\displaystyle \Delta _{S^{2}}f} is the Laplace-Beltrami operator on the unit sphere.
In general curvilinear coordinates ( ξ 1 , ξ 2 , ξ 3 ): Δ = ∇ ξ m ⋅ ∇ ξ n ∂ 2 ∂ ξ m ∂ ξ n + ∇ 2 ξ m ∂ ∂ ξ m = g m n ( ∂ 2 ∂ ξ m ∂ ξ n − Γ m n l ∂ ∂ ξ l ) , {\displaystyle \Delta =\nabla \xi ^{m}\cdot \nabla \xi ^{n}{\frac {\partial ^{2}}{\partial \xi ^{m}\,\partial \xi ^{n}}}+\nabla ^{2}\xi ^{m}{\frac {\partial }{\partial \xi ^{m}}}=g^{mn}\left({\frac {\partial ^{2}}{\partial \xi ^{m}\,\partial \xi ^{n}}}-\Gamma _{mn}^{l}{\frac {\partial }{\partial \xi ^{l}}}\right),}
where summation over the repeated indices is implied , g mn is the inverse metric tensor and Γ l mn are the Christoffel symbols for the selected coordinates.
In arbitrary curvilinear coordinates in N dimensions ( ξ 1 , ..., ξ N ), we can write the Laplacian in terms of the inverse metric tensor , g i j {\displaystyle g^{ij}} : Δ = 1 det g ∂ ∂ ξ i ( det g g i j ∂ ∂ ξ j ) , {\displaystyle \Delta ={\frac {1}{\sqrt {\det g}}}{\frac {\partial }{\partial \xi ^{i}}}\left({\sqrt {\det g}}\,g^{ij}{\frac {\partial }{\partial \xi ^{j}}}\right),} from the Voss - Weyl formula [ 4 ] for the divergence .
In spherical coordinates in N dimensions , with the parametrization x = rθ ∈ R N with r representing a positive real radius and θ an element of the unit sphere S N −1 , Δ f = ∂ 2 f ∂ r 2 + N − 1 r ∂ f ∂ r + 1 r 2 Δ S N − 1 f {\displaystyle \Delta f={\frac {\partial ^{2}f}{\partial r^{2}}}+{\frac {N-1}{r}}{\frac {\partial f}{\partial r}}+{\frac {1}{r^{2}}}\Delta _{S^{N-1}}f} where Δ S N −1 is the Laplace–Beltrami operator on the ( N − 1) -sphere, known as the spherical Laplacian. The two radial derivative terms can be equivalently rewritten as: 1 r N − 1 ∂ ∂ r ( r N − 1 ∂ f ∂ r ) . {\displaystyle {\frac {1}{r^{N-1}}}{\frac {\partial }{\partial r}}\left(r^{N-1}{\frac {\partial f}{\partial r}}\right).}
As a consequence, the spherical Laplacian of a function defined on S N −1 ⊂ R N can be computed as the ordinary Laplacian of the function extended to R N ∖{0} so that it is constant along rays, i.e., homogeneous of degree zero.
The Laplacian is invariant under all Euclidean transformations : rotations and translations . In two dimensions, for example, this means that: Δ ( f ( x cos θ − y sin θ + a , x sin θ + y cos θ + b ) ) = ( Δ f ) ( x cos θ − y sin θ + a , x sin θ + y cos θ + b ) {\displaystyle \Delta (f(x\cos \theta -y\sin \theta +a,x\sin \theta +y\cos \theta +b))=(\Delta f)(x\cos \theta -y\sin \theta +a,x\sin \theta +y\cos \theta +b)} for all θ , a , and b . In arbitrary dimensions, Δ ( f ∘ ρ ) = ( Δ f ) ∘ ρ {\displaystyle \Delta (f\circ \rho )=(\Delta f)\circ \rho } whenever ρ is a rotation, and likewise: Δ ( f ∘ τ ) = ( Δ f ) ∘ τ {\displaystyle \Delta (f\circ \tau )=(\Delta f)\circ \tau } whenever τ is a translation. (More generally, this remains true when ρ is an orthogonal transformation such as a reflection .)
In fact, the algebra of all scalar linear differential operators, with constant coefficients, that commute with all Euclidean transformations, is the polynomial algebra generated by the Laplace operator.
The spectrum of the Laplace operator consists of all eigenvalues λ for which there is a corresponding eigenfunction f with: − Δ f = λ f . {\displaystyle -\Delta f=\lambda f.}
This is known as the Helmholtz equation .
If Ω is a bounded domain in R n , then the eigenfunctions of the Laplacian are an orthonormal basis for the Hilbert space L 2 (Ω) . This result essentially follows from the spectral theorem on compact self-adjoint operators , applied to the inverse of the Laplacian (which is compact, by the Poincaré inequality and the Rellich–Kondrachov theorem ). [ 5 ] It can also be shown that the eigenfunctions are infinitely differentiable functions. [ 6 ] More generally, these results hold for the Laplace–Beltrami operator on any compact Riemannian manifold with boundary, or indeed for the Dirichlet eigenvalue problem of any elliptic operator with smooth coefficients on a bounded domain. When Ω is the n -sphere , the eigenfunctions of the Laplacian are the spherical harmonics .
The vector Laplace operator , also denoted by ∇ 2 {\displaystyle \nabla ^{2}} , is a differential operator defined over a vector field . [ 7 ] The vector Laplacian is similar to the scalar Laplacian; whereas the scalar Laplacian applies to a scalar field and returns a scalar quantity, the vector Laplacian applies to a vector field , returning a vector quantity. When computed in orthonormal Cartesian coordinates , the returned vector field is equal to the vector field of the scalar Laplacian applied to each vector component.
The vector Laplacian of a vector field A {\displaystyle \mathbf {A} } is defined as ∇ 2 A = ∇ ( ∇ ⋅ A ) − ∇ × ( ∇ × A ) . {\displaystyle \nabla ^{2}\mathbf {A} =\nabla (\nabla \cdot \mathbf {A} )-\nabla \times (\nabla \times \mathbf {A} ).} This definition can be seen as the Helmholtz decomposition of the vector Laplacian.
In Cartesian coordinates , this reduces to the much simpler expression ∇ 2 A = ( ∇ 2 A x , ∇ 2 A y , ∇ 2 A z ) , {\displaystyle \nabla ^{2}\mathbf {A} =(\nabla ^{2}A_{x},\nabla ^{2}A_{y},\nabla ^{2}A_{z}),} where A x {\displaystyle A_{x}} , A y {\displaystyle A_{y}} , and A z {\displaystyle A_{z}} are the components of the vector field A {\displaystyle \mathbf {A} } , and ∇ 2 {\displaystyle \nabla ^{2}} just on the left of each vector field component is the (scalar) Laplace operator. This can be seen to be a special case of Lagrange's formula; see Vector triple product .
For expressions of the vector Laplacian in other coordinate systems see Del in cylindrical and spherical coordinates .
The Laplacian of any tensor field T {\displaystyle \mathbf {T} } ("tensor" includes scalar and vector) is defined as the divergence of the gradient of the tensor: ∇ 2 T = ( ∇ ⋅ ∇ ) T . {\displaystyle \nabla ^{2}\mathbf {T} =(\nabla \cdot \nabla )\mathbf {T} .}
For the special case where T {\displaystyle \mathbf {T} } is a scalar (a tensor of degree zero), the Laplacian takes on the familiar form.
If T {\displaystyle \mathbf {T} } is a vector (a tensor of first degree), the gradient is a covariant derivative which results in a tensor of second degree, and the divergence of this is again a vector. The formula for the vector Laplacian above may be used to avoid tensor math and may be shown to be equivalent to the divergence of the Jacobian matrix shown below for the gradient of a vector: ∇ T = ( ∇ T x , ∇ T y , ∇ T z ) = [ T x x T x y T x z T y x T y y T y z T z x T z y T z z ] , where T u v ≡ ∂ T u ∂ v . {\displaystyle \nabla \mathbf {T} =(\nabla T_{x},\nabla T_{y},\nabla T_{z})={\begin{bmatrix}T_{xx}&T_{xy}&T_{xz}\\T_{yx}&T_{yy}&T_{yz}\\T_{zx}&T_{zy}&T_{zz}\end{bmatrix}},{\text{ where }}T_{uv}\equiv {\frac {\partial T_{u}}{\partial v}}.}
And, in the same manner, a dot product , which evaluates to a vector, of a vector by the gradient of another vector (a tensor of 2nd degree) can be seen as a product of matrices: A ⋅ ∇ B = [ A x A y A z ] ∇ B = [ A ⋅ ∇ B x A ⋅ ∇ B y A ⋅ ∇ B z ] . {\displaystyle \mathbf {A} \cdot \nabla \mathbf {B} ={\begin{bmatrix}A_{x}&A_{y}&A_{z}\end{bmatrix}}\nabla \mathbf {B} ={\begin{bmatrix}\mathbf {A} \cdot \nabla B_{x}&\mathbf {A} \cdot \nabla B_{y}&\mathbf {A} \cdot \nabla B_{z}\end{bmatrix}}.} This identity is a coordinate dependent result, and is not general.
An example of the usage of the vector Laplacian is the Navier-Stokes equations for a Newtonian incompressible flow : ρ ( ∂ v ∂ t + ( v ⋅ ∇ ) v ) = ρ f − ∇ p + μ ( ∇ 2 v ) , {\displaystyle \rho \left({\frac {\partial \mathbf {v} }{\partial t}}+(\mathbf {v} \cdot \nabla )\mathbf {v} \right)=\rho \mathbf {f} -\nabla p+\mu \left(\nabla ^{2}\mathbf {v} \right),} where the term with the vector Laplacian of the velocity field μ ( ∇ 2 v ) {\displaystyle \mu \left(\nabla ^{2}\mathbf {v} \right)} represents the viscous stresses in the fluid.
Another example is the wave equation for the electric field that can be derived from Maxwell's equations in the absence of charges and currents: ∇ 2 E − μ 0 ϵ 0 ∂ 2 E ∂ t 2 = 0. {\displaystyle \nabla ^{2}\mathbf {E} -\mu _{0}\epsilon _{0}{\frac {\partial ^{2}\mathbf {E} }{\partial t^{2}}}=0.}
This equation can also be written as: ◻ E = 0 , {\displaystyle \Box \,\mathbf {E} =0,} where ◻ ≡ 1 c 2 ∂ 2 ∂ t 2 − ∇ 2 , {\displaystyle \Box \equiv {\frac {1}{c^{2}}}{\frac {\partial ^{2}}{\partial t^{2}}}-\nabla ^{2},} is the D'Alembertian , used in the Klein–Gordon equation .
First of all, we say that a smooth function u : Ω ⊂ R N → R {\displaystyle u\colon \Omega \subset \mathbb {R} ^{N}\to \mathbb {R} } is superharmonic whenever − Δ u ≥ 0 {\displaystyle -\Delta u\geq 0} .
Let u : Ω → R {\displaystyle u\colon \Omega \to \mathbb {R} } be a smooth function, and let K ⊂ Ω {\displaystyle K\subset \Omega } be a connected compact set. If u {\displaystyle u} is superharmonic, then, for every x ∈ K {\displaystyle x\in K} , we have u ( x ) ≥ inf Ω u + c ‖ u ‖ L 1 ( K ) , {\displaystyle u(x)\geq \inf _{\Omega }u+c\lVert u\rVert _{L^{1}(K)}\;,} for some constant c > 0 {\displaystyle c>0} depending on Ω {\displaystyle \Omega } and K {\displaystyle K} . [ 8 ]
A version of the Laplacian can be defined wherever the Dirichlet energy functional makes sense, which is the theory of Dirichlet forms . For spaces with additional structure, one can give more explicit descriptions of the Laplacian, as follows.
The Laplacian also can be generalized to an elliptic operator called the Laplace–Beltrami operator defined on a Riemannian manifold . The Laplace–Beltrami operator, when applied to a function, is the trace ( tr ) of the function's Hessian : Δ f = tr ( H ( f ) ) {\displaystyle \Delta f=\operatorname {tr} {\big (}H(f){\big )}} where the trace is taken with respect to the inverse of the metric tensor . The Laplace–Beltrami operator also can be generalized to an operator (also called the Laplace–Beltrami operator) which operates on tensor fields , by a similar formula.
Another generalization of the Laplace operator that is available on pseudo-Riemannian manifolds uses the exterior derivative , in terms of which the "geometer's Laplacian" is expressed as Δ f = δ d f . {\displaystyle \Delta f=\delta df.}
Here δ is the codifferential , which can also be expressed in terms of the Hodge star and the exterior derivative. This operator differs in sign from the "analyst's Laplacian" defined above. More generally, the "Hodge" Laplacian is defined on differential forms α by Δ α = δ d α + d δ α . {\displaystyle \Delta \alpha =\delta d\alpha +d\delta \alpha .}
This is known as the Laplace–de Rham operator , which is related to the Laplace–Beltrami operator by the Weitzenböck identity .
The Laplacian can be generalized in certain ways to non-Euclidean spaces, where it may be elliptic , hyperbolic , or ultrahyperbolic .
In Minkowski space the Laplace–Beltrami operator becomes the D'Alembert operator ◻ {\displaystyle \Box } or D'Alembertian: ◻ = 1 c 2 ∂ 2 ∂ t 2 − ∂ 2 ∂ x 2 − ∂ 2 ∂ y 2 − ∂ 2 ∂ z 2 . {\displaystyle \square ={\frac {1}{c^{2}}}{\frac {\partial ^{2}}{\partial t^{2}}}-{\frac {\partial ^{2}}{\partial x^{2}}}-{\frac {\partial ^{2}}{\partial y^{2}}}-{\frac {\partial ^{2}}{\partial z^{2}}}.}
It is the generalization of the Laplace operator in the sense that it is the differential operator which is invariant under the isometry group of the underlying space and it reduces to the Laplace operator if restricted to time-independent functions. The overall sign of the metric here is chosen such that the spatial parts of the operator admit a negative sign, which is the usual convention in high-energy particle physics . The D'Alembert operator is also known as the wave operator because it is the differential operator appearing in the wave equations , and it is also part of the Klein–Gordon equation , which reduces to the wave equation in the massless case.
The additional factor of c in the metric is needed in physics if space and time are measured in different units; a similar factor would be required if, for example, the x direction were measured in meters while the y direction were measured in centimeters. Indeed, theoretical physicists usually work in units such that c = 1 in order to simplify the equation.
The d'Alembert operator generalizes to a hyperbolic operator on pseudo-Riemannian manifolds .
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Vector Product Format (VPF) is a military standard for vector -based digital map products produced by the U.S. Department of Defense (DOD). It has been adopted as part of the Digital Geographic Exchange Standard (DIGEST) in the form of Vector Relational Format (VRF), so VPF can be considered to be an international standard as well. [ 1 ]
This computing article is a stub . You can help Wikipedia by expanding it .
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The following are important identities in vector algebra . Identities that only involve the magnitude of a vector ‖ A ‖ {\displaystyle \|\mathbf {A} \|} and the dot product (scalar product) of two vectors A · B , apply to vectors in any dimension, while identities that use the cross product (vector product) A × B only apply in three dimensions, since the cross product is only defined there. [ nb 1 ] [ 1 ] Most of these relations can be dated to founder of vector calculus Josiah Willard Gibbs , if not earlier. [ 2 ]
The magnitude of a vector A can be expressed using the dot product:
In three-dimensional Euclidean space , the magnitude of a vector is determined from its three components using Pythagoras' theorem :
The vector product and the scalar product of two vectors define the angle between them, say θ : [ 1 ] [ 3 ]
To satisfy the right-hand rule , for positive θ , vector B is counter-clockwise from A , and for negative θ it is clockwise.
The Pythagorean trigonometric identity then provides:
If a vector A = ( A x , A y , A z ) makes angles α , β , γ with an orthogonal set of x- , y- and z- axes, then:
and analogously for angles β, γ. Consequently:
with i ^ , j ^ , k ^ {\displaystyle {\hat {\mathbf {i} }},\ {\hat {\mathbf {j} }},\ {\hat {\mathbf {k} }}} unit vectors along the axis directions.
The area Σ of a parallelogram with sides A and B containing the angle θ is:
which will be recognized as the magnitude of the vector cross product of the vectors A and B lying along the sides of the parallelogram. That is:
(If A , B are two-dimensional vectors, this is equal to the determinant of the 2 × 2 matrix with rows A , B .) The square of this expression is: [ 4 ]
where Γ( A , B ) is the Gram determinant of A and B defined by:
In a similar fashion, the squared volume V of a parallelepiped spanned by the three vectors A , B , C is given by the Gram determinant of the three vectors: [ 4 ]
Since A , B, C are three-dimensional vectors, this is equal to the square of the scalar triple product det [ A , B , C ] = | A , B , C | {\displaystyle \det[\mathbf {A} ,\mathbf {B} ,\mathbf {C} ]=|\mathbf {A} ,\mathbf {B} ,\mathbf {C} |} below.
This process can be extended to n -dimensions.
The name "quadruple product" is used for two different products, [ 5 ] the scalar-valued scalar quadruple product and the vector-valued vector quadruple product or vector product of four vectors .
The scalar quadruple product is defined as the dot product of two cross products :
where a, b, c, d are vectors in three-dimensional Euclidean space. [ 6 ] It can be evaluated using the Binet-Cauchy identity : [ 6 ]
or using the determinant :
The vector quadruple product is defined as the cross product of two cross products:
where a, b, c, d are vectors in three-dimensional Euclidean space. [ 2 ] It can be evaluated using the identity: [ 7 ]
Equivalent forms can be obtained using the identity: [ 8 ] [ 9 ] [ 10 ]
This identity can also be written using tensor notation and the Einstein summation convention as follows:
where ε ijk is the Levi-Civita symbol .
Related relationships:
These relations are useful for deriving various formulas in spherical and Euclidean geometry. For example, if four points are chosen on the unit sphere, A, B, C, D , and unit vectors drawn from the center of the sphere to the four points, a, b, c, d respectively, the identity:
in conjunction with the relation for the magnitude of the cross product:
and the dot product:
where a = b = 1 for the unit sphere, results in the identity among the angles attributed to Gauss:
where x is the angle between a × b and c × d , or equivalently, between the planes defined by these vectors. [ 2 ]
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In particle physics , a vector boson is a boson whose spin equals one. Vector bosons that are also elementary particles are gauge bosons , the force carriers of fundamental interactions . Some composite particles are vector bosons, for instance any vector meson ( quark and antiquark ). During the 1970s and 1980s, intermediate vector bosons (the W and Z bosons, which mediate the weak interaction) drew much attention in particle physics . [ 1 ] [ 2 ]
A pseudovector boson is a vector boson that has even parity , whereas "regular" vector bosons have odd parity. There are no fundamental pseudovector bosons, but there are pseudovector mesons . [ 3 ]
The W and Z particles interact with the Higgs boson as shown in the Feynman diagram . [ 4 ]
The name vector boson arises from quantum field theory . The component of such a particle's spin along any axis has the three eigenvalues − ħ , 0, and + ħ (where ħ is the reduced Planck constant ), meaning that any measurement of its spin can only yield one of these values. (This is true for massive vector bosons; the situation differs for massless particles such as the photon, for reasons beyond the scope of this article. See Wigner's classification . [ 5 ] )
The space of spin states therefore is a discrete degree of freedom consisting of three states, the same as the number of components of a vector in three-dimensional space. Quantum superpositions of these states can be taken such that they transform under rotations just like the spatial components of a rotating vector [ 6 ] (the so-called 3 representation of SU(2) ). If the vector boson is taken to be the quantum of a field, the field is a vector field , hence the name.
The boson part of the name arises from the spin-statistics relation , which requires that all integer spin particles be bosons. [ 7 ]
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The following are important identities involving derivatives and integrals in vector calculus .
For a function f ( x , y , z ) {\displaystyle f(x,y,z)} in three-dimensional Cartesian coordinate variables, the gradient is the vector field:
where i , j , k are the standard unit vectors for the x , y , z -axes. More generally, for a function of n variables ψ ( x 1 , … , x n ) {\displaystyle \psi (x_{1},\ldots ,x_{n})} , also called a scalar field, the gradient is the vector field : ∇ ψ = ( ∂ ∂ x 1 , … , ∂ ∂ x n ) ψ = ∂ ψ ∂ x 1 e 1 + ⋯ + ∂ ψ ∂ x n e n {\displaystyle \nabla \psi ={\begin{pmatrix}\displaystyle {\frac {\partial }{\partial x_{1}}},\ldots ,{\frac {\partial }{\partial x_{n}}}\end{pmatrix}}\psi ={\frac {\partial \psi }{\partial x_{1}}}\mathbf {e} _{1}+\dots +{\frac {\partial \psi }{\partial x_{n}}}\mathbf {e} _{n}} where e i ( i = 1 , 2 , . . . , n ) {\displaystyle \mathbf {e} _{i}\,(i=1,2,...,n)} are mutually orthogonal unit vectors.
As the name implies, the gradient is proportional to, and points in the direction of, the function's most rapid (positive) change.
For a vector field A = ( A 1 , … , A n ) {\displaystyle \mathbf {A} =\left(A_{1},\ldots ,A_{n}\right)} , also called a tensor field of order 1, the gradient or total derivative is the n × n Jacobian matrix : [ 1 ] J A = d A = ( ∇ A ) T = ( ∂ A i ∂ x j ) i j . {\displaystyle \mathbf {J} _{\mathbf {A} }=d\mathbf {A} =(\nabla \!\mathbf {A} )^{\textsf {T}}=\left({\frac {\partial A_{i}}{\partial x_{j}}}\right)_{\!ij}.}
For a tensor field T {\displaystyle \mathbf {T} } of any order k , the gradient grad ( T ) = d T = ( ∇ T ) T {\displaystyle \operatorname {grad} (\mathbf {T} )=d\mathbf {T} =(\nabla \mathbf {T} )^{\textsf {T}}} is a tensor field of order k + 1.
For a tensor field T {\displaystyle \mathbf {T} } of order k > 0, the tensor field ∇ T {\displaystyle \nabla \mathbf {T} } of order k + 1 is defined by the recursive relation ( ∇ T ) ⋅ C = ∇ ( T ⋅ C ) {\displaystyle (\nabla \mathbf {T} )\cdot \mathbf {C} =\nabla (\mathbf {T} \cdot \mathbf {C} )} where C {\displaystyle \mathbf {C} } is an arbitrary constant vector.
In Cartesian coordinates, the divergence of a continuously differentiable vector field F = F x i + F y j + F z k {\displaystyle \mathbf {F} =F_{x}\mathbf {i} +F_{y}\mathbf {j} +F_{z}\mathbf {k} } is the scalar-valued function: div F = ∇ ⋅ F = ( ∂ ∂ x , ∂ ∂ y , ∂ ∂ z ) ⋅ ( F x , F y , F z ) = ∂ F x ∂ x + ∂ F y ∂ y + ∂ F z ∂ z . {\displaystyle \operatorname {div} \mathbf {F} =\nabla \cdot \mathbf {F} ={\begin{pmatrix}\displaystyle {\frac {\partial }{\partial x}},\ {\frac {\partial }{\partial y}},\ {\frac {\partial }{\partial z}}\end{pmatrix}}\cdot {\begin{pmatrix}F_{x},\ F_{y},\ F_{z}\end{pmatrix}}={\frac {\partial F_{x}}{\partial x}}+{\frac {\partial F_{y}}{\partial y}}+{\frac {\partial F_{z}}{\partial z}}.}
As the name implies, the divergence is a (local) measure of the degree to which vectors in the field diverge.
The divergence of a tensor field T {\displaystyle \mathbf {T} } of non-zero order k is written as div ( T ) = ∇ ⋅ T {\displaystyle \operatorname {div} (\mathbf {T} )=\nabla \cdot \mathbf {T} } , a contraction of a tensor field of order k − 1. Specifically, the divergence of a vector is a scalar. The divergence of a higher-order tensor field may be found by decomposing the tensor field into a sum of outer products and using the identity, ∇ ⋅ ( A ⊗ T ) = T ( ∇ ⋅ A ) + ( A ⋅ ∇ ) T {\displaystyle \nabla \cdot \left(\mathbf {A} \otimes \mathbf {T} \right)=\mathbf {T} (\nabla \cdot \mathbf {A} )+(\mathbf {A} \cdot \nabla )\mathbf {T} } where A ⋅ ∇ {\displaystyle \mathbf {A} \cdot \nabla } is the directional derivative in the direction of A {\displaystyle \mathbf {A} } multiplied by its magnitude. Specifically, for the outer product of two vectors, [ 2 ] ∇ ⋅ ( A B T ) = B ( ∇ ⋅ A ) + ( A ⋅ ∇ ) B . {\displaystyle \nabla \cdot \left(\mathbf {A} \mathbf {B} ^{\textsf {T}}\right)=\mathbf {B} (\nabla \cdot \mathbf {A} )+(\mathbf {A} \cdot \nabla )\mathbf {B} .}
For a tensor field T {\displaystyle \mathbf {T} } of order k > 1, the tensor field ∇ ⋅ T {\displaystyle \nabla \cdot \mathbf {T} } of order k − 1 is defined by the recursive relation ( ∇ ⋅ T ) ⋅ C = ∇ ⋅ ( T ⋅ C ) {\displaystyle (\nabla \cdot \mathbf {T} )\cdot \mathbf {C} =\nabla \cdot (\mathbf {T} \cdot \mathbf {C} )} where C {\displaystyle \mathbf {C} } is an arbitrary constant vector.
In Cartesian coordinates, for F = F x i + F y j + F z k {\displaystyle \mathbf {F} =F_{x}\mathbf {i} +F_{y}\mathbf {j} +F_{z}\mathbf {k} } the curl is the vector field: curl F = ∇ × F = ( ∂ ∂ x , ∂ ∂ y , ∂ ∂ z ) × ( F x , F y , F z ) = | i j k ∂ ∂ x ∂ ∂ y ∂ ∂ z F x F y F z | = ( ∂ F z ∂ y − ∂ F y ∂ z ) i + ( ∂ F x ∂ z − ∂ F z ∂ x ) j + ( ∂ F y ∂ x − ∂ F x ∂ y ) k {\displaystyle {\begin{aligned}\operatorname {curl} \mathbf {F} &=\nabla \times \mathbf {F} ={\begin{pmatrix}\displaystyle {\frac {\partial }{\partial x}},\ {\frac {\partial }{\partial y}},\ {\frac {\partial }{\partial z}}\end{pmatrix}}\times {\begin{pmatrix}F_{x},\ F_{y},\ F_{z}\end{pmatrix}}={\begin{vmatrix}\mathbf {i} &\mathbf {j} &\mathbf {k} \\{\frac {\partial }{\partial x}}&{\frac {\partial }{\partial y}}&{\frac {\partial }{\partial z}}\\F_{x}&F_{y}&F_{z}\end{vmatrix}}\\[1em]&=\left({\frac {\partial F_{z}}{\partial y}}-{\frac {\partial F_{y}}{\partial z}}\right)\mathbf {i} +\left({\frac {\partial F_{x}}{\partial z}}-{\frac {\partial F_{z}}{\partial x}}\right)\mathbf {j} +\left({\frac {\partial F_{y}}{\partial x}}-{\frac {\partial F_{x}}{\partial y}}\right)\mathbf {k} \end{aligned}}} where i , j , and k are the unit vectors for the x -, y -, and z -axes, respectively.
As the name implies the curl is a measure of how much nearby vectors tend in a circular direction.
In Einstein notation , the vector field F = ( F 1 , F 2 , F 3 ) {\displaystyle \mathbf {F} ={\begin{pmatrix}F_{1},\ F_{2},\ F_{3}\end{pmatrix}}} has curl given by: ∇ × F = ε i j k e i ∂ F k ∂ x j {\displaystyle \nabla \times \mathbf {F} =\varepsilon ^{ijk}\mathbf {e} _{i}{\frac {\partial F_{k}}{\partial x_{j}}}} where ε {\displaystyle \varepsilon } = ±1 or 0 is the Levi-Civita parity symbol .
For a tensor field T {\displaystyle \mathbf {T} } of order k > 1, the tensor field ∇ × T {\displaystyle \nabla \times \mathbf {T} } of order k is defined by the recursive relation ( ∇ × T ) ⋅ C = ∇ × ( T ⋅ C ) {\displaystyle (\nabla \times \mathbf {T} )\cdot \mathbf {C} =\nabla \times (\mathbf {T} \cdot \mathbf {C} )} where C {\displaystyle \mathbf {C} } is an arbitrary constant vector.
A tensor field of order greater than one may be decomposed into a sum of outer products , and then the following identity may be used: ∇ × ( A ⊗ T ) = ( ∇ × A ) ⊗ T − A × ( ∇ T ) . {\displaystyle \nabla \times \left(\mathbf {A} \otimes \mathbf {T} \right)=(\nabla \times \mathbf {A} )\otimes \mathbf {T} -\mathbf {A} \times (\nabla \mathbf {T} ).} Specifically, for the outer product of two vectors, [ 3 ] ∇ × ( A B T ) = ( ∇ × A ) B T − A × ( ∇ B ) . {\displaystyle \nabla \times \left(\mathbf {A} \mathbf {B} ^{\textsf {T}}\right)=(\nabla \times \mathbf {A} )\mathbf {B} ^{\textsf {T}}-\mathbf {A} \times (\nabla \mathbf {B} ).}
In Cartesian coordinates , the Laplacian of a function f ( x , y , z ) {\displaystyle f(x,y,z)} is Δ f = ∇ 2 f = ( ∇ ⋅ ∇ ) f = ∂ 2 f ∂ x 2 + ∂ 2 f ∂ y 2 + ∂ 2 f ∂ z 2 . {\displaystyle \Delta f=\nabla ^{2}\!f=(\nabla \cdot \nabla )f={\frac {\partial ^{2}\!f}{\partial x^{2}}}+{\frac {\partial ^{2}\!f}{\partial y^{2}}}+{\frac {\partial ^{2}\!f}{\partial z^{2}}}.}
The Laplacian is a measure of how much a function is changing over a small sphere centered at the point.
When the Laplacian is equal to 0, the function is called a harmonic function . That is, Δ f = 0. {\displaystyle \Delta f=0.}
For a tensor field , T {\displaystyle \mathbf {T} } , the Laplacian is generally written as: Δ T = ∇ 2 T = ( ∇ ⋅ ∇ ) T {\displaystyle \Delta \mathbf {T} =\nabla ^{2}\mathbf {T} =(\nabla \cdot \nabla )\mathbf {T} } and is a tensor field of the same order.
For a tensor field T {\displaystyle \mathbf {T} } of order k > 0, the tensor field ∇ 2 T {\displaystyle \nabla ^{2}\mathbf {T} } of order k is defined by the recursive relation ( ∇ 2 T ) ⋅ C = ∇ 2 ( T ⋅ C ) {\displaystyle \left(\nabla ^{2}\mathbf {T} \right)\cdot \mathbf {C} =\nabla ^{2}(\mathbf {T} \cdot \mathbf {C} )} where C {\displaystyle \mathbf {C} } is an arbitrary constant vector.
In Feynman subscript notation , ∇ B ( A ⋅ B ) = A × ( ∇ × B ) + ( A ⋅ ∇ ) B {\displaystyle \nabla _{\mathbf {B} }\!\left(\mathbf {A{\cdot }B} \right)=\mathbf {A} {\times }\!\left(\nabla {\times }\mathbf {B} \right)+\left(\mathbf {A} {\cdot }\nabla \right)\mathbf {B} } where the notation ∇ B means the subscripted gradient operates on only the factor B . [ 4 ] [ 5 ] [ 6 ]
More general but similar is the Hestenes overdot notation in geometric algebra . [ 7 ] [ 8 ] The above identity is then expressed as: ∇ ˙ ( A ⋅ B ˙ ) = A × ( ∇ × B ) + ( A ⋅ ∇ ) B {\displaystyle {\dot {\nabla }}\left(\mathbf {A} {\cdot }{\dot {\mathbf {B} }}\right)=\mathbf {A} {\times }\!\left(\nabla {\times }\mathbf {B} \right)+\left(\mathbf {A} {\cdot }\nabla \right)\mathbf {B} } where overdots define the scope of the vector derivative. The dotted vector, in this case B , is differentiated, while the (undotted) A is held constant.
The utility of the Feynman subscript notation lies in its use in the derivation of vector and tensor derivative identities, as in the following example which uses the algebraic identity C ⋅( A × B ) = ( C × A )⋅ B :
An alternative method is to use the Cartesian components of the del operator as follows (with implicit summation over the index i ):
Another method of deriving vector and tensor derivative identities is to replace all occurrences of a vector in an algebraic identity by the del operator, provided that no variable occurs both inside and outside the scope of an operator or both inside the scope of one operator in a term and outside the scope of another operator in the same term (i.e., the operators must be nested). The validity of this rule follows from the validity of the Feynman method, for one may always substitute a subscripted del and then immediately drop the subscript under the condition of the rule.
For example, from the identity A ⋅( B × C ) = ( A × B )⋅ C we may derive A ⋅(∇× C ) = ( A ×∇)⋅ C but not ∇⋅( B × C ) = (∇× B )⋅ C ,
nor from A ⋅( B × A ) = 0 may we derive A ⋅(∇× A ) = 0.
On the other hand, a subscripted del operates on all occurrences of the subscript in the term, so that A ⋅(∇ A × A ) = ∇ A ⋅( A × A ) = ∇⋅( A × A ) = 0.
Also, from A ×( A × C ) = A ( A ⋅ C ) − ( A ⋅ A ) C we may derive ∇×(∇× C ) = ∇(∇⋅ C ) − ∇ 2 C ,
but from ( A ψ )⋅( A φ ) = ( A ⋅ A )( ψφ ) we may not derive (∇ ψ )⋅(∇ φ ) = ∇ 2 ( ψφ ).
A subscript c on a quantity indicates that it is temporarily considered to be a constant. Since a constant is not a variable, when the substitution rule (see the preceding paragraph) is used it, unlike a variable, may be moved into or out of the scope of a del operator, as in the following example: [ 9 ]
Another way to indicate that a quantity is a constant is to affix it as a subscript to the scope of a del operator, as follows: [ 10 ] ∇ ( A ⋅ B ) A = A × ( ∇ × B ) + ( A ⋅ ∇ ) B {\displaystyle \nabla \left(\mathbf {A{\cdot }B} \right)_{\mathbf {A} }=\mathbf {A} {\times }\!\left(\nabla {\times }\mathbf {B} \right)+\left(\mathbf {A} {\cdot }\nabla \right)\mathbf {B} }
For the remainder of this article, Feynman subscript notation will be used where appropriate.
For scalar fields ψ {\displaystyle \psi } , ϕ {\displaystyle \phi } and vector fields A {\displaystyle \mathbf {A} } , B {\displaystyle \mathbf {B} } , we have the following derivative identities.
We have the following generalizations of the product rule in single-variable calculus .
Let f ( x ) {\displaystyle f(x)} be a one-variable function from scalars to scalars, r ( t ) = ( x 1 ( t ) , … , x n ( t ) ) {\displaystyle \mathbf {r} (t)=(x_{1}(t),\ldots ,x_{n}(t))} a parametrized curve, ϕ : R n → R {\displaystyle \phi \!:\mathbb {R} ^{n}\to \mathbb {R} } a function from vectors to scalars, and A : R n → R n {\displaystyle \mathbf {A} \!:\mathbb {R} ^{n}\to \mathbb {R} ^{n}} a vector field. We have the following special cases of the multi-variable chain rule .
For a vector transformation x : R n → R n {\displaystyle \mathbf {x} \!:\mathbb {R} ^{n}\to \mathbb {R} ^{n}} we have:
Here we take the trace of the dot product of two second-order tensors, which corresponds to the product of their matrices.
where J A = ( ∇ A ) T = ( ∂ A i / ∂ x j ) i j {\displaystyle \mathbf {J} _{\mathbf {A} }=(\nabla \!\mathbf {A} )^{\textsf {T}}=(\partial A_{i}/\partial x_{j})_{ij}} denotes the Jacobian matrix of the vector field A = ( A 1 , … , A n ) {\displaystyle \mathbf {A} =(A_{1},\ldots ,A_{n})} .
Alternatively, using Feynman subscript notation,
See these notes. [ 11 ]
As a special case, when A = B ,
The generalization of the dot product formula to Riemannian manifolds is a defining property of a Riemannian connection , which differentiates a vector field to give a vector-valued 1-form .
Note that the matrix J B − J B T {\displaystyle \mathbf {J} _{\mathbf {B} }\,-\,\mathbf {J} _{\mathbf {B} }^{\textsf {T}}} is antisymmetric.
The divergence of the curl of any continuously twice-differentiable vector field A is always zero: ∇ ⋅ ( ∇ × A ) = 0 {\displaystyle \nabla \cdot (\nabla \times \mathbf {A} )=0}
This is a special case of the vanishing of the square of the exterior derivative in the De Rham chain complex .
The Laplacian of a scalar field is the divergence of its gradient: Δ ψ = ∇ 2 ψ = ∇ ⋅ ( ∇ ψ ) {\displaystyle \Delta \psi =\nabla ^{2}\psi =\nabla \cdot (\nabla \psi )} The result is a scalar quantity.
The divergence of a vector field A is a scalar, and the divergence of a scalar quantity is undefined. Therefore, ∇ ⋅ ( ∇ ⋅ A ) is undefined. {\displaystyle \nabla \cdot (\nabla \cdot \mathbf {A} ){\text{ is undefined.}}}
The curl of the gradient of any continuously twice-differentiable scalar field φ {\displaystyle \varphi } (i.e., differentiability class C 2 {\displaystyle C^{2}} ) is always the zero vector : ∇ × ( ∇ φ ) = 0 . {\displaystyle \nabla \times (\nabla \varphi )=\mathbf {0} .}
It can be easily proved by expressing ∇ × ( ∇ φ ) {\displaystyle \nabla \times (\nabla \varphi )} in a Cartesian coordinate system with Schwarz's theorem (also called Clairaut's theorem on equality of mixed partials). This result is a special case of the vanishing of the square of the exterior derivative in the De Rham chain complex .
∇ × ( ∇ × A ) = ∇ ( ∇ ⋅ A ) − ∇ 2 A {\displaystyle \nabla \times \left(\nabla \times \mathbf {A} \right)\ =\ \nabla (\nabla {\cdot }\mathbf {A} )\,-\,\nabla ^{2\!}\mathbf {A} }
Here ∇ 2 is the vector Laplacian operating on the vector field A .
The divergence of a vector field A is a scalar, and the curl of a scalar quantity is undefined. Therefore, ∇ × ( ∇ ⋅ A ) is undefined. {\displaystyle \nabla \times (\nabla \cdot \mathbf {A} ){\text{ is undefined.}}}
The figure to the right is a mnemonic for some of these identities. The abbreviations used are:
Each arrow is labeled with the result of an identity, specifically, the result of applying the operator at the arrow's tail to the operator at its head. The blue circle in the middle means curl of curl exists, whereas the other two red circles (dashed) mean that DD and GG do not exist.
Below, the curly symbol ∂ means " boundary of " a surface or solid.
In the following surface–volume integral theorems, V denotes a three-dimensional volume with a corresponding two-dimensional boundary S = ∂ V (a closed surface ):
In the following curve–surface integral theorems, S denotes a 2d open surface with a corresponding 1d boundary C = ∂ S (a closed curve ):
Integration around a closed curve in the clockwise sense is the negative of the same line integral in the counterclockwise sense (analogous to interchanging the limits in a definite integral ):
In the following endpoint–curve integral theorems, P denotes a 1d open path with signed 0d boundary points q − p = ∂ P {\displaystyle \mathbf {q} -\mathbf {p} =\partial P} and integration along P is from p {\displaystyle \mathbf {p} } to q {\displaystyle \mathbf {q} } :
A tensor form of a vector integral theorem may be obtained by replacing the vector (or one of them) by a tensor, provided that the vector is first made to appear only as the right-most vector of each integrand. For example, Stokes' theorem becomes [ 18 ]
A scalar field may also be treated as a vector and replaced by a vector or tensor. For example, Green's first identity becomes
Similar rules apply to algebraic and differentiation formulas. For algebraic formulas one may alternatively use the left-most vector position.
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Note: This page uses common physics notation for spherical coordinates, in which θ {\displaystyle \theta } is the angle between the z axis and the radius vector connecting the origin to the point in question, while ϕ {\displaystyle \phi } is the angle between the projection of the radius vector onto the x-y plane and the x axis. Several other definitions are in use, and so care must be taken in comparing different sources. [ 1 ]
Vectors are defined in cylindrical coordinates by ( ρ , φ , z ), where
( ρ , φ , z ) is given in Cartesian coordinates by:
[ ρ ϕ z ] = [ x 2 + y 2 arctan ( y / x ) z ] , 0 ≤ ϕ < 2 π , {\displaystyle {\begin{bmatrix}\rho \\\phi \\z\end{bmatrix}}={\begin{bmatrix}{\sqrt {x^{2}+y^{2}}}\\\operatorname {arctan} (y/x)\\z\end{bmatrix}},\ \ \ 0\leq \phi <2\pi ,}
or inversely by: [ x y z ] = [ ρ cos ϕ ρ sin ϕ z ] . {\displaystyle {\begin{bmatrix}x\\y\\z\end{bmatrix}}={\begin{bmatrix}\rho \cos \phi \\\rho \sin \phi \\z\end{bmatrix}}.}
Any vector field can be written in terms of the unit vectors as: A = A x x ^ + A y y ^ + A z z ^ = A ρ ρ ^ + A ϕ ϕ ^ + A z z ^ {\displaystyle \mathbf {A} =A_{x}\mathbf {\hat {x}} +A_{y}\mathbf {\hat {y}} +A_{z}\mathbf {\hat {z}} =A_{\rho }\mathbf {\hat {\rho }} +A_{\phi }{\boldsymbol {\hat {\phi }}}+A_{z}\mathbf {\hat {z}} } The cylindrical unit vectors are related to the Cartesian unit vectors by: [ ρ ^ ϕ ^ z ^ ] = [ cos ϕ sin ϕ 0 − sin ϕ cos ϕ 0 0 0 1 ] [ x ^ y ^ z ^ ] {\displaystyle {\begin{bmatrix}{\boldsymbol {\hat {\rho }}}\\{\boldsymbol {\hat {\phi }}}\\\mathbf {\hat {z}} \end{bmatrix}}={\begin{bmatrix}\cos \phi &\sin \phi &0\\-\sin \phi &\cos \phi &0\\0&0&1\end{bmatrix}}{\begin{bmatrix}\mathbf {\hat {x}} \\\mathbf {\hat {y}} \\\mathbf {\hat {z}} \end{bmatrix}}}
Note: the matrix is an orthogonal matrix , that is, its inverse is simply its transpose .
To find out how the vector field A changes in time, the time derivatives should be calculated.
For this purpose Newton's notation will be used for the time derivative ( A ˙ {\displaystyle {\dot {\mathbf {A} }}} ).
In Cartesian coordinates this is simply: A ˙ = A ˙ x x ^ + A ˙ y y ^ + A ˙ z z ^ {\displaystyle {\dot {\mathbf {A} }}={\dot {A}}_{x}{\hat {\mathbf {x} }}+{\dot {A}}_{y}{\hat {\mathbf {y} }}+{\dot {A}}_{z}{\hat {\mathbf {z} }}}
However, in cylindrical coordinates this becomes: A ˙ = A ˙ ρ ρ ^ + A ρ ρ ^ ˙ + A ˙ ϕ ϕ ^ + A ϕ ϕ ^ ˙ + A ˙ z z ^ + A z z ^ ˙ {\displaystyle {\dot {\mathbf {A} }}={\dot {A}}_{\rho }{\hat {\boldsymbol {\rho }}}+A_{\rho }{\dot {\hat {\boldsymbol {\rho }}}}+{\dot {A}}_{\phi }{\hat {\boldsymbol {\phi }}}+A_{\phi }{\dot {\hat {\boldsymbol {\phi }}}}+{\dot {A}}_{z}{\hat {\boldsymbol {z}}}+A_{z}{\dot {\hat {\boldsymbol {z}}}}}
The time derivatives of the unit vectors are needed.
They are given by: ρ ^ ˙ = ϕ ˙ ϕ ^ ϕ ^ ˙ = − ϕ ˙ ρ ^ z ^ ˙ = 0 {\displaystyle {\begin{aligned}{\dot {\hat {\boldsymbol {\rho }}}}&={\dot {\phi }}{\hat {\boldsymbol {\phi }}}\\{\dot {\hat {\boldsymbol {\phi }}}}&=-{\dot {\phi }}{\hat {\boldsymbol {\rho }}}\\{\dot {\hat {\mathbf {z} }}}&=0\end{aligned}}}
So the time derivative simplifies to: A ˙ = ρ ^ ( A ˙ ρ − A ϕ ϕ ˙ ) + ϕ ^ ( A ˙ ϕ + A ρ ϕ ˙ ) + z ^ A ˙ z {\displaystyle {\dot {\mathbf {A} }}={\hat {\boldsymbol {\rho }}}\left({\dot {A}}_{\rho }-A_{\phi }{\dot {\phi }}\right)+{\hat {\boldsymbol {\phi }}}\left({\dot {A}}_{\phi }+A_{\rho }{\dot {\phi }}\right)+{\hat {\mathbf {z} }}{\dot {A}}_{z}}
The second time derivative is of interest in physics , as it is found in equations of motion for classical mechanical systems.
The second time derivative of a vector field in cylindrical coordinates is given by: A ¨ = ρ ^ ( A ¨ ρ − A ϕ ϕ ¨ − 2 A ˙ ϕ ϕ ˙ − A ρ ϕ ˙ 2 ) + ϕ ^ ( A ¨ ϕ + A ρ ϕ ¨ + 2 A ˙ ρ ϕ ˙ − A ϕ ϕ ˙ 2 ) + z ^ A ¨ z {\displaystyle {\ddot {\mathbf {A} }}=\mathbf {\hat {\rho }} \left({\ddot {A}}_{\rho }-A_{\phi }{\ddot {\phi }}-2{\dot {A}}_{\phi }{\dot {\phi }}-A_{\rho }{\dot {\phi }}^{2}\right)+{\boldsymbol {\hat {\phi }}}\left({\ddot {A}}_{\phi }+A_{\rho }{\ddot {\phi }}+2{\dot {A}}_{\rho }{\dot {\phi }}-A_{\phi }{\dot {\phi }}^{2}\right)+\mathbf {\hat {z}} {\ddot {A}}_{z}}
To understand this expression, A is substituted for P , where P is the vector ( ρ , φ , z ).
This means that A = P = ρ ρ ^ + z z ^ {\displaystyle \mathbf {A} =\mathbf {P} =\rho \mathbf {\hat {\rho }} +z\mathbf {\hat {z}} } .
After substituting, the result is given: P ¨ = ρ ^ ( ρ ¨ − ρ ϕ ˙ 2 ) + ϕ ^ ( ρ ϕ ¨ + 2 ρ ˙ ϕ ˙ ) + z ^ z ¨ {\displaystyle {\ddot {\mathbf {P} }}=\mathbf {\hat {\rho }} \left({\ddot {\rho }}-\rho {\dot {\phi }}^{2}\right)+{\boldsymbol {\hat {\phi }}}\left(\rho {\ddot {\phi }}+2{\dot {\rho }}{\dot {\phi }}\right)+\mathbf {\hat {z}} {\ddot {z}}}
In mechanics, the terms of this expression are called:
Vectors are defined in spherical coordinates by ( r , θ , φ ), where
( r , θ , φ ) is given in Cartesian coordinates by: [ r θ ϕ ] = [ x 2 + y 2 + z 2 arccos ( z / x 2 + y 2 + z 2 ) arctan ( y / x ) ] , 0 ≤ θ ≤ π , 0 ≤ ϕ < 2 π , {\displaystyle {\begin{bmatrix}r\\\theta \\\phi \end{bmatrix}}={\begin{bmatrix}{\sqrt {x^{2}+y^{2}+z^{2}}}\\\arccos(z/{\sqrt {x^{2}+y^{2}+z^{2}}})\\\arctan(y/x)\end{bmatrix}},\ \ \ 0\leq \theta \leq \pi ,\ \ \ 0\leq \phi <2\pi ,} or inversely by: [ x y z ] = [ r sin θ cos ϕ r sin θ sin ϕ r cos θ ] . {\displaystyle {\begin{bmatrix}x\\y\\z\end{bmatrix}}={\begin{bmatrix}r\sin \theta \cos \phi \\r\sin \theta \sin \phi \\r\cos \theta \end{bmatrix}}.}
Any vector field can be written in terms of the unit vectors as: A = A x x ^ + A y y ^ + A z z ^ = A r r ^ + A θ θ ^ + A ϕ ϕ ^ {\displaystyle \mathbf {A} =A_{x}\mathbf {\hat {x}} +A_{y}\mathbf {\hat {y}} +A_{z}\mathbf {\hat {z}} =A_{r}{\boldsymbol {\hat {r}}}+A_{\theta }{\boldsymbol {\hat {\theta }}}+A_{\phi }{\boldsymbol {\hat {\phi }}}}
The spherical basis vectors are related to the Cartesian basis vectors by the Jacobian matrix:
[ r ^ θ ^ ϕ ^ ] = [ ∂ x ∂ r ∂ y ∂ r ∂ z ∂ r ∂ x ∂ θ ∂ y ∂ θ ∂ z ∂ θ ∂ x ∂ ϕ ∂ y ∂ ϕ ∂ z ∂ ϕ ] [ x ^ y ^ z ^ ] {\displaystyle {\begin{bmatrix}{\boldsymbol {\hat {r}}}\\{\boldsymbol {\hat {\theta }}}\\{\boldsymbol {\hat {\phi }}}\end{bmatrix}}={\begin{bmatrix}{\frac {\partial x}{\partial r}}&{\frac {\partial y}{\partial r}}&{\frac {\partial z}{\partial r}}\\{\frac {\partial x}{\partial \theta }}&{\frac {\partial y}{\partial \theta }}&{\frac {\partial z}{\partial \theta }}\\{\frac {\partial x}{\partial \phi }}&{\frac {\partial y}{\partial \phi }}&{\frac {\partial z}{\partial \phi }}\end{bmatrix}}{\begin{bmatrix}\mathbf {\hat {x}} \\\mathbf {\hat {y}} \\\mathbf {\hat {z}} \end{bmatrix}}}
Normalizing the Jacobian matrix so that the spherical basis vectors have unit length we get:
[ r ^ θ ^ ϕ ^ ] = [ sin θ cos ϕ sin θ sin ϕ cos θ cos θ cos ϕ cos θ sin ϕ − sin θ − sin ϕ cos ϕ 0 ] [ x ^ y ^ z ^ ] {\displaystyle {\begin{bmatrix}{\boldsymbol {\hat {r}}}\\{\boldsymbol {\hat {\theta }}}\\{\boldsymbol {\hat {\phi }}}\end{bmatrix}}={\begin{bmatrix}\sin \theta \cos \phi &\sin \theta \sin \phi &\cos \theta \\\cos \theta \cos \phi &\cos \theta \sin \phi &-\sin \theta \\-\sin \phi &\cos \phi &0\end{bmatrix}}{\begin{bmatrix}\mathbf {\hat {x}} \\\mathbf {\hat {y}} \\\mathbf {\hat {z}} \end{bmatrix}}}
Note: the matrix is an orthogonal matrix , that is, its inverse is simply its transpose .
The Cartesian unit vectors are thus related to the spherical unit vectors by:
[ x ^ y ^ z ^ ] = [ sin θ cos ϕ cos θ cos ϕ − sin ϕ sin θ sin ϕ cos θ sin ϕ cos ϕ cos θ − sin θ 0 ] [ r ^ θ ^ ϕ ^ ] {\displaystyle {\begin{bmatrix}\mathbf {\hat {x}} \\\mathbf {\hat {y}} \\\mathbf {\hat {z}} \end{bmatrix}}={\begin{bmatrix}\sin \theta \cos \phi &\cos \theta \cos \phi &-\sin \phi \\\sin \theta \sin \phi &\cos \theta \sin \phi &\cos \phi \\\cos \theta &-\sin \theta &0\end{bmatrix}}{\begin{bmatrix}{\boldsymbol {\hat {r}}}\\{\boldsymbol {\hat {\theta }}}\\{\boldsymbol {\hat {\phi }}}\end{bmatrix}}}
To find out how the vector field A changes in time, the time derivatives should be calculated.
In Cartesian coordinates this is simply: A ˙ = A ˙ x x ^ + A ˙ y y ^ + A ˙ z z ^ {\displaystyle \mathbf {\dot {A}} ={\dot {A}}_{x}\mathbf {\hat {x}} +{\dot {A}}_{y}\mathbf {\hat {y}} +{\dot {A}}_{z}\mathbf {\hat {z}} } However, in spherical coordinates this becomes: A ˙ = A ˙ r r ^ + A r r ^ ˙ + A ˙ θ θ ^ + A θ θ ^ ˙ + A ˙ ϕ ϕ ^ + A ϕ ϕ ^ ˙ {\displaystyle \mathbf {\dot {A}} ={\dot {A}}_{r}{\boldsymbol {\hat {r}}}+A_{r}{\boldsymbol {\dot {\hat {r}}}}+{\dot {A}}_{\theta }{\boldsymbol {\hat {\theta }}}+A_{\theta }{\boldsymbol {\dot {\hat {\theta }}}}+{\dot {A}}_{\phi }{\boldsymbol {\hat {\phi }}}+A_{\phi }{\boldsymbol {\dot {\hat {\phi }}}}} The time derivatives of the unit vectors are needed. They are given by: r ^ ˙ = θ ˙ θ ^ + ϕ ˙ sin θ ϕ ^ θ ^ ˙ = − θ ˙ r ^ + ϕ ˙ cos θ ϕ ^ ϕ ^ ˙ = − ϕ ˙ sin θ r ^ − ϕ ˙ cos θ θ ^ {\displaystyle {\begin{aligned}{\boldsymbol {\dot {\hat {r}}}}&={\dot {\theta }}{\boldsymbol {\hat {\theta }}}+{\dot {\phi }}\sin \theta {\boldsymbol {\hat {\phi }}}\\{\boldsymbol {\dot {\hat {\theta }}}}&=-{\dot {\theta }}{\boldsymbol {\hat {r}}}+{\dot {\phi }}\cos \theta {\boldsymbol {\hat {\phi }}}\\{\boldsymbol {\dot {\hat {\phi }}}}&=-{\dot {\phi }}\sin \theta {\boldsymbol {\hat {r}}}-{\dot {\phi }}\cos \theta {\boldsymbol {\hat {\theta }}}\end{aligned}}} Thus the time derivative becomes: A ˙ = r ^ ( A ˙ r − A θ θ ˙ − A ϕ ϕ ˙ sin θ ) + θ ^ ( A ˙ θ + A r θ ˙ − A ϕ ϕ ˙ cos θ ) + ϕ ^ ( A ˙ ϕ + A r ϕ ˙ sin θ + A θ ϕ ˙ cos θ ) {\displaystyle \mathbf {\dot {A}} ={\boldsymbol {\hat {r}}}\left({\dot {A}}_{r}-A_{\theta }{\dot {\theta }}-A_{\phi }{\dot {\phi }}\sin \theta \right)+{\boldsymbol {\hat {\theta }}}\left({\dot {A}}_{\theta }+A_{r}{\dot {\theta }}-A_{\phi }{\dot {\phi }}\cos \theta \right)+{\boldsymbol {\hat {\phi }}}\left({\dot {A}}_{\phi }+A_{r}{\dot {\phi }}\sin \theta +A_{\theta }{\dot {\phi }}\cos \theta \right)}
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https://en.wikipedia.org/wiki/Vector_fields_in_cylindrical_and_spherical_coordinates
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In mathematics , the discussion of vector fields on spheres was a classical problem of differential topology , beginning with the hairy ball theorem , and early work on the classification of division algebras .
Specifically, the question is how many linearly independent smooth nowhere-zero vector fields can be constructed on a sphere in n {\displaystyle n} -dimensional Euclidean space . A definitive answer was provided in 1962 by Frank Adams . It was already known, [ 1 ] by direct construction using Clifford algebras , that there were at least ρ ( n ) − 1 {\displaystyle \rho (n)-1} such fields (see definition below). Adams applied homotopy theory and topological K-theory [ 2 ] to prove that no more independent vector fields could be found. Hence ρ ( n ) − 1 {\displaystyle \rho (n)-1} is the exact number of pointwise linearly independent vector fields that exist on an ( n − 1 {\displaystyle n-1} )-dimensional sphere.
In detail, the question applies to the 'round spheres' and to their tangent bundles : in fact since all exotic spheres have isomorphic tangent bundles, the Radon–Hurwitz numbers ρ ( n ) {\displaystyle \rho (n)} determine the maximum number of linearly independent sections of the tangent bundle of any homotopy sphere. The case of n {\displaystyle n} odd is taken care of by the Poincaré–Hopf index theorem (see hairy ball theorem ), so the case n {\displaystyle n} even is an extension of that. Adams showed that the maximum number of continuous ( smooth would be no different here) pointwise linearly-independent vector fields on the ( n − 1 {\displaystyle n-1} )-sphere is exactly ρ ( n ) − 1 {\displaystyle \rho (n)-1} .
The construction of the fields is related to the real Clifford algebras , which is a theory with a periodicity modulo 8 that also shows up here. By the Gram–Schmidt process , it is the same to ask for (pointwise) linear independence or fields that give an orthonormal basis at each point.
The Radon–Hurwitz numbers ρ ( n ) {\displaystyle \rho (n)} occur in earlier work of Johann Radon (1922) and Adolf Hurwitz (1923) on the Hurwitz problem on quadratic forms . [ 3 ] For n {\displaystyle n} written as the product of an odd number A {\displaystyle A} and a power of two 2 B {\displaystyle 2^{B}} , write
Then [ 3 ]
The first few values of ρ ( 2 n ) {\displaystyle \rho (2n)} are (from (sequence A053381 in the OEIS )):
For odd n {\displaystyle n} , the value of the function ρ ( n ) {\displaystyle \rho (n)} is one.
These numbers occur also in other, related areas. In matrix theory , the Radon–Hurwitz number counts the maximum size of a linear subspace of the real n × n {\displaystyle n\times n} matrices, for which each non-zero matrix is a similarity transformation , i.e. a product of an orthogonal matrix and a scalar matrix . In quadratic forms , the Hurwitz problem asks for multiplicative identities between quadratic forms. The classical results were revisited in 1952 by Beno Eckmann . They are now applied in areas including coding theory and theoretical physics .
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https://en.wikipedia.org/wiki/Vector_fields_on_spheres
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In physics, vector meson dominance (VMD) was a model developed by J. J. Sakurai [ 1 ] in the 1960s before the introduction of quantum chromodynamics to describe interactions between energetic photons and hadronic matter.
In particular, the hadronic components of the physical photon consist of the lightest vector mesons, ρ {\displaystyle \rho } , ω {\displaystyle \omega } and ϕ {\displaystyle \phi } . Therefore, interactions between photons and hadronic matter occur by the exchange of a hadron between the dressed photon and the hadronic target.
Measurements of the interaction between energetic photons and hadrons show that the interaction is much more intense than expected by the interaction of merely photons with the hadron's electric charge. Furthermore, the interaction of energetic photons with protons is similar to the interaction of photons with neutrons [ 2 ] in spite of the fact that the electric charge structures of protons and neutrons are substantially different.
According to VMD, the photon is a superposition of the pure electromagnetic photon (which interacts only with electric charges) and vector meson.
Just after 1970, when more accurate data on the above processes became available, some discrepancies with the VMD predictions appeared and new extensions of the model were published. [ 3 ] These theories are known as Generalized Vector Meson Dominance theories (GVMD).
Whilst the ultraviolet description of the standard model is based on QCD, work over many decades has involved writing a low energy effective description of QCD, and further, positing a possible "dual" description. One such popular description is that of the hidden local symmetry. [ 4 ] The dual description is based on the idea of emergence of gauge symmetries in the infrared of strongly coupled theories. Gauge symmetries are not really physical symmetries (only the global elements of the local gauge group are physical). This emergent property of gauge symmetries was demonstrated in Seiberg duality [ 5 ] and later in the development of the AdS/CFT correspondence . [ 6 ] In its generalised form, Vector Meson Dominance appears in AdS/CFT, AdS/QCD, AdS/condensed matter and some Seiberg dual constructions. It is therefore a common place idea within the theoretical physics community.
Measurements of the photon-hadron interactions in higher energy levels show that VMD cannot predict the interaction in such levels. In his Nobel lecture [ 7 ] J.I. Friedman summarizes the situation of VMD as follows: "...this eliminated the model [VMD] as a possible description of deep inelastic scattering... calculations of the generalized vector-dominance failed in general to describe the data over the full kinematic range..."
The Vector Meson Dominance model still sometimes makes significantly more accurate predictions of hadronic decays of excited light mesons involving photons than subsequent models such as the relativistic quark model for the meson wave function and the covariant oscillator quark model. [ 8 ] Similarly, the Vector Meson Dominance model has outperformed perturbative QCD in making predictions of transitional form factors of the neutral pion meson, the eta meson , and the eta prime meson, that are "hard to explain within QCD." [ 9 ] And, the model accurately reproduces recent experimental data for rho meson decays. [ 10 ] Generalizations of the Vector Meson Dominance model to higher energies, or to consider additional factors present in cases where VMD fails, have been proposed to address the shortcomings identified by Friedman and others. [ 11 ] [ 12 ]
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https://en.wikipedia.org/wiki/Vector_meson_dominance
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In vector calculus , a vector potential is a vector field whose curl is a given vector field. This is analogous to a scalar potential , which is a scalar field whose gradient is a given vector field.
Formally, given a vector field v {\displaystyle \mathbf {v} } , a vector potential is a C 2 {\displaystyle C^{2}} vector field A {\displaystyle \mathbf {A} } such that v = ∇ × A . {\displaystyle \mathbf {v} =\nabla \times \mathbf {A} .}
If a vector field v {\displaystyle \mathbf {v} } admits a vector potential A {\displaystyle \mathbf {A} } , then from the equality ∇ ⋅ ( ∇ × A ) = 0 {\displaystyle \nabla \cdot (\nabla \times \mathbf {A} )=0} ( divergence of the curl is zero) one obtains ∇ ⋅ v = ∇ ⋅ ( ∇ × A ) = 0 , {\displaystyle \nabla \cdot \mathbf {v} =\nabla \cdot (\nabla \times \mathbf {A} )=0,} which implies that v {\displaystyle \mathbf {v} } must be a solenoidal vector field .
Let v : R 3 → R 3 {\displaystyle \mathbf {v} :\mathbb {R} ^{3}\to \mathbb {R} ^{3}} be a solenoidal vector field which is twice continuously differentiable . Assume that v ( x ) {\displaystyle \mathbf {v} (\mathbf {x} )} decreases at least as fast as 1 / ‖ x ‖ {\displaystyle 1/\|\mathbf {x} \|} for ‖ x ‖ → ∞ {\displaystyle \|\mathbf {x} \|\to \infty } . Define A ( x ) = 1 4 π ∫ R 3 ∇ y × v ( y ) ‖ x − y ‖ d 3 y {\displaystyle \mathbf {A} (\mathbf {x} )={\frac {1}{4\pi }}\int _{\mathbb {R} ^{3}}{\frac {\nabla _{y}\times \mathbf {v} (\mathbf {y} )}{\left\|\mathbf {x} -\mathbf {y} \right\|}}\,d^{3}\mathbf {y} } where ∇ y × {\displaystyle \nabla _{y}\times } denotes curl with respect to variable y {\displaystyle \mathbf {y} } . Then A {\displaystyle \mathbf {A} } is a vector potential for v {\displaystyle \mathbf {v} } . That is, ∇ × A = v . {\displaystyle \nabla \times \mathbf {A} =\mathbf {v} .}
The integral domain can be restricted to any simply connected region Ω {\displaystyle \mathbf {\Omega } } . That is, A ′ {\displaystyle \mathbf {A'} } also is a vector potential of v {\displaystyle \mathbf {v} } , where A ′ ( x ) = 1 4 π ∫ Ω ∇ y × v ( y ) ‖ x − y ‖ d 3 y . {\displaystyle \mathbf {A'} (\mathbf {x} )={\frac {1}{4\pi }}\int _{\Omega }{\frac {\nabla _{y}\times \mathbf {v} (\mathbf {y} )}{\left\|\mathbf {x} -\mathbf {y} \right\|}}\,d^{3}\mathbf {y} .}
A generalization of this theorem is the Helmholtz decomposition theorem, which states that any vector field can be decomposed as a sum of a solenoidal vector field and an irrotational vector field .
By analogy with the Biot-Savart law , A ″ ( x ) {\displaystyle \mathbf {A''} (\mathbf {x} )} also qualifies as a vector potential for v {\displaystyle \mathbf {v} } , where
Substituting j {\displaystyle \mathbf {j} } ( current density ) for v {\displaystyle \mathbf {v} } and H {\displaystyle \mathbf {H} } ( H-field ) for A {\displaystyle \mathbf {A} } , yields the Biot-Savart law.
Let Ω {\displaystyle \mathbf {\Omega } } be a star domain centered at the point p {\displaystyle \mathbf {p} } , where p ∈ R 3 {\displaystyle \mathbf {p} \in \mathbb {R} ^{3}} . Applying Poincaré's lemma for differential forms to vector fields, then A ‴ ( x ) {\displaystyle \mathbf {A'''} (\mathbf {x} )} also is a vector potential for v {\displaystyle \mathbf {v} } , where
A ‴ ( x ) = ∫ 0 1 s ( ( x − p ) × ( v ( s x + ( 1 − s ) p ) ) d s {\displaystyle \mathbf {A'''} (\mathbf {x} )=\int _{0}^{1}s((\mathbf {x} -\mathbf {p} )\times (\mathbf {v} (s\mathbf {x} +(1-s)\mathbf {p} ))\ ds}
The vector potential admitted by a solenoidal field is not unique. If A {\displaystyle \mathbf {A} } is a vector potential for v {\displaystyle \mathbf {v} } , then so is A + ∇ f , {\displaystyle \mathbf {A} +\nabla f,} where f {\displaystyle f} is any continuously differentiable scalar function. This follows from the fact that the curl of the gradient is zero.
This nonuniqueness leads to a degree of freedom in the formulation of electrodynamics, or gauge freedom, and requires choosing a gauge .
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https://en.wikipedia.org/wiki/Vector_potential
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The vector projection (also known as the vector component or vector resolution ) of a vector a on (or onto) a nonzero vector b is the orthogonal projection of a onto a straight line parallel to b .
The projection of a onto b is often written as proj b a {\displaystyle \operatorname {proj} _{\mathbf {b} }\mathbf {a} } or a ∥ b .
The vector component or vector resolute of a perpendicular to b , sometimes also called the vector rejection of a from b (denoted oproj b a {\displaystyle \operatorname {oproj} _{\mathbf {b} }\mathbf {a} } or a ⊥ b ), [ 1 ] is the orthogonal projection of a onto the plane (or, in general, hyperplane ) that is orthogonal to b . Since both proj b a {\displaystyle \operatorname {proj} _{\mathbf {b} }\mathbf {a} } and oproj b a {\displaystyle \operatorname {oproj} _{\mathbf {b} }\mathbf {a} } are vectors, and their sum is equal to a , the rejection of a from b is given by: oproj b a = a − proj b a . {\displaystyle \operatorname {oproj} _{\mathbf {b} }\mathbf {a} =\mathbf {a} -\operatorname {proj} _{\mathbf {b} }\mathbf {a} .}
To simplify notation, this article defines a 1 := proj b a {\displaystyle \mathbf {a} _{1}:=\operatorname {proj} _{\mathbf {b} }\mathbf {a} } and a 2 := oproj b a . {\displaystyle \mathbf {a} _{2}:=\operatorname {oproj} _{\mathbf {b} }\mathbf {a} .} Thus, the vector a 1 {\displaystyle \mathbf {a} _{1}} is parallel to b , {\displaystyle \mathbf {b} ,} the vector a 2 {\displaystyle \mathbf {a} _{2}} is orthogonal to b , {\displaystyle \mathbf {b} ,} and a = a 1 + a 2 . {\displaystyle \mathbf {a} =\mathbf {a} _{1}+\mathbf {a} _{2}.}
The projection of a onto b can be decomposed into a direction and a scalar magnitude by writing it as a 1 = a 1 b ^ {\displaystyle \mathbf {a} _{1}=a_{1}\mathbf {\hat {b}} } where a 1 {\displaystyle a_{1}} is a scalar, called the scalar projection of a onto b , and b̂ is the unit vector in the direction of b . The scalar projection is defined as [ 2 ] a 1 = ‖ a ‖ cos θ = a ⋅ b ^ {\displaystyle a_{1}=\left\|\mathbf {a} \right\|\cos \theta =\mathbf {a} \cdot \mathbf {\hat {b}} } where the operator ⋅ denotes a dot product , ‖ a ‖ is the length of a , and θ is the angle between a and b .
The scalar projection is equal in absolute value to the length of the vector projection, with a minus sign if the direction of the projection is opposite to the direction of b , that is, if the angle between the vectors is more than 90 degrees.
The vector projection can be calculated using the dot product of a {\displaystyle \mathbf {a} } and b {\displaystyle \mathbf {b} } as: proj b a = ( a ⋅ b ^ ) b ^ = a ⋅ b ‖ b ‖ b ‖ b ‖ = a ⋅ b ‖ b ‖ 2 b = a ⋅ b b ⋅ b b . {\displaystyle \operatorname {proj} _{\mathbf {b} }\mathbf {a} =\left(\mathbf {a} \cdot \mathbf {\hat {b}} \right)\mathbf {\hat {b}} ={\frac {\mathbf {a} \cdot \mathbf {b} }{\left\|\mathbf {b} \right\|}}{\frac {\mathbf {b} }{\left\|\mathbf {b} \right\|}}={\frac {\mathbf {a} \cdot \mathbf {b} }{\left\|\mathbf {b} \right\|^{2}}}{\mathbf {b} }={\frac {\mathbf {a} \cdot \mathbf {b} }{\mathbf {b} \cdot \mathbf {b} }}{\mathbf {b} }~.}
This article uses the convention that vectors are denoted in a bold font (e.g. a 1 ), and scalars are written in normal font (e.g. a 1 ).
The dot product of vectors a and b is written as a ⋅ b {\displaystyle \mathbf {a} \cdot \mathbf {b} } , the norm of a is written ‖ a ‖, the angle between a and b is denoted θ .
The scalar projection of a on b is a scalar equal to a 1 = ‖ a ‖ cos θ , {\displaystyle a_{1}=\left\|\mathbf {a} \right\|\cos \theta ,} where θ is the angle between a and b .
A scalar projection can be used as a scale factor to compute the corresponding vector projection.
The vector projection of a on b is a vector whose magnitude is the scalar projection of a on b with the same direction as b . Namely, it is defined as a 1 = a 1 b ^ = ( ‖ a ‖ cos θ ) b ^ {\displaystyle \mathbf {a} _{1}=a_{1}\mathbf {\hat {b}} =(\left\|\mathbf {a} \right\|\cos \theta )\mathbf {\hat {b}} } where a 1 {\displaystyle a_{1}} is the corresponding scalar projection, as defined above, and b ^ {\displaystyle \mathbf {\hat {b}} } is the unit vector with the same direction as b : b ^ = b ‖ b ‖ {\displaystyle \mathbf {\hat {b}} ={\frac {\mathbf {b} }{\left\|\mathbf {b} \right\|}}}
By definition, the vector rejection of a on b is: a 2 = a − a 1 {\displaystyle \mathbf {a} _{2}=\mathbf {a} -\mathbf {a} _{1}}
Hence, a 2 = a − ( ‖ a ‖ cos θ ) b ^ {\displaystyle \mathbf {a} _{2}=\mathbf {a} -\left(\left\|\mathbf {a} \right\|\cos \theta \right)\mathbf {\hat {b}} }
When θ is not known, the cosine of θ can be computed in terms of a and b , by the following property of the dot product a ⋅ b a ⋅ b = ‖ a ‖ ‖ b ‖ cos θ {\displaystyle \mathbf {a} \cdot \mathbf {b} =\left\|\mathbf {a} \right\|\left\|\mathbf {b} \right\|\cos \theta }
By the above-mentioned property of the dot product, the definition of the scalar projection becomes: [ 2 ]
In two dimensions, this becomes a 1 = a x b x + a y b y ‖ b ‖ . {\displaystyle a_{1}={\frac {\mathbf {a} _{x}\mathbf {b} _{x}+\mathbf {a} _{y}\mathbf {b} _{y}}{\left\|\mathbf {b} \right\|}}.}
Similarly, the definition of the vector projection of a onto b becomes: [ 2 ] a 1 = a 1 b ^ = a ⋅ b ‖ b ‖ b ‖ b ‖ , {\displaystyle \mathbf {a} _{1}=a_{1}\mathbf {\hat {b}} ={\frac {\mathbf {a} \cdot \mathbf {b} }{\left\|\mathbf {b} \right\|}}{\frac {\mathbf {b} }{\left\|\mathbf {b} \right\|}},} which is equivalent to either a 1 = ( a ⋅ b ^ ) b ^ , {\displaystyle \mathbf {a} _{1}=\left(\mathbf {a} \cdot \mathbf {\hat {b}} \right)\mathbf {\hat {b}} ,} or [ 3 ] a 1 = a ⋅ b ‖ b ‖ 2 b = a ⋅ b b ⋅ b b . {\displaystyle \mathbf {a} _{1}={\frac {\mathbf {a} \cdot \mathbf {b} }{\left\|\mathbf {b} \right\|^{2}}}{\mathbf {b} }={\frac {\mathbf {a} \cdot \mathbf {b} }{\mathbf {b} \cdot \mathbf {b} }}{\mathbf {b} }~.}
In two dimensions, the scalar rejection is equivalent to the projection of a onto b ⊥ = ( − b y b x ) {\displaystyle \mathbf {b} ^{\perp }={\begin{pmatrix}-\mathbf {b} _{y}&\mathbf {b} _{x}\end{pmatrix}}} , which is b = ( b x b y ) {\displaystyle \mathbf {b} ={\begin{pmatrix}\mathbf {b} _{x}&\mathbf {b} _{y}\end{pmatrix}}} rotated 90° to the left. Hence, a 2 = ‖ a ‖ sin θ = a ⋅ b ⊥ ‖ b ‖ = a y b x − a x b y ‖ b ‖ . {\displaystyle a_{2}=\left\|\mathbf {a} \right\|\sin \theta ={\frac {\mathbf {a} \cdot \mathbf {b} ^{\perp }}{\left\|\mathbf {b} \right\|}}={\frac {\mathbf {a} _{y}\mathbf {b} _{x}-\mathbf {a} _{x}\mathbf {b} _{y}}{\left\|\mathbf {b} \right\|}}.}
Such a dot product is called the "perp dot product."
By definition, a 2 = a − a 1 {\displaystyle \mathbf {a} _{2}=\mathbf {a} -\mathbf {a} _{1}}
Hence, a 2 = a − a ⋅ b b ⋅ b b . {\displaystyle \mathbf {a} _{2}=\mathbf {a} -{\frac {\mathbf {a} \cdot \mathbf {b} }{\mathbf {b} \cdot \mathbf {b} }}{\mathbf {b} }.}
By using the Scalar rejection using the perp dot product this gives
a 2 = a ⋅ b ⊥ b ⋅ b b ⊥ {\displaystyle \mathbf {a} _{2}={\frac {\mathbf {a} \cdot \mathbf {b} ^{\perp }}{\mathbf {b} \cdot \mathbf {b} }}\mathbf {b} ^{\perp }}
The scalar projection a on b is a scalar which has a negative sign if 90 degrees < θ ≤ 180 degrees . It coincides with the length ‖ c ‖ of the vector projection if the angle is smaller than 90°. More exactly:
The vector projection of a on b is a vector a 1 which is either null or parallel to b . More exactly:
The vector rejection of a on b is a vector a 2 which is either null or orthogonal to b . More exactly:
The orthogonal projection can be represented by a projection matrix . To project a vector onto the unit vector a = ( a x , a y , a z ) , it would need to be multiplied with this projection matrix:
The vector projection is an important operation in the Gram–Schmidt orthonormalization of vector space bases . It is also used in the separating axis theorem to detect whether two convex shapes intersect.
Since the notions of vector length and angle between vectors can be generalized to any n -dimensional inner product space , this is also true for the notions of orthogonal projection of a vector, projection of a vector onto another, and rejection of a vector from another.
In some cases, the inner product coincides with the dot product. Whenever they don't coincide, the inner product is used instead of the dot product in the formal definitions of projection and rejection. For a three-dimensional inner product space , the notions of projection of a vector onto another and rejection of a vector from another can be generalized to the notions of projection of a vector onto a plane , and rejection of a vector from a plane. [ 4 ] The projection of a vector on a plane is its orthogonal projection on that plane. The rejection of a vector from a plane is its orthogonal projection on a straight line which is orthogonal to that plane. Both are vectors. The first is parallel to the plane, the second is orthogonal.
For a given vector and plane, the sum of projection and rejection is equal to the original vector. Similarly, for inner product spaces with more than three dimensions, the notions of projection onto a vector and rejection from a vector can be generalized to the notions of projection onto a hyperplane , and rejection from a hyperplane . In geometric algebra , they can be further generalized to the notions of projection and rejection of a general multivector onto/from any invertible k -blade.
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https://en.wikipedia.org/wiki/Vector_projection
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The vector projection (also known as the vector component or vector resolution ) of a vector a on (or onto) a nonzero vector b is the orthogonal projection of a onto a straight line parallel to b .
The projection of a onto b is often written as proj b a {\displaystyle \operatorname {proj} _{\mathbf {b} }\mathbf {a} } or a ∥ b .
The vector component or vector resolute of a perpendicular to b , sometimes also called the vector rejection of a from b (denoted oproj b a {\displaystyle \operatorname {oproj} _{\mathbf {b} }\mathbf {a} } or a ⊥ b ), [ 1 ] is the orthogonal projection of a onto the plane (or, in general, hyperplane ) that is orthogonal to b . Since both proj b a {\displaystyle \operatorname {proj} _{\mathbf {b} }\mathbf {a} } and oproj b a {\displaystyle \operatorname {oproj} _{\mathbf {b} }\mathbf {a} } are vectors, and their sum is equal to a , the rejection of a from b is given by: oproj b a = a − proj b a . {\displaystyle \operatorname {oproj} _{\mathbf {b} }\mathbf {a} =\mathbf {a} -\operatorname {proj} _{\mathbf {b} }\mathbf {a} .}
To simplify notation, this article defines a 1 := proj b a {\displaystyle \mathbf {a} _{1}:=\operatorname {proj} _{\mathbf {b} }\mathbf {a} } and a 2 := oproj b a . {\displaystyle \mathbf {a} _{2}:=\operatorname {oproj} _{\mathbf {b} }\mathbf {a} .} Thus, the vector a 1 {\displaystyle \mathbf {a} _{1}} is parallel to b , {\displaystyle \mathbf {b} ,} the vector a 2 {\displaystyle \mathbf {a} _{2}} is orthogonal to b , {\displaystyle \mathbf {b} ,} and a = a 1 + a 2 . {\displaystyle \mathbf {a} =\mathbf {a} _{1}+\mathbf {a} _{2}.}
The projection of a onto b can be decomposed into a direction and a scalar magnitude by writing it as a 1 = a 1 b ^ {\displaystyle \mathbf {a} _{1}=a_{1}\mathbf {\hat {b}} } where a 1 {\displaystyle a_{1}} is a scalar, called the scalar projection of a onto b , and b̂ is the unit vector in the direction of b . The scalar projection is defined as [ 2 ] a 1 = ‖ a ‖ cos θ = a ⋅ b ^ {\displaystyle a_{1}=\left\|\mathbf {a} \right\|\cos \theta =\mathbf {a} \cdot \mathbf {\hat {b}} } where the operator ⋅ denotes a dot product , ‖ a ‖ is the length of a , and θ is the angle between a and b .
The scalar projection is equal in absolute value to the length of the vector projection, with a minus sign if the direction of the projection is opposite to the direction of b , that is, if the angle between the vectors is more than 90 degrees.
The vector projection can be calculated using the dot product of a {\displaystyle \mathbf {a} } and b {\displaystyle \mathbf {b} } as: proj b a = ( a ⋅ b ^ ) b ^ = a ⋅ b ‖ b ‖ b ‖ b ‖ = a ⋅ b ‖ b ‖ 2 b = a ⋅ b b ⋅ b b . {\displaystyle \operatorname {proj} _{\mathbf {b} }\mathbf {a} =\left(\mathbf {a} \cdot \mathbf {\hat {b}} \right)\mathbf {\hat {b}} ={\frac {\mathbf {a} \cdot \mathbf {b} }{\left\|\mathbf {b} \right\|}}{\frac {\mathbf {b} }{\left\|\mathbf {b} \right\|}}={\frac {\mathbf {a} \cdot \mathbf {b} }{\left\|\mathbf {b} \right\|^{2}}}{\mathbf {b} }={\frac {\mathbf {a} \cdot \mathbf {b} }{\mathbf {b} \cdot \mathbf {b} }}{\mathbf {b} }~.}
This article uses the convention that vectors are denoted in a bold font (e.g. a 1 ), and scalars are written in normal font (e.g. a 1 ).
The dot product of vectors a and b is written as a ⋅ b {\displaystyle \mathbf {a} \cdot \mathbf {b} } , the norm of a is written ‖ a ‖, the angle between a and b is denoted θ .
The scalar projection of a on b is a scalar equal to a 1 = ‖ a ‖ cos θ , {\displaystyle a_{1}=\left\|\mathbf {a} \right\|\cos \theta ,} where θ is the angle between a and b .
A scalar projection can be used as a scale factor to compute the corresponding vector projection.
The vector projection of a on b is a vector whose magnitude is the scalar projection of a on b with the same direction as b . Namely, it is defined as a 1 = a 1 b ^ = ( ‖ a ‖ cos θ ) b ^ {\displaystyle \mathbf {a} _{1}=a_{1}\mathbf {\hat {b}} =(\left\|\mathbf {a} \right\|\cos \theta )\mathbf {\hat {b}} } where a 1 {\displaystyle a_{1}} is the corresponding scalar projection, as defined above, and b ^ {\displaystyle \mathbf {\hat {b}} } is the unit vector with the same direction as b : b ^ = b ‖ b ‖ {\displaystyle \mathbf {\hat {b}} ={\frac {\mathbf {b} }{\left\|\mathbf {b} \right\|}}}
By definition, the vector rejection of a on b is: a 2 = a − a 1 {\displaystyle \mathbf {a} _{2}=\mathbf {a} -\mathbf {a} _{1}}
Hence, a 2 = a − ( ‖ a ‖ cos θ ) b ^ {\displaystyle \mathbf {a} _{2}=\mathbf {a} -\left(\left\|\mathbf {a} \right\|\cos \theta \right)\mathbf {\hat {b}} }
When θ is not known, the cosine of θ can be computed in terms of a and b , by the following property of the dot product a ⋅ b a ⋅ b = ‖ a ‖ ‖ b ‖ cos θ {\displaystyle \mathbf {a} \cdot \mathbf {b} =\left\|\mathbf {a} \right\|\left\|\mathbf {b} \right\|\cos \theta }
By the above-mentioned property of the dot product, the definition of the scalar projection becomes: [ 2 ]
In two dimensions, this becomes a 1 = a x b x + a y b y ‖ b ‖ . {\displaystyle a_{1}={\frac {\mathbf {a} _{x}\mathbf {b} _{x}+\mathbf {a} _{y}\mathbf {b} _{y}}{\left\|\mathbf {b} \right\|}}.}
Similarly, the definition of the vector projection of a onto b becomes: [ 2 ] a 1 = a 1 b ^ = a ⋅ b ‖ b ‖ b ‖ b ‖ , {\displaystyle \mathbf {a} _{1}=a_{1}\mathbf {\hat {b}} ={\frac {\mathbf {a} \cdot \mathbf {b} }{\left\|\mathbf {b} \right\|}}{\frac {\mathbf {b} }{\left\|\mathbf {b} \right\|}},} which is equivalent to either a 1 = ( a ⋅ b ^ ) b ^ , {\displaystyle \mathbf {a} _{1}=\left(\mathbf {a} \cdot \mathbf {\hat {b}} \right)\mathbf {\hat {b}} ,} or [ 3 ] a 1 = a ⋅ b ‖ b ‖ 2 b = a ⋅ b b ⋅ b b . {\displaystyle \mathbf {a} _{1}={\frac {\mathbf {a} \cdot \mathbf {b} }{\left\|\mathbf {b} \right\|^{2}}}{\mathbf {b} }={\frac {\mathbf {a} \cdot \mathbf {b} }{\mathbf {b} \cdot \mathbf {b} }}{\mathbf {b} }~.}
In two dimensions, the scalar rejection is equivalent to the projection of a onto b ⊥ = ( − b y b x ) {\displaystyle \mathbf {b} ^{\perp }={\begin{pmatrix}-\mathbf {b} _{y}&\mathbf {b} _{x}\end{pmatrix}}} , which is b = ( b x b y ) {\displaystyle \mathbf {b} ={\begin{pmatrix}\mathbf {b} _{x}&\mathbf {b} _{y}\end{pmatrix}}} rotated 90° to the left. Hence, a 2 = ‖ a ‖ sin θ = a ⋅ b ⊥ ‖ b ‖ = a y b x − a x b y ‖ b ‖ . {\displaystyle a_{2}=\left\|\mathbf {a} \right\|\sin \theta ={\frac {\mathbf {a} \cdot \mathbf {b} ^{\perp }}{\left\|\mathbf {b} \right\|}}={\frac {\mathbf {a} _{y}\mathbf {b} _{x}-\mathbf {a} _{x}\mathbf {b} _{y}}{\left\|\mathbf {b} \right\|}}.}
Such a dot product is called the "perp dot product."
By definition, a 2 = a − a 1 {\displaystyle \mathbf {a} _{2}=\mathbf {a} -\mathbf {a} _{1}}
Hence, a 2 = a − a ⋅ b b ⋅ b b . {\displaystyle \mathbf {a} _{2}=\mathbf {a} -{\frac {\mathbf {a} \cdot \mathbf {b} }{\mathbf {b} \cdot \mathbf {b} }}{\mathbf {b} }.}
By using the Scalar rejection using the perp dot product this gives
a 2 = a ⋅ b ⊥ b ⋅ b b ⊥ {\displaystyle \mathbf {a} _{2}={\frac {\mathbf {a} \cdot \mathbf {b} ^{\perp }}{\mathbf {b} \cdot \mathbf {b} }}\mathbf {b} ^{\perp }}
The scalar projection a on b is a scalar which has a negative sign if 90 degrees < θ ≤ 180 degrees . It coincides with the length ‖ c ‖ of the vector projection if the angle is smaller than 90°. More exactly:
The vector projection of a on b is a vector a 1 which is either null or parallel to b . More exactly:
The vector rejection of a on b is a vector a 2 which is either null or orthogonal to b . More exactly:
The orthogonal projection can be represented by a projection matrix . To project a vector onto the unit vector a = ( a x , a y , a z ) , it would need to be multiplied with this projection matrix:
The vector projection is an important operation in the Gram–Schmidt orthonormalization of vector space bases . It is also used in the separating axis theorem to detect whether two convex shapes intersect.
Since the notions of vector length and angle between vectors can be generalized to any n -dimensional inner product space , this is also true for the notions of orthogonal projection of a vector, projection of a vector onto another, and rejection of a vector from another.
In some cases, the inner product coincides with the dot product. Whenever they don't coincide, the inner product is used instead of the dot product in the formal definitions of projection and rejection. For a three-dimensional inner product space , the notions of projection of a vector onto another and rejection of a vector from another can be generalized to the notions of projection of a vector onto a plane , and rejection of a vector from a plane. [ 4 ] The projection of a vector on a plane is its orthogonal projection on that plane. The rejection of a vector from a plane is its orthogonal projection on a straight line which is orthogonal to that plane. Both are vectors. The first is parallel to the plane, the second is orthogonal.
For a given vector and plane, the sum of projection and rejection is equal to the original vector. Similarly, for inner product spaces with more than three dimensions, the notions of projection onto a vector and rejection from a vector can be generalized to the notions of projection onto a hyperplane , and rejection from a hyperplane . In geometric algebra , they can be further generalized to the notions of projection and rejection of a general multivector onto/from any invertible k -blade.
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https://en.wikipedia.org/wiki/Vector_projection_on_a_plane
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In spectroscopy and radiometry , vector radiative transfer (VRT) is a method of modelling the propagation of polarized electromagnetic radiation in low density media. In contrast to scalar radiative transfer (RT), which models only the first Stokes component , the intensity, VRT models all four components through vector methods.
For a single frequency, ν {\displaystyle \nu } , the VRT equation for a scattering media can be written as follows:
d d s I → ( n ^ , ν ) = − K I → + a → B ( ν , T ) + ∫ 4 π Z ( n ^ , n ^ ′ , ν ) I → d n ^ ′ {\displaystyle {\frac {d}{ds}}{\vec {I}}({\hat {n}},\nu )=-\mathbf {K} {\vec {I}}+{\vec {a}}B(\nu ,T)+\int _{4\pi }\mathbf {Z} ({\hat {n}},{\hat {n}}',\nu )\,{\vec {I}}\,d{\hat {n}}'}
where s is the path, n ^ {\displaystyle {\hat {n}}} is the propagation vector, K is the extinction matrix, a → {\displaystyle {\vec {a}}} is the absorption vector, B is the Planck function and Z is the scattering phase matrix.
All the coefficient matrices, K , a → {\displaystyle {\vec {a}}} and Z , will vary depending on the density of absorbers/scatterers present and must be calculated from their density-independent quantities, that is the attenuation coefficient vector, a → {\displaystyle {\vec {a}}} , is calculated from the mass absorption coefficient vector times the density of the absorber.
Moreover, it is typical for media to have multiple species causing extinction, absorption and scattering, thus these coefficient matrices must be summed up over all the different species.
Extinction is caused both by simple absorption as well as from scattering out of the line-of-sight, n ^ {\displaystyle {\hat {n}}} , therefore we calculate the extinction matrix from the combination of the absorption vector and the scattering phase matrix:
K ( n ^ , ν ) = a → ( ν ) I + ∫ 4 π Z ( n ^ ′ , n ^ , ν ) d n ^ ′ {\displaystyle \mathbf {K} ({\hat {n}},\nu )={\vec {a}}(\nu )\mathbf {I} +\int _{4\pi }\mathbf {Z} ({\hat {n}}^{\prime },{\hat {n}},\nu )\,d{\hat {n}}'}
where I is the identity matrix.
The four-component radiation vector, I → = ( I , Q , U , V ) {\displaystyle {\vec {I}}=(I,Q,U,V)} where I , Q , U , and V are the first through fourth elements of the Stokes parameters , respectively, fully describes the polarization state of the electromagnetic radiation.
It is this vector-nature that considerably complicates the equation.
Absorption will be different for each of the four components, moreover, whenever the radiation is scattered, there can be a complex transfer between the different Stokes components—see polarization mixing —thus the scattering phase function has 4×4=16 components. It is, in fact, a rank-two tensor .
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https://en.wikipedia.org/wiki/Vector_radiative_transfer
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A vector signal analyzer is an instrument that measures the magnitude and phase of the input signal at a single frequency within the IF bandwidth of the instrument. The primary use is to make in-channel measurements, such as error vector magnitude , code domain power, and spectral flatness , on known signals.
Vector signal analyzers are useful in measuring and demodulating digitally modulated signals like W-CDMA , LTE , and WLAN . [ 1 ] These measurements are used to determine the quality of modulation and can be used for design validation and compliance testing of electronic devices.
The vector signal analyzer spectrum analysis process typically has a down-convert & digitizing stage and a DSP & display stage.
A vector signal analyzer operates by first down-converting the signal spectra by using superheterodyne techniques .
A portion of the input signal spectrum is down-converted [ broken anchor ] (using a voltage-controlled oscillator and a mixer ) to the center frequency of a band-pass filter . The use of a voltage-controlled oscillator allows consideration of different carrier frequencies.
After the conversion to an intermediate frequency , the signal is filtered in order to band-limit the signal and prevent aliasing . The signal is then digitized using an analog-to-digital converter . Sampling rate is often varied in relation to the frequency span under consideration.
Once the signal is digitized, it is separated into quadrature and in-phase components ( I/Q data ) using a quadrature detector, which is typically [ citation needed ] implemented with a discrete Hilbert transform . Several measurements are made and displayed using these signal components and various DSP processes, such as the ones below.
A FFT is used to compute the frequency spectrum of the signal. Usually there is a windowing function option to limit spectral leakage and enhance frequency resolution. [ 2 ] This window is implemented by multiplying it with the digitized values of the sample period before computing the FFT.
A constellation diagram represents a signal modulated by a digital modulation scheme such as quadrature amplitude modulation or phase-shift keying . This diagram maps the magnitude of the quadrature and in-phase components to the vertical and horizontal directions respectively. Qualitative assessments of signal integrity can be made based on interpretation of this diagram .
By representing the quadrature and in-phase components as the vertical and horizontal axes, the error vector magnitude can be computed as the distance between the ideal and measured constellation points on the diagram. This requires knowledge of the modulated signal in order to compare the received signal with the ideal signal.
Typical vector signal analyzer displays feature the spectrum of the signal measured within the IF bandwidth , a constellation diagram of the demodulated signal, error vector magnitude measurements, and a time-domain plot of the signal. Many more measurement results can be displayed depending on the type of modulation being used (symbol decoding, MIMO measurements, radio frame summary, etc.).
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https://en.wikipedia.org/wiki/Vector_signal_analyzer
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In mathematics , vector spherical harmonics ( VSH ) are an extension of the scalar spherical harmonics for use with vector fields . The components of the VSH are complex-valued functions expressed in the spherical coordinate basis vectors .
Several conventions have been used to define the VSH. [ 1 ] [ 2 ] [ 3 ] [ 4 ] [ 5 ] We follow that of Barrera et al. . Given a scalar spherical harmonic Y ℓm ( θ , φ ) , we define three VSH:
with r ^ {\displaystyle {\hat {\mathbf {r} }}} being the unit vector along the radial direction in spherical coordinates and r {\displaystyle \mathbf {r} } the vector along the radial direction with the same norm as the radius, i.e., r = r r ^ {\displaystyle \mathbf {r} =r{\hat {\mathbf {r} }}} . The radial factors are included to guarantee that the dimensions of the VSH are the same as those of the ordinary spherical harmonics and that the VSH do not depend on the radial spherical coordinate.
The interest of these new vector fields is to separate the radial dependence from the angular one when using spherical coordinates, so that a vector field admits a multipole expansion
E = ∑ ℓ = 0 ∞ ∑ m = − ℓ ℓ ( E ℓ m r ( r ) Y ℓ m + E ℓ m ( 1 ) ( r ) Ψ ℓ m + E ℓ m ( 2 ) ( r ) Φ ℓ m ) . {\displaystyle \mathbf {E} =\sum _{\ell =0}^{\infty }\sum _{m=-\ell }^{\ell }\left(E_{\ell m}^{r}(r)\mathbf {Y} _{\ell m}+E_{\ell m}^{(1)}(r)\mathbf {\Psi } _{\ell m}+E_{\ell m}^{(2)}(r)\mathbf {\Phi } _{\ell m}\right).}
The labels on the components reflect that E ℓ m r {\displaystyle E_{\ell m}^{r}} is the radial component of the vector field, while E ℓ m ( 1 ) {\displaystyle E_{\ell m}^{(1)}} and E ℓ m ( 2 ) {\displaystyle E_{\ell m}^{(2)}} are transverse components (with respect to the radius vector r {\displaystyle \mathbf {r} } ).
Like the scalar spherical harmonics, the VSH satisfy
Y ℓ , − m = ( − 1 ) m Y ℓ m ∗ , Ψ ℓ , − m = ( − 1 ) m Ψ ℓ m ∗ , Φ ℓ , − m = ( − 1 ) m Φ ℓ m ∗ , {\displaystyle {\begin{aligned}\mathbf {Y} _{\ell ,-m}&=(-1)^{m}\mathbf {Y} _{\ell m}^{*},\\\mathbf {\Psi } _{\ell ,-m}&=(-1)^{m}\mathbf {\Psi } _{\ell m}^{*},\\\mathbf {\Phi } _{\ell ,-m}&=(-1)^{m}\mathbf {\Phi } _{\ell m}^{*},\end{aligned}}}
which cuts the number of independent functions roughly in half. The star indicates complex conjugation .
The VSH are orthogonal in the usual three-dimensional way at each point r {\displaystyle \mathbf {r} } :
Y ℓ m ( r ) ⋅ Ψ ℓ m ( r ) = 0 , Y ℓ m ( r ) ⋅ Φ ℓ m ( r ) = 0 , Ψ ℓ m ( r ) ⋅ Φ ℓ m ( r ) = 0. {\displaystyle {\begin{aligned}\mathbf {Y} _{\ell m}(\mathbf {r} )\cdot \mathbf {\Psi } _{\ell m}(\mathbf {r} )&=0,\\\mathbf {Y} _{\ell m}(\mathbf {r} )\cdot \mathbf {\Phi } _{\ell m}(\mathbf {r} )&=0,\\\mathbf {\Psi } _{\ell m}(\mathbf {r} )\cdot \mathbf {\Phi } _{\ell m}(\mathbf {r} )&=0.\end{aligned}}}
They are also orthogonal in Hilbert space:
∫ Y ℓ m ⋅ Y ℓ ′ m ′ ∗ d Ω = δ ℓ ℓ ′ δ m m ′ , ∫ Ψ ℓ m ⋅ Ψ ℓ ′ m ′ ∗ d Ω = ℓ ( ℓ + 1 ) δ ℓ ℓ ′ δ m m ′ , ∫ Φ ℓ m ⋅ Φ ℓ ′ m ′ ∗ d Ω = ℓ ( ℓ + 1 ) δ ℓ ℓ ′ δ m m ′ , ∫ Y ℓ m ⋅ Ψ ℓ ′ m ′ ∗ d Ω = 0 , ∫ Y ℓ m ⋅ Φ ℓ ′ m ′ ∗ d Ω = 0 , ∫ Ψ ℓ m ⋅ Φ ℓ ′ m ′ ∗ d Ω = 0. {\displaystyle {\begin{aligned}\int \mathbf {Y} _{\ell m}\cdot \mathbf {Y} _{\ell 'm'}^{*}\,d\Omega &=\delta _{\ell \ell '}\delta _{mm'},\\\int \mathbf {\Psi } _{\ell m}\cdot \mathbf {\Psi } _{\ell 'm'}^{*}\,d\Omega &=\ell (\ell +1)\delta _{\ell \ell '}\delta _{mm'},\\\int \mathbf {\Phi } _{\ell m}\cdot \mathbf {\Phi } _{\ell 'm'}^{*}\,d\Omega &=\ell (\ell +1)\delta _{\ell \ell '}\delta _{mm'},\\\int \mathbf {Y} _{\ell m}\cdot \mathbf {\Psi } _{\ell 'm'}^{*}\,d\Omega &=0,\\\int \mathbf {Y} _{\ell m}\cdot \mathbf {\Phi } _{\ell 'm'}^{*}\,d\Omega &=0,\\\int \mathbf {\Psi } _{\ell m}\cdot \mathbf {\Phi } _{\ell 'm'}^{*}\,d\Omega &=0.\end{aligned}}}
An additional result at a single point r {\displaystyle \mathbf {r} } (not reported in Barrera et al, 1985) is, for all ℓ , m , ℓ ′ , m ′ {\displaystyle \ell ,m,\ell ',m'} ,
Y ℓ m ( r ) ⋅ Ψ ℓ ′ m ′ ( r ) = 0 , Y ℓ m ( r ) ⋅ Φ ℓ ′ m ′ ( r ) = 0. {\displaystyle {\begin{aligned}\mathbf {Y} _{\ell m}(\mathbf {r} )\cdot \mathbf {\Psi } _{\ell 'm'}(\mathbf {r} )&=0,\\\mathbf {Y} _{\ell m}(\mathbf {r} )\cdot \mathbf {\Phi } _{\ell 'm'}(\mathbf {r} )&=0.\end{aligned}}}
The orthogonality relations allow one to compute the spherical multipole moments of a vector field as
E ℓ m r = ∫ E ⋅ Y ℓ m ∗ d Ω , E ℓ m ( 1 ) = 1 ℓ ( ℓ + 1 ) ∫ E ⋅ Ψ ℓ m ∗ d Ω , E ℓ m ( 2 ) = 1 ℓ ( ℓ + 1 ) ∫ E ⋅ Φ ℓ m ∗ d Ω . {\displaystyle {\begin{aligned}E_{\ell m}^{r}&=\int \mathbf {E} \cdot \mathbf {Y} _{\ell m}^{*}\,d\Omega ,\\E_{\ell m}^{(1)}&={\frac {1}{\ell (\ell +1)}}\int \mathbf {E} \cdot \mathbf {\Psi } _{\ell m}^{*}\,d\Omega ,\\E_{\ell m}^{(2)}&={\frac {1}{\ell (\ell +1)}}\int \mathbf {E} \cdot \mathbf {\Phi } _{\ell m}^{*}\,d\Omega .\end{aligned}}}
Given the multipole expansion of a scalar field
ϕ = ∑ ℓ = 0 ∞ ∑ m = − ℓ ℓ ϕ ℓ m ( r ) Y ℓ m ( θ , ϕ ) , {\displaystyle \phi =\sum _{\ell =0}^{\infty }\sum _{m=-\ell }^{\ell }\phi _{\ell m}(r)Y_{\ell m}(\theta ,\phi ),}
we can express its gradient in terms of the VSH as
∇ ϕ = ∑ ℓ = 0 ∞ ∑ m = − ℓ ℓ ( d ϕ ℓ m d r Y ℓ m + ϕ ℓ m r Ψ ℓ m ) . {\displaystyle \nabla \phi =\sum _{\ell =0}^{\infty }\sum _{m=-\ell }^{\ell }\left({\frac {d\phi _{\ell m}}{dr}}\mathbf {Y} _{\ell m}+{\frac {\phi _{\ell m}}{r}}\mathbf {\Psi } _{\ell m}\right).}
For any multipole field we have
∇ ⋅ ( f ( r ) Y ℓ m ) = ( d f d r + 2 r f ) Y ℓ m , ∇ ⋅ ( f ( r ) Ψ ℓ m ) = − ℓ ( ℓ + 1 ) r f Y ℓ m , ∇ ⋅ ( f ( r ) Φ ℓ m ) = 0. {\displaystyle {\begin{aligned}\nabla \cdot \left(f(r)\mathbf {Y} _{\ell m}\right)&=\left({\frac {df}{dr}}+{\frac {2}{r}}f\right)Y_{\ell m},\\\nabla \cdot \left(f(r)\mathbf {\Psi } _{\ell m}\right)&=-{\frac {\ell (\ell +1)}{r}}fY_{\ell m},\\\nabla \cdot \left(f(r)\mathbf {\Phi } _{\ell m}\right)&=0.\end{aligned}}}
By superposition we obtain the divergence of any vector field:
∇ ⋅ E = ∑ ℓ = 0 ∞ ∑ m = − ℓ ℓ ( d E ℓ m r d r + 2 r E ℓ m r − ℓ ( ℓ + 1 ) r E ℓ m ( 1 ) ) Y ℓ m . {\displaystyle \nabla \cdot \mathbf {E} =\sum _{\ell =0}^{\infty }\sum _{m=-\ell }^{\ell }\left({\frac {dE_{\ell m}^{r}}{dr}}+{\frac {2}{r}}E_{\ell m}^{r}-{\frac {\ell (\ell +1)}{r}}E_{\ell m}^{(1)}\right)Y_{\ell m}.}
We see that the component on Φ ℓm is always solenoidal .
For any multipole field we have
∇ × ( f ( r ) Y ℓ m ) = − 1 r f Φ ℓ m , ∇ × ( f ( r ) Ψ ℓ m ) = ( d f d r + 1 r f ) Φ ℓ m , ∇ × ( f ( r ) Φ ℓ m ) = − ℓ ( ℓ + 1 ) r f Y ℓ m − ( d f d r + 1 r f ) Ψ ℓ m . {\displaystyle {\begin{aligned}\nabla \times \left(f(r)\mathbf {Y} _{\ell m}\right)&=-{\frac {1}{r}}f\mathbf {\Phi } _{\ell m},\\\nabla \times \left(f(r)\mathbf {\Psi } _{\ell m}\right)&=\left({\frac {df}{dr}}+{\frac {1}{r}}f\right)\mathbf {\Phi } _{\ell m},\\\nabla \times \left(f(r)\mathbf {\Phi } _{\ell m}\right)&=-{\frac {\ell (\ell +1)}{r}}f\mathbf {Y} _{\ell m}-\left({\frac {df}{dr}}+{\frac {1}{r}}f\right)\mathbf {\Psi } _{\ell m}.\end{aligned}}}
By superposition we obtain the curl of any vector field:
∇ × E = ∑ ℓ = 0 ∞ ∑ m = − ℓ ℓ ( − ℓ ( ℓ + 1 ) r E ℓ m ( 2 ) Y ℓ m − ( d E ℓ m ( 2 ) d r + 1 r E ℓ m ( 2 ) ) Ψ ℓ m + ( − 1 r E ℓ m r + d E ℓ m ( 1 ) d r + 1 r E ℓ m ( 1 ) ) Φ ℓ m ) . {\displaystyle \nabla \times \mathbf {E} =\sum _{\ell =0}^{\infty }\sum _{m=-\ell }^{\ell }\left(-{\frac {\ell (\ell +1)}{r}}E_{\ell m}^{(2)}\mathbf {Y} _{\ell m}-\left({\frac {dE_{\ell m}^{(2)}}{dr}}+{\frac {1}{r}}E_{\ell m}^{(2)}\right)\mathbf {\Psi } _{\ell m}+\left(-{\frac {1}{r}}E_{\ell m}^{r}+{\frac {dE_{\ell m}^{(1)}}{dr}}+{\frac {1}{r}}E_{\ell m}^{(1)}\right)\mathbf {\Phi } _{\ell m}\right).}
The action of the Laplace operator Δ = ∇ ⋅ ∇ {\displaystyle \Delta =\nabla \cdot \nabla } separates as follows:
Δ ( f ( r ) Z ℓ m ) = ( 1 r 2 ∂ ∂ r r 2 ∂ f ∂ r ) Z ℓ m + f ( r ) Δ Z ℓ m , {\displaystyle \Delta \left(f(r)\mathbf {Z} _{\ell m}\right)=\left({\frac {1}{r^{2}}}{\frac {\partial }{\partial r}}r^{2}{\frac {\partial f}{\partial r}}\right)\mathbf {Z} _{\ell m}+f(r)\Delta \mathbf {Z} _{\ell m},} where Z ℓ m = Y ℓ m , Ψ ℓ m , Φ ℓ m {\displaystyle \mathbf {Z} _{\ell m}=\mathbf {Y} _{\ell m},\mathbf {\Psi } _{\ell m},\mathbf {\Phi } _{\ell m}} and
Δ Y ℓ m = − 1 r 2 ( 2 + ℓ ( ℓ + 1 ) ) Y ℓ m + 2 r 2 Ψ ℓ m , Δ Ψ ℓ m = 2 r 2 ℓ ( ℓ + 1 ) Y ℓ m − 1 r 2 ℓ ( ℓ + 1 ) Ψ ℓ m , Δ Φ ℓ m = − 1 r 2 ℓ ( ℓ + 1 ) Φ ℓ m . {\displaystyle {\begin{aligned}\Delta \mathbf {Y} _{\ell m}&=-{\frac {1}{r^{2}}}(2+\ell (\ell +1))\mathbf {Y} _{\ell m}+{\frac {2}{r^{2}}}\mathbf {\Psi } _{\ell m},\\\Delta \mathbf {\Psi } _{\ell m}&={\frac {2}{r^{2}}}\ell (\ell +1)\mathbf {Y} _{\ell m}-{\frac {1}{r^{2}}}\ell (\ell +1)\mathbf {\Psi } _{\ell m},\\\Delta \mathbf {\Phi } _{\ell m}&=-{\frac {1}{r^{2}}}\ell (\ell +1)\mathbf {\Phi } _{\ell m}.\end{aligned}}}
Also note that this action becomes symmetric , i.e. the off-diagonal coefficients are equal to 2 r 2 ℓ ( ℓ + 1 ) {\textstyle {\frac {2}{r^{2}}}{\sqrt {\ell (\ell +1)}}} , for properly normalized VSH.
Expressions for negative values of m are obtained by applying the symmetry relations.
The VSH are especially useful in the study of multipole radiation fields . For instance, a magnetic multipole is due to an oscillating current with angular frequency ω {\displaystyle \omega } and complex amplitude
J ^ = J ( r ) Φ ℓ m , {\displaystyle {\hat {\mathbf {J} }}=J(r)\mathbf {\Phi } _{\ell m},}
and the corresponding electric and magnetic fields, can be written as
E ^ = E ( r ) Φ ℓ m , B ^ = B r ( r ) Y ℓ m + B ( 1 ) ( r ) Ψ ℓ m . {\displaystyle {\begin{aligned}{\hat {\mathbf {E} }}&=E(r)\mathbf {\Phi } _{\ell m},\\{\hat {\mathbf {B} }}&=B^{r}(r)\mathbf {Y} _{\ell m}+B^{(1)}(r)\mathbf {\Psi } _{\ell m}.\end{aligned}}}
Substituting into Maxwell equations, Gauss's law is automatically satisfied
∇ ⋅ E ^ = 0 , {\displaystyle \nabla \cdot {\hat {\mathbf {E} }}=0,}
while Faraday's law decouples as
∇ × E ^ = − i ω B ^ ⇒ { ℓ ( ℓ + 1 ) r E = i ω B r , d E d r + E r = i ω B ( 1 ) . {\displaystyle \nabla \times {\hat {\mathbf {E} }}=-i\omega {\hat {\mathbf {B} }}\quad \Rightarrow \quad {\begin{cases}{\dfrac {\ell (\ell +1)}{r}}E=i\omega B^{r},\\{\dfrac {dE}{dr}}+{\dfrac {E}{r}}=i\omega B^{(1)}.\end{cases}}}
Gauss' law for the magnetic field implies
∇ ⋅ B ^ = 0 ⇒ d B r d r + 2 r B r − ℓ ( ℓ + 1 ) r B ( 1 ) = 0 , {\displaystyle \nabla \cdot {\hat {\mathbf {B} }}=0\quad \Rightarrow \quad {\frac {dB^{r}}{dr}}+{\frac {2}{r}}B^{r}-{\frac {\ell (\ell +1)}{r}}B^{(1)}=0,}
and Ampère–Maxwell's equation gives
∇ × B ^ = μ 0 J ^ + i μ 0 ε 0 ω E ^ ⇒ − B r r + d B ( 1 ) d r + B ( 1 ) r = μ 0 J + i ω μ 0 ε 0 E . {\displaystyle \nabla \times {\hat {\mathbf {B} }}=\mu _{0}{\hat {\mathbf {J} }}+i\mu _{0}\varepsilon _{0}\omega {\hat {\mathbf {E} }}\quad \Rightarrow \quad -{\frac {B^{r}}{r}}+{\frac {dB^{(1)}}{dr}}+{\frac {B^{(1)}}{r}}=\mu _{0}J+i\omega \mu _{0}\varepsilon _{0}E.}
In this way, the partial differential equations have been transformed into a set of ordinary differential equations.
In many applications, vector spherical harmonics are defined as fundamental set of the solutions of vector Helmholtz equation in spherical coordinates. [ 6 ] [ 7 ]
In this case, vector spherical harmonics are generated by scalar functions, which are solutions of scalar Helmholtz equation with the wavevector k {\displaystyle \mathbf {k} } . ψ e m n = cos m φ P n m ( cos ϑ ) z n ( k r ) ψ o m n = sin m φ P n m ( cos ϑ ) z n ( k r ) {\displaystyle {\begin{array}{l}{\psi _{emn}=\cos m\varphi P_{n}^{m}(\cos \vartheta )z_{n}({k}r)}\\{\psi _{omn}=\sin m\varphi P_{n}^{m}(\cos \vartheta )z_{n}({k}r)}\end{array}}} here P n m ( cos θ ) {\displaystyle P_{n}^{m}(\cos \theta )} are the associated Legendre polynomials , and z n ( k r ) {\displaystyle z_{n}({k}r)} are any of the spherical Bessel functions .
Vector spherical harmonics are defined as:
Here we use harmonics real-valued angular part, where m ≥ 0 {\displaystyle m\geq 0} , but complex functions can be introduced in the same way.
Let us introduce the notation ρ = k r {\displaystyle \rho =kr} . In the component form vector spherical harmonics are written as: M e m n ( k , r ) = − m sin ( θ ) sin ( m φ ) P n m ( cos ( θ ) ) z n ( ρ ) e θ − cos ( m φ ) d P n m ( cos ( θ ) ) d θ z n ( ρ ) e φ {\displaystyle {\begin{aligned}{\mathbf {M} _{emn}(k,\mathbf {r} )=\qquad {{\frac {-m}{\sin(\theta )}}\sin(m\varphi )P_{n}^{m}(\cos(\theta ))}z_{n}(\rho )\mathbf {e} _{\theta }}\\{{}-\cos(m\varphi ){\frac {dP_{n}^{m}(\cos(\theta ))}{d\theta }}}z_{n}(\rho )\mathbf {e} _{\varphi }\end{aligned}}} M o m n ( k , r ) = m sin ( θ ) cos ( m φ ) P n m ( cos ( θ ) ) z n ( ρ ) e θ − sin ( m φ ) d P n m ( cos ( θ ) ) d θ z n ( ρ ) e φ {\displaystyle {\begin{aligned}{\mathbf {M} _{omn}(k,\mathbf {r} )=\qquad {{\frac {m}{\sin(\theta )}}\cos(m\varphi )P_{n}^{m}(\cos(\theta ))}}z_{n}(\rho )\mathbf {e} _{\theta }\\{{}-\sin(m\varphi ){\frac {dP_{n}^{m}(\cos(\theta ))}{d\theta }}z_{n}(\rho )\mathbf {e} _{\varphi }}\end{aligned}}}
N e m n ( k , r ) = z n ( ρ ) ρ cos ( m φ ) n ( n + 1 ) P n m ( cos ( θ ) ) e r + cos ( m φ ) d P n m ( cos ( θ ) ) d θ 1 ρ d d ρ [ ρ z n ( ρ ) ] e θ − m sin ( m φ ) P n m ( cos ( θ ) ) sin ( θ ) 1 ρ d d ρ [ ρ z n ( ρ ) ] e φ {\displaystyle {\begin{aligned}{\mathbf {N} _{emn}(k,\mathbf {r} )=\qquad {\frac {z_{n}(\rho )}{\rho }}\cos(m\varphi )n(n+1)P_{n}^{m}(\cos(\theta ))\mathbf {e} _{\mathbf {r} }}\\{{}+\cos(m\varphi ){\frac {dP_{n}^{m}(\cos(\theta ))}{d\theta }}}{\frac {1}{\rho }}{\frac {d}{d\rho }}\left[\rho z_{n}(\rho )\right]\mathbf {e} _{\theta }\\{{}-m\sin(m\varphi ){\frac {P_{n}^{m}(\cos(\theta ))}{\sin(\theta )}}}{\frac {1}{\rho }}{\frac {d}{d\rho }}\left[\rho z_{n}(\rho )\right]\mathbf {e} _{\varphi }\end{aligned}}}
N o m n ( k , r ) = z n ( ρ ) ρ sin ( m φ ) n ( n + 1 ) P n m ( cos ( θ ) ) e r + sin ( m φ ) d P n m ( cos ( θ ) ) d θ 1 ρ d d ρ [ ρ z n ( ρ ) ] e θ + m cos ( m φ ) P n m ( cos ( θ ) ) sin ( θ ) 1 ρ d d ρ [ ρ z n ( ρ ) ] e φ {\displaystyle {\begin{aligned}\mathbf {N} _{omn}(k,\mathbf {r} )=\qquad {\frac {z_{n}(\rho )}{\rho }}\sin(m\varphi )n(n+1)P_{n}^{m}(\cos(\theta ))\mathbf {e} _{\mathbf {r} }\\{}+\sin(m\varphi ){\frac {dP_{n}^{m}(\cos(\theta ))}{d\theta }}{\frac {1}{\rho }}{\frac {d}{d\rho }}\left[\rho z_{n}(\rho )\right]\mathbf {e} _{\theta }\\{}+{m\cos(m\varphi ){\frac {P_{n}^{m}(\cos(\theta ))}{\sin(\theta )}}}{\frac {1}{\rho }}{\frac {d}{d\rho }}\left[\rho z_{n}(\rho )\right]\mathbf {e} _{\varphi }\end{aligned}}} There is no radial part for magnetic harmonics. For electric harmonics, the radial part decreases faster than angular, and for big ρ {\displaystyle \rho } can be neglected. We can also see that for electric and magnetic harmonics angular parts are the same up to permutation of the polar and azimuthal unit vectors, so for big ρ {\displaystyle \rho } electric and magnetic harmonics vectors are equal in value and perpendicular to each other.
Longitudinal harmonics: L o e m n ( k , r ) = ∂ ∂ r z n ( k r ) P n m ( cos θ ) sin cos m φ e r + 1 r z n ( k r ) ∂ ∂ θ P n m ( cos θ ) sin cos m φ e θ ∓ m r sin θ z n ( k r ) P n m ( cos θ ) cos sin m φ e φ {\displaystyle {\begin{aligned}\mathbf {L} _{^{e}_{o}{mn}}(k,\mathbf {r} ){}=\qquad &{\frac {\partial }{\partial r}}z_{n}(kr)P_{n}^{m}(\cos \theta ){^{\cos }_{\sin }}{m\varphi }\mathbf {e} _{r}\\{}+{}&{\frac {1}{r}}z_{n}(kr){\frac {\partial }{\partial \theta }}P_{n}^{m}(\cos \theta ){^{\cos }_{\sin }}m\varphi \mathbf {e} _{\theta }\\{}\mp {}&{\frac {m}{r\sin \theta }}z_{n}(kr)P_{n}^{m}(\cos \theta ){^{\sin }_{\cos }}m\varphi \mathbf {e} _{\varphi }\end{aligned}}}
The solutions of the Helmholtz vector equation obey the following orthogonality relations: [ 7 ] ∫ 0 2 π ∫ 0 π L o e m n ⋅ L o e m n sin ϑ d ϑ d φ = ( 1 + δ m , 0 ) 2 π ( 2 n + 1 ) 2 ( n + m ) ! ( n − m ) ! k 2 { n [ z n − 1 ( k r ) ] 2 + ( n + 1 ) [ z n + 1 ( k r ) ] 2 } ∫ 0 2 π ∫ 0 π M o e m n ⋅ M o e m n sin ϑ d ϑ d φ = ( 1 + δ m , 0 ) 2 π 2 n + 1 ( n + m ) ! ( n − m ) ! n ( n + 1 ) [ z n ( k r ) ] 2 ∫ 0 2 π ∫ 0 π N o e m n ⋅ N o e m n sin ϑ d ϑ d φ = ( 1 + δ m , 0 ) 2 π ( 2 n + 1 ) 2 ( n + m ) ! ( n − m ) ! n ( n + 1 ) { ( n + 1 ) [ z n − 1 ( k r ) ] 2 + n [ z n + 1 ( k r ) ] 2 } ∫ 0 π ∫ 0 2 π L o e m n ⋅ N o e m n sin ϑ d ϑ d φ = ( 1 + δ m , 0 ) 2 π ( 2 n + 1 ) 2 ( n + m ) ! ( n − m ) ! n ( n + 1 ) k { [ z n − 1 ( k r ) ] 2 − [ z n + 1 ( k r ) ] 2 } {\displaystyle {\begin{aligned}\int _{0}^{2\pi }\int _{0}^{\pi }\mathbf {L} _{^{e}_{o}mn}\cdot \mathbf {L} _{^{e}_{o}mn}\sin \vartheta d\vartheta d\varphi &=(1+\delta _{m,0}){\frac {2\pi }{(2n+1)^{2}}}{\frac {(n+m)!}{(n-m)!}}k^{2}\left\{n\left[z_{n-1}(kr)\right]^{2}+(n+1)\left[z_{n+1}(kr)\right]^{2}\right\}\\[3pt]\int _{0}^{2\pi }\int _{0}^{\pi }\mathbf {M} _{^{e}_{o}mn}\cdot \mathbf {M} _{^{e}_{o}mn}\sin \vartheta d\vartheta d\varphi &=(1+\delta _{m,0}){\frac {2\pi }{2n+1}}{\frac {(n+m)!}{(n-m)!}}n(n+1)\left[z_{n}(kr)\right]^{2}\\[3pt]\int _{0}^{2\pi }\int _{0}^{\pi }\mathbf {N} _{^{e}_{o}mn}\cdot \mathbf {N} _{^{e}_{o}mn}\sin \vartheta d\vartheta d\varphi &=(1+\delta _{m,0}){\frac {2\pi }{(2n+1)^{2}}}{\frac {(n+m)!}{(n-m)!}}n(n+1)\left\{(n+1)\left[z_{n-1}(kr)\right]^{2}+n\left[z_{n+1}(kr)\right]^{2}\right\}\\[3pt]\int _{0}^{\pi }\int _{0}^{2\pi }\mathbf {L} _{^{e}_{o}mn}\cdot \mathbf {N} _{^{e}_{o}mn}\sin \vartheta d\vartheta d\varphi &=(1+\delta _{m,0}){\frac {2\pi }{(2n+1)^{2}}}{\frac {(n+m)!}{(n-m)!}}n(n+1)k\left\{\left[z_{n-1}(kr)\right]^{2}-\left[z_{n+1}(kr)\right]^{2}\right\}\end{aligned}}}
All other integrals over the angles between different functions or functions with different indices are equal to zero.
Under rotation, vector spherical harmonics are transformed through each other in the same way as the corresponding scalar spherical functions , which are generating for a specific type of vector harmonics. For example, if the generating functions are the usual spherical harmonics , then the vector harmonics will also be transformed through the Wigner D-matrices [ 8 ] [ 9 ] [ 10 ] D ^ ( α , β , γ ) Y J M ( s ) ( θ , φ ) = ∑ M ′ = − J J [ D M M ′ ( J ) ( α , β , γ ) ] ∗ Y J M ′ ( s ) ( θ , φ ) , {\displaystyle {\hat {D}}(\alpha ,\beta ,\gamma )\mathbf {Y} _{JM}^{(s)}(\theta ,\varphi )=\sum _{M'=-J}^{J}[D_{MM'}^{(J)}(\alpha ,\beta ,\gamma )]^{*}\mathbf {Y} _{JM'}^{(s)}(\theta ,\varphi ),} The behavior under rotations is the same for electrical, magnetic and longitudinal harmonics.
Under inversion, electric and longitudinal spherical harmonics behave in the same way as scalar spherical functions, i.e. I ^ N J M ( θ , φ ) = ( − 1 ) J N J M ( θ , φ ) , {\displaystyle {\hat {I}}\mathbf {N} _{JM}(\theta ,\varphi )=(-1)^{J}\mathbf {N} _{JM}(\theta ,\varphi ),} and magnetic ones have the opposite parity: I ^ M J M ( θ , φ ) = ( − 1 ) J + 1 M J M ( θ , φ ) , {\displaystyle {\hat {I}}\mathbf {M} _{JM}(\theta ,\varphi )=(-1)^{J+1}\mathbf {M} _{JM}(\theta ,\varphi ),}
In the calculation of the Stokes' law for the drag that a viscous fluid exerts on a small spherical particle, the velocity distribution obeys Navier–Stokes equations neglecting inertia, i.e.,
0 = ∇ ⋅ v , 0 = − ∇ p + η ∇ 2 v , {\displaystyle {\begin{aligned}0&=\nabla \cdot \mathbf {v} ,\\\mathbf {0} &=-\nabla p+\eta \nabla ^{2}\mathbf {v} ,\end{aligned}}}
with the boundary conditions
v = { 0 r = a , − U 0 r → ∞ . {\displaystyle \mathbf {v} ={\begin{cases}\mathbf {0} &r=a,\\-\mathbf {U} _{0}&r\to \infty .\end{cases}}}
where U is the relative velocity of the particle to the fluid far from the particle. In spherical coordinates this velocity at infinity can be written as
U 0 = U 0 ( cos θ r ^ − sin θ θ ^ ) = U 0 ( Y 10 + Ψ 10 ) . {\displaystyle \mathbf {U} _{0}=U_{0}\left(\cos \theta \,{\hat {\mathbf {r} }}-\sin \theta \,{\hat {\mathbf {\theta } }}\right)=U_{0}\left(\mathbf {Y} _{10}+\mathbf {\Psi } _{10}\right).}
The last expression suggests an expansion in spherical harmonics for the liquid velocity and the pressure
p = p ( r ) Y 10 , v = v r ( r ) Y 10 + v ( 1 ) ( r ) Ψ 10 . {\displaystyle {\begin{aligned}p&=p(r)Y_{10},\\\mathbf {v} &=v^{r}(r)\mathbf {Y} _{10}+v^{(1)}(r)\mathbf {\Psi } _{10}.\end{aligned}}}
Substitution in the Navier–Stokes equations produces a set of ordinary differential equations for the coefficients.
Here the following definitions are used:
Y e m n = cos m φ P n m ( cos θ ) Y o m n = sin m φ P n m ( cos θ ) {\displaystyle {\begin{aligned}Y_{emn}&=\cos m\varphi P_{n}^{m}(\cos \theta )\\Y_{omn}&=\sin m\varphi P_{n}^{m}(\cos \theta )\end{aligned}}}
X o e m n ( k k ) = ∇ × ( k Y e o m n ( k k ) ) {\displaystyle \mathbf {X} _{^{e}_{o}mn}\left({\frac {\mathbf {k} }{k}}\right)=\nabla \times \left(\mathbf {k} Y_{^{o}_{e}mn}\left({\frac {\mathbf {k} }{k}}\right)\right)}
Z e o m n ( k k ) = i k k × X o e m n ( k k ) {\displaystyle \mathbf {Z} _{^{o}_{e}mn}\left({\frac {\mathbf {k} }{k}}\right)=i{\frac {\mathbf {k} }{k}}\times \mathbf {X} _{^{e}_{o}mn}\left({\frac {\mathbf {k} }{k}}\right)} In case, when instead of z n {\displaystyle z_{n}} are spherical Bessel functions , with help of plane wave expansion one can obtain the following integral relations: [ 11 ]
N p m n ( 1 ) ( k , r ) = i − n 4 π ∫ Z p m n ( k k ) e i k ⋅ r d Ω k {\displaystyle \mathbf {N} _{pmn}^{(1)}(k,\mathbf {r} )={\frac {i^{-n}}{4\pi }}\int \mathbf {Z} _{pmn}\left({\frac {\mathbf {k} }{k}}\right)e^{i\mathbf {k} \cdot \mathbf {r} }d\Omega _{k}}
M p m n ( 1 ) ( k , r ) = i − n 4 π ∫ X p m n ( k k ) e i k ⋅ r d Ω k {\displaystyle \mathbf {M} _{pmn}^{(1)}(k,\mathbf {r} )={\frac {i^{-n}}{4\pi }}\int \mathbf {X} _{pmn}\left({\frac {\mathbf {k} }{k}}\right)e^{i\mathbf {k} \cdot \mathbf {r} }d\Omega _{k}}
In case, when z n {\displaystyle z_{n}} are spherical Hankel functions, one should use the different formulae. [ 12 ] [ 11 ] For vector spherical harmonics the following relations are obtained:
M p m n ( 3 ) ( k , r ) = i − n 2 π k ∬ − ∞ ∞ d k ‖ e i ( k x x + k y y ± k z z ) k z X p m n ( k k ) {\displaystyle \mathbf {M} _{pmn}^{(3)}(k,\mathbf {r} )={\frac {i^{-n}}{2\pi k}}\iint _{-\infty }^{\infty }dk_{\|}{\frac {e^{i\left(k_{x}x+k_{y}y\pm k_{z}z\right)}}{k_{z}}}\mathbf {X} _{pmn}\left({\frac {\mathbf {k} }{k}}\right)}
N p m n ( 3 ) ( k , r ) = i − n 2 π k ∬ − ∞ ∞ d k ‖ e i ( k x x + k y y ± k z z ) k z Z p m n ( k k ) {\displaystyle \mathbf {N} _{pmn}^{(3)}(k,\mathbf {r} )={\frac {i^{-n}}{2\pi k}}\iint _{-\infty }^{\infty }dk_{\|}{\frac {e^{i\left(k_{x}x+k_{y}y\pm k_{z}z\right)}}{k_{z}}}\mathbf {Z} _{pmn}\left({\frac {\mathbf {k} }{k}}\right)} where k z = k 2 − k x 2 − k y 2 {\textstyle k_{z}={\sqrt {k^{2}-k_{x}^{2}-k_{y}^{2}}}} , index ( 3 ) {\displaystyle (3)} means, that spherical Hankel functions are used.
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Vectorette PCR is a variation of polymerase chain reaction (PCR) designed in 1988. [ 1 ] The original PCR was created and also patented during the 1980s. [ 2 ] Vectorette PCR was first noted and described in an article in 1990 by John H. Riley and his team. [ 3 ] Since then, multiple variants of PCR have been created. Vectorette PCR focuses on amplifying a specific sequence obtained from an internal sequence that is originally known until the fragment end. [ 4 ] Multiple researches have taken this method as an opportunity to conduct experiments in order to uncover the potential uses that can be derived from Vectorette PCR. [ 1 ]
Vectorette PCR is similar to PCR with the difference being that it is capable of obtaining the sequence desired for amplification from an already known primer site. [ 5 ] While PCR needs information of already known sequences at both ends, Vectorette PCR only requires previous knowledge of one. [ 1 ] This means that is able to apply the method of PCR which needs sequence information from both ends to fragments of DNA that contain the information of the sequence at only one end and not the other. [ 6 ] [ 7 ] In order to achieve this, there are specific steps that this method must first go through. These steps have been researched for the purpose of discovering the scientific uses of Vectorette PCR and how they can be applied. [ 1 ]
Vectorette PCR can develop a strategy to bring about PCR amplification that is unidirectional. [ 8 ] Vectorette PCR comprises three main steps . [ 1 ] The first step includes utilizing a restriction enzyme in order to accomplish digestion of the sample DNA. [ 1 ] [ 6 ] The DNA that is to be utilized for the purpose of investigation has to be capable of being digested by restriction enzymes that are appropriate for that gene otherwise the DNA fragments that form the general population cannot be created. [ 9 ] After that is completed, a Vectorette library is brought together by ligating the Vectorette units to the appropriate DNA fragments which were previously digested. [ 1 ] [ 6 ] Ligation is the act of binding two things together. [ 10 ] A Vectorette unit is only partially not completely double stranded with a mismatched section located in the center of the unit. [ 11 ] The reason it is mismatched is to help it avoid Vectorette primers’ attempts at causing it to undergo first strand synthesis. By doing this any priming that is nonspecific is also avoided. [ 11 ] This ligation brings together the vectorette which is double stranded and the ends of the restriction fragments which were previously made in the first step. [ 12 ] By doing this, the known sequence which is used to prime the PCR reaction at one side is introduced while the other is primed on the genomic sequence which is already known to the user. [ 12 ] The third and last step has two parts to it. This is due to there being two primers, the initiating primer (IP) and the Vectorette primer (VP), that act in different stages. During the first part, the IP works on amplifying the primer extension while the VP remains hybridized with the product; thus, any background amplification is not carried out at this stage. However, this changes during the last and following part of PCR as the priming that is performed comes from both the IP and the VP. [ 6 ]
A lot of research has been conducted on Vectorette PCR and the applications it has in the field of biology. Scientists used Vectorette PCR to take the transgene flanking DNA and isolate it. They used this technique on the DNA belonging to mice that was next to transgene sections. From this the scientists were able to show that the use of Vectorettes is capable of facilitating the recovery and mapping of sequences in complex genomes . They have also found that Vectorette PCR can help in the analysis of sequences by subvectoretting when PCR products of a large size are the subject at hand. [ 5 ]
Other work has looked at developing a method using Vectorette PCR in order to accomplish genomic walking. By using Vectorette PCR, scientists were able to acquire single-stranded DNA which were obtained from PCR products in order to sequence them. From this an approach was identified in which the amplification of sequences which were previously uncharacterized was possible. This research demonstrates how novel sequences can be rapidly developed when only a known sequence of DNA is used to start. [ 6 ]
Further research has experimented with the creation of a method that progresses the isolation of microsatellite repeats. By using Vectorette PCR, researchers have found a rapid technique to accomplish this with novel, microsatellite repeats. They have attempted and succeeded in using this technique to isolate an amount of six microsatellite repeats. [ 13 ]
Vectorette PCR has also been used to not only identify genomic positions of insertion sequences (IS) but also to map them. Research on this has shed light on a way to complete the typing of microbial stains and the identification and mapping of things like IS insertion sites that reside in microbial genomes. Vectorette PCR proves useful when it comes to rapidly and simply surveying genomes’ IS elements. [ 14 ]
Transposable element , transposon, or TE is a variation of genetic elements that is capable of changing its location in a genome by a process called “jumping”. [ 15 ] TE display is designed to present the different variations of TE insertion sites which helps to make numerous dominant markers. [ 16 ] A problem that arose in the original method was finding a PCR method that was capable of being specific and efficient in its output of the transposon within the genome. [ 16 ] Researchers have found a solution for this problem by using Vectorette PCR as the PCR method. Since Vectorette PCR is capable of being specific with its isolation and amplification of genes, this helped with their research and aided in improving the method of TE display by saving both time and costs. [ 16 ] The researchers were then able to produce numerous dominant markers with the use of Vectorette PCR that is based on a TE display that is nonradioactive. [ 16 ]
Thyroid lymphoma is an illness which leads to the transformation of the lymphocytes belonging to the thyroid into cells of a cancerous nature. [ 17 ] Researchers have tested a new method that aids in the diagnosis of this condition. The use of Vectorette PCR was combined with restriction enzyme digestion, and it was found that Vectorette PCR proved to be useful in their study and aided in the diagnosis of thyroid lymphoma. [ 18 ]
Researchers have looked into the potential use of Vectorette PCR in the examination of the genes of diseases. They have taken two methods, trinucleotide repeats which are specifically used for the targeting of transcribed regions and Vectorette PCR, to obtain simple sequence repeats or SSRs. [ 19 ] It is believed that genetic markers can be made from these SSRs. The outcome from this research is hoped to aid researchers attempt the derivation of genetic markers which are transportable from unknown genomes. Vectorette PCR was used to uncover SSRs which flank the trinucleotide repeat that was targeted for testing. [ 19 ] This is also known as TNR or trinucleotide Vectorette PCR. They believe that their TNR method combined with the amplification provided by Vectorette PCR can be used in eukaryotes to create molecular markers that are based on simple repeat sequences. The researchers also think that this method will be of value when attempting to isolate genes that are able to bring about diseases. [ 19 ]
The uses that have been derived from Vectorette PCR are many and have been useful to the science of biology . For example, it gives rise to methods that can help during the outbreaks of diseases by making it easier to subtype pathogens that are similar or closely related. [ 14 ] It can also be used to help diagnose certain diseases. [ 18 ] Earlier in this page it was noted that Vectorette PCR can give rise to multiple functions that can be performed on novel DNA sequences located near a sequence that is already known. These functions like isolating DNA, amplifying it, and analyzing it are behind the uses for Vectorette PCR. [ 1 ] These uses are things like genome walking, DNA sequencing for the termini of Yeast Artificial Chromosomes (YAC) and cosmid inserts, being able to map introns and promoters in genomic DNA and regions with mutations, facilitating the sequencing of clones of a large size, and filling in the gaps that arise during the mapping of genomes. [ 1 ]
An intron is a DNA sequence that is flanked by exons and therefore located in between them. [ 20 ] It is the region that gets cut out while exons are expressed, and so introns do not affect the code of amino acids. Gene expression can be affected by only a number of intronic sequences. [ 20 ] Vectorette PCR has been found to be beneficial when it comes to the characterization of these intronic sequences when they are found to be next to known sequences. [ 21 ]
cDNA or complementary DNA is a DNA sequence which is complementary to the RNA that is the template when synthesizing DNA during the reverse transcriptase process. [ 22 ] Vectorette PCR that utilizes the primers that originate from cDNA gives rise to a method that is capable of acquiring intron sequences which are located adjacent to exons and aiding in the development of the structure of genes. [ 23 ] It is able to achieve this when initializing the process with a sequence of cDNA and a clone of a genome. [ 23 ]
Vectorette PCR also gives the user an advantage than if he/she were using other existing technologies. The user will be able to carry out tasks like gene manipulation that is cell-free, Vectorette PCR with minimal material to start with, and performing Vectorette PCR with DNA that needs not be of high purity. These advantages allow the user to save time and resources while increasing the range of DNA that can be targeted. [ 1 ]
Chromosome walking can be used for the purpose of cloning a gene. [ 24 ] It does this by using the known gene’s markers that are closest and can therefore be used in techniques like isolating DNA sequences and aiding in the sequencing and cloning of the DNA of organisms. Chromosome walking is also useful when it comes to filling in the gaps that may be present in genomes by locating clones that overlap with a library clone end. This means that for chromosome walking to be carried out, it requires a clone library of a genomic format. This is why Vectorette PCR is one of the methods that can be used to create this library for chromosome walking to occur. Vectorette PCR comes in handy when it is necessary to obtain the regions that are both upstream and downstream and flank a sequence that is already known. By obtaining these regions, it provides the library of a genomic format that chromosome walking requires. [ citation needed ]
Yeast artificial chromosome or YAC is a DNA molecule that is developed by humans to take the DNA sequences that belong to yeast cells and clone them. [ 25 ] Yeast artificial chromosomes can be inserted with fragments of DNA from the organism of interest. Yeast cells will then assimilate the yeast artificial chromosome that contains the DNA from the organism of interest. [ 25 ] The yeast cells then multiply in number and this brings about the amplification of the DNA that has been incorporated into it which is then isolated for the purpose of things like sequencing and mapping of the DNA desired i.e. the DNA originally inserted into the yeast artificial chromosome. [ 25 ] Vectorette PCR helps with this process by bringing about not only the isolation of the yeast artificial chromosome’s ends but also the amplification of the ends. [ 26 ]
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Gene therapy utilizes the delivery of DNA into cells, which can be accomplished by several methods, summarized below. The two major classes of methods are those that use recombinant viruses (sometimes called biological nanoparticles or viral vectors) and those that use naked DNA or DNA complexes (non-viral methods).
All viruses bind to their hosts and introduce their genetic material into the host cell as part of their replication cycle. This genetic material contains basic 'instructions' of how to produce more copies of these viruses, hacking the body's normal production machinery to serve the needs of the virus. The host cell will carry out these instructions and produce additional copies of the virus, leading to more and more cells becoming infected. Some types of viruses insert their genome into the host's cytoplasm, but do not actually enter the cell. Others penetrate the cell membrane disguised as protein molecules and enter the cell.
There are two main types of virus infection: lytic and lysogenic . Shortly after inserting its DNA, viruses of the lytic cycle quickly produce more viruses, burst from the cell and infect more cells. Lysogenic viruses integrate their DNA into the DNA of the host cell and may live in the body for many years before responding to a trigger. The virus reproduces as the cell does and does not inflict bodily harm until it is triggered. The trigger releases the DNA from that of the host and employs it to create new viruses. [ citation needed ]
The genetic material in retroviruses is in the form of RNA molecules, while the genetic material of their hosts is in the form of DNA. When a retrovirus infects a host cell, it will introduce its RNA together with some enzymes, namely reverse transcriptase and integrase , into the cell. This RNA molecule from the retrovirus must produce a DNA copy from its RNA molecule before it can be integrated into the genetic material of the host cell. The process of producing a DNA copy from an RNA molecule is termed reverse transcription . It is carried out by one of the enzymes carried in the virus, called reverse transcriptase . After this DNA copy is produced and is free in the nucleus of the host cell, it must be incorporated into the genome of the host cell. That is, it must be inserted into the large DNA molecules in the cell (the chromosomes). This process is done by another enzyme carried in the virus called integrase . [ citation needed ]
Now that the genetic material of the virus has been inserted, it can be said that the host cell has been modified to contain new genes. If this host cell divides later, its descendants will all contain the new genes. Sometimes the genes of the retrovirus do not express their information immediately. [ citation needed ]
One of the problems of gene therapy using retroviruses is that the integrase enzyme can insert the genetic material of the virus into any arbitrary position in the genome of the host; it randomly inserts the genetic material into a chromosome. If genetic material happens to be inserted in the middle of one of the original genes of the host cell, this gene will be disrupted ( insertional mutagenesis ). If the gene happens to be one regulating cell division, uncontrolled cell division (i.e., cancer ) can occur. This problem has recently begun to be addressed by utilizing zinc finger nucleases [ 1 ] or by including certain sequences such as the beta-globin locus control region to direct the site of integration to specific chromosomal sites.
Gene therapy trials using retroviral vectors to treat X-linked severe combined immunodeficiency (X-SCID) represent the most successful application of gene therapy to date. More than twenty patients have been treated in France and Britain, with a high rate of immune system reconstitution observed. Similar trials were restricted or halted in the US when leukemia was reported in patients treated in the French X-SCID gene therapy trial. [ 2 ] To date, four children in the French trial and one in the British trial have developed leukemia as a result of insertional mutagenesis by the retroviral vector. All but one of these children responded well to conventional anti-leukemia treatment. Gene therapy trials to treat SCID due to deficiency of the Adenosine Deaminase ( ADA ) enzyme (one form of SCID) [ 3 ] continue with relative success in the US, Britain, Ireland, Italy and Japan. [ citation needed ]
Adenoviruses are viruses that carry their genetic material in the form of double-stranded DNA. They cause respiratory, intestinal, and eye infections in humans (especially the common cold). When these viruses infect a host cell, they introduce their DNA molecule into the host. The genetic material of the adenoviruses is not incorporated (transient) into the host cell's genetic material. The DNA molecule is left free in the nucleus of the host cell, and the instructions in this extra DNA molecule are transcribed just like any other gene. The only difference is that these extra genes are not replicated when the cell is about to undergo cell division so the descendants of that cell will not have the extra gene. [ citation needed ]
As a result, treatment with the adenovirus will require re-administration in a growing cell population although the absence of integration into the host cell's genome should prevent the type of cancer seen in the SCID trials. This vector system has been promoted for treating cancer and indeed the first gene therapy product to be licensed to treat cancer, Gendicine , is an adenovirus. Gendicine, an adenoviral p53-based gene therapy was approved by the Chinese food and drug regulators in 2003 for treatment of head and neck cancer. Advexin, a similar gene therapy approach from Introgen, was turned down by the US Food and Drug Administration (FDA) in 2008. [ 4 ]
Concerns about the safety of adenovirus vectors were raised after the 1999 death of Jesse Gelsinger while participating in a gene therapy trial. Since then, work using adenovirus vectors has focused on genetically limited versions of the virus. [ citation needed ]
Cytomegalovirus (CMV) is part of the β-herpesvirus subfamily that includes roseoloviruses. CMV coevolved with an assortment of mammalian hosts, including human CMV (HCMV), murine CMV (MCMV) and rhesus CMV (RhCMV). CMVs are characterized by large DNA genomes and typically asymptomatic infection in healthy hosts.
The first investigation into cytomegalovirus (CMV) as a gene therapy vector was published in 2000. CMV's tropism for hematopoietic progenitor cells and its large genome (230 kbp) initially attracted researchers. [ 5 ] CMV-based vaccine vectors have since been used to induce T Cell response. [ 6 ] More recently, CMV containing telomerase and follistatin was intravenously and intranasally delivered in mouse studies with the intention of extending healthspan. [ 7 ]
The viral vectors described above have natural host cell populations that they infect most efficiently. Retroviruses have limited natural host cell ranges, and although adenovirus and adeno-associated virus are able to infect a relatively broader range of cells efficiently, some cell types are resistant to infection by these viruses as well. Attachment to and entry into a susceptible cell is mediated by the protein envelope on the surface of a virus. Retroviruses and adeno-associated viruses have a single protein coating their membrane, while adenoviruses are coated with both an envelope protein and fibers that extend away from the surface of the virus. The envelope proteins on each of these viruses bind to cell-surface molecules such as heparin sulfate , which localizes them upon the surface of the potential host, as well as with the specific protein receptor that either induces entry-promoting structural changes in the viral protein, or localizes the virus in endosomes wherein acidification of the lumen induces this refolding of the viral coat . In either case, entry into potential host cells requires a favorable interaction between a protein on the surface of the virus and a protein on the surface of the cell. [ citation needed ]
For the purposes of gene therapy, one might either want to limit or expand the range of cells susceptible to transduction by a gene therapy vector. To this end, many vectors have been developed in which the endogenous viral envelope proteins have been replaced by either envelope proteins from other viruses, or by chimeric proteins. Such chimera would consist of those parts of the viral protein necessary for incorporation into the virion as well as sequences meant to interact with specific host cell proteins. Viruses in which the envelope proteins have been replaced as described are referred to as pseudotyped viruses . For example, the most popular retroviral vector for use in gene therapy trials has been the lentivirus Simian immunodeficiency virus coated with the envelope proteins, G-protein , from Vesicular stomatitis virus . This vector is referred to as VSV G-pseudotyped lentivirus , and infects an almost universal set of cells. This tropism is characteristic of the VSV G-protein with which this vector is coated. Many attempts have been made to limit the tropism of viral vectors to one or a few host cell populations. This advance would allow for the systemic administration of a relatively small amount of vector. The potential for off-target cell modification would be limited, and many concerns from the medical community would be alleviated. Most attempts to limit tropism have used chimeric envelope proteins bearing antibody fragments. These vectors show great promise for the development of "magic bullet" gene therapies. [ citation needed ]
A replication-competent vector called ONYX-015 is used in replicating tumor cells. It was found that in the absence of the E1B-55Kd viral protein, adenovirus caused very rapid apoptosis of infected, p53(+) cells, and this results in dramatically reduced virus progeny and no subsequent spread. Apoptosis was mainly the result of the ability of EIA to inactivate p300. In p53(-) cells, deletion of E1B 55kd has no consequence in terms of apoptosis, and viral replication is similar to that of wild-type virus, resulting in massive killing of cells. [ citation needed ]
A replication-defective vector deletes some essential genes. These deleted genes are still necessary in the body so they are replaced with either a helper virus or a DNA molecule. [ 8 ]
Replication-defective vectors always contain a "transfer construct". The transfer construct carries the gene to be transduced or "transgene". The transfer construct also carries the sequences which are necessary for the general functioning of the viral genome: packaging sequence, repeats for replication and, when needed, priming of reverse transcription. These are denominated cis-acting elements, because they need to be on the same piece of DNA as the viral genome and the gene of interest. Trans-acting elements are viral elements, which can be encoded on a different DNA molecule. For example, the viral structural proteins can be expressed from a different genetic element than the viral genome. [ 8 ]
The herpes simplex virus is a human neurotropic virus. This is mostly examined for gene transfer in the nervous system. The wild type HSV-1 virus is able to infect neurons and evade the host immune response, but may still become reactivated and produce a lytic cycle of viral replication. Therefore, it is typical to use mutant strains of HSV-1 that are deficient in their ability to replicate. Though the latent virus is not transcriptionally apparent, it does possess neuron specific promoters that can continue to function normally. [ further explanation needed ] Antibodies to HSV-1 are common in humans, however complications due to herpes infection are somewhat rare. [ 9 ] Caution for rare cases of encephalitis must be taken and this provides some rationale to using HSV-2 as a viral vector as it generally has tropism for neuronal cells innervating the urogenital area of the body and could then spare the host of severe pathology in the brain. [ citation needed ]
Non-viral methods present certain advantages over viral methods, with simple large scale production and low host immunogenicity being just two. Previously, low levels of transfection and expression of the gene held non-viral methods at a disadvantage; however, recent advances in vector technology have yielded molecules and techniques with transfection efficiencies similar to those of viruses. [ 10 ]
This is the simplest method of non-viral transfection. Clinical trials carried out of intramuscular injection of a naked DNA plasmid have occurred with some success; however, the expression has been very low in comparison to other methods of transfection. In addition to trials with plasmids, there have been trials with naked PCR product, which have had similar or greater success. Cellular uptake of naked DNA is generally inefficient. Research efforts focusing on improving the efficiency of naked DNA uptake have yielded several novel methods, such as electroporation , sonoporation , and the use of a " gene gun ", which shoots DNA coated gold particles into the cell using high pressure gas. [ 11 ]
Electroporation is a method that uses short pulses of high voltage to carry DNA across the cell membrane. This shock is thought to cause temporary formation of pores in the cell membrane, allowing DNA molecules to pass through. Electroporation is generally efficient and works across a broad range of cell types. However, a high rate of cell death following electroporation has limited its use, including clinical applications.
More recently a newer method of electroporation, termed electron-avalanche transfection, has been used in gene therapy experiments. By using a high-voltage plasma discharge, DNA was efficiently delivered following very short (microsecond) pulses. Compared to electroporation, the technique resulted in greatly increased efficiency and less cellular damage.
The use of particle bombardment, or the gene gun , is another physical method of DNA transfection. In this technique, DNA is coated onto gold particles and loaded into a device which generates a force to achieve penetration of the DNA into the cells, leaving the gold behind on a "stopping" disk.
Sonoporation uses ultrasonic frequencies to deliver DNA into cells. The process of acoustic cavitation is thought to disrupt the cell membrane and allow DNA to move into cells.
In a method termed magnetofection , DNA is complexed to magnetic particles, and a magnet is placed underneath the tissue culture dish to bring DNA complexes into contact with a cell monolayer.
Hydrodynamic delivery involves rapid injection of a high volume of a solution into vasculature (such as into the inferior vena cava , bile duct , or tail vein ). The solution contains molecules that are to be inserted into cells, such as DNA plasmids or siRNA , and transfer of these molecules into cells is assisted by the elevated hydrostatic pressure caused by the high volume of injected solution. [ 12 ] [ 13 ] [ 14 ]
The use of synthetic oligonucleotides in gene therapy is to deactivate the genes involved in the disease process. There are several methods by which this is achieved. One strategy uses antisense specific to the target gene to disrupt the transcription of the faulty gene. Another uses small molecules of RNA called siRNA to signal the cell to cleave specific unique sequences in the mRNA transcript of the faulty gene, disrupting translation of the faulty mRNA, and therefore expression of the gene. A further strategy uses double stranded oligodeoxynucleotides as a decoy for the transcription factors that are required to activate the transcription of the target gene. The transcription factors bind to the decoys instead of the promoter of the faulty gene, which reduces the transcription of the target gene, lowering expression. Additionally, single stranded DNA oligonucleotides have been used to direct a single base change within a mutant gene. The oligonucleotide is designed to anneal with complementarity to the target gene with the exception of a central base, the target base, which serves as the template base for repair. This technique is referred to as oligonucleotide mediated gene repair, targeted gene repair, or targeted nucleotide alteration.
To improve the delivery of the new DNA into the cell, the DNA must be protected from damage and positively charged. Initially, anionic and neutral lipids were used for the construction of lipoplexes for synthetic vectors. However, in spite of the facts that there is little toxicity associated with them, that they are compatible with body fluids and that there was a possibility of adapting them to be tissue specific; they are complicated and time-consuming to produce so attention was turned to the cationic versions.
Cationic lipids , due to their positive charge, were first used to condense negatively charged DNA molecules so as to facilitate the encapsulation of DNA into liposomes. Later it was found that the use of cationic lipids significantly enhanced the stability of lipoplexes. Also as a result of their charge, cationic liposomes interact with the cell membrane, endocytosis was widely believed as the major route by which cells uptake lipoplexes. Endosomes are formed as the results of endocytosis, however, if genes can not be released into cytoplasm by breaking the membrane of endosome, they will be sent to lysosomes where all DNA will be destroyed before they could achieve their functions. It was also found that although cationic lipids themselves could condense and encapsulate DNA into liposomes, the transfection efficiency is very low due to the lack of ability in terms of "endosomal escaping". However, when helper lipids (usually electroneutral lipids, such as DOPE) were added to form lipoplexes, much higher transfection efficiency was observed. Later on, it was discovered that certain lipids have the ability to destabilize endosomal membranes so as to facilitate the escape of DNA from endosome, therefore those lipids are called fusogenic lipids. Although cationic liposomes have been widely used as an alternative for gene delivery vectors, a dose dependent toxicity of cationic lipids were also observed which could limit their therapeutic usages. [ 15 ]
The most common use of lipoplexes has been in gene transfer into cancer cells, where the supplied genes have activated tumor suppressor control genes in the cell and decrease the activity of oncogenes. Recent studies have shown lipoplexes to be useful in transfecting respiratory epithelial cells .
Polymersomes are synthetic versions of liposomes ( vesicles with a lipid bilayer ), made of amphiphilic block copolymers . They can encapsulate either hydrophilic or hydrophobic contents and can be used to deliver cargo such as DNA, proteins, or drugs to cells. Advantages of polymersomes over liposomes include greater stability, mechanical strength, blood circulation time, and storage capacity. [ 16 ] [ 17 ] [ 18 ]
Complexes of polymers with DNA are called polyplexes. [ 15 ] [ 19 ] Most polyplexes consist of cationic polymers and their fabrication is based on self-assembly by ionic interactions. One important difference between the methods of action of polyplexes and lipoplexes is that polyplexes cannot directly release their DNA load into the cytoplasm. As a result, co-transfection with endosome-lytic agents such as inactivated adenovirus was required to facilitate nanoparticle escape from the endocytic vesicle made during particle uptake. However, a better understanding of the mechanisms by which DNA can escape from endolysosomal pathway, i.e. proton sponge effect, [ 20 ] has triggered new polymer synthesis strategies such as incorporation of protonable residues in polymer backbone and has revitalized research on polycation-based systems. [ 21 ]
Due to their low toxicity, high loading capacity, and ease of fabrication, polycationic nanocarriers demonstrate great promise compared to their rivals such as viral vectors which show high immunogenicity and potential carcinogenicity, and lipid-based vectors which cause dose dependence toxicity. Polyethyleneimine [ 22 ] and chitosan are among the polymeric carriers that have been extensively studied for development of gene delivery therapeutics. Other polycationic carriers such as poly(beta-amino esters) [ 23 ] and polyphosphoramidate [ 24 ] are being added to the library of potential gene carriers. In addition to the variety of polymers and copolymers, the ease of controlling the size, shape, surface chemistry of these polymeric nano-carriers gives them an edge in targeting capability and taking advantage of enhanced permeability and retention effect. [ 25 ]
A dendrimer is a highly branched macromolecule with a spherical shape. The surface of the particle may be functionalized in many ways and many of the properties of the resulting construct are determined by its surface.
In particular it is possible to construct a cationic dendrimer, i.e. one with a positive surface charge. When in the presence of genetic material such as DNA or RNA, charge complementarity leads to a temporary association of the nucleic acid with the cationic dendrimer. On reaching its destination the dendrimer-nucleic acid complex is then taken into the cell via endocytosis.
In recent years the benchmark for transfection agents has been cationic lipids. Limitations of these competing reagents have been reported to include: the lack of ability to transfect some cell types, the lack of robust active targeting capabilities, incompatibility with animal models, and toxicity. Dendrimers offer robust covalent construction and extreme control over molecule structure, and therefore size. Together these give compelling advantages compared to existing approaches.
Producing dendrimers has historically been a slow and expensive process consisting of numerous slow reactions, an obstacle that severely curtailed their commercial development. The Michigan-based company Dendritic Nanotechnologies discovered a method to produce dendrimers using kinetically driven chemistry, a process that not only reduced cost by a magnitude of three, but also cut reaction time from over a month to several days. These new "Priostar" dendrimers can be specifically constructed to carry a DNA or RNA payload that transfects cells at a high efficiency with little or no toxicity. [ citation needed ]
Inorganic nanoparticles, such as gold , silica , iron oxide (ex. magnetofection ) and calcium phosphates have been shown to be capable of gene delivery. [ 26 ] Some of the benefits of inorganic vectors is in their storage stability, low manufacturing cost and often time, low immunogenicity, and resistance to microbial attack. Nanosized materials less than 100 nm have been shown to efficiently trap the DNA or RNA and allows its escape from the endosome without degradation. Inorganics have also been shown to exhibit improved in vitro transfection for attached cell lines due to their increased density and preferential location on the base of the culture dish. Quantum dots have also been used successfully and permits the coupling of gene therapy with a stable fluorescence marker. Engineered organic nanoparticles are also under development, which could be used for co-delivery of genes and therapeutic agents. [ 27 ]
Cell-penetrating peptides (CPPs), also known as peptide transduction domains (PTDs), are short peptides (< 40 amino acids) that efficiently pass through cell membranes while being covalently or non-covalently bound to various molecules, thus facilitating these molecules' entry into cells. Cell entry occurs primarily by endocytosis but other entry mechanisms also exist. Examples of cargo molecules of CPPs include nucleic acids , liposomes , and drugs of low molecular weight. [ 28 ] [ 29 ]
CPP cargo can be directed into specific cell organelles by incorporating localization sequences into CPP sequences. For example, nuclear localization sequences are commonly used to guide CPP cargo into the nucleus. [ 30 ] For guidance into mitochondria, a mitochondrial targeting sequence can be used; this method is used in protofection (a technique that allows for foreign mitochondrial DNA to be inserted into cells' mitochondria). [ 31 ] [ 32 ]
Due to every method of gene transfer having shortcomings, there have been some hybrid methods developed that combine two or more techniques. Virosomes are one example; they combine liposomes with an inactivated HIV or influenza virus . This has been shown to have more efficient gene transfer in respiratory epithelial cells than either viral or liposomal methods alone. Other methods involve mixing other viral vectors with cationic lipids or hybridising viruses . [ citation needed ]
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https://en.wikipedia.org/wiki/Vectors_in_gene_therapy
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A vectorscope is a special type of oscilloscope used in both audio and video applications. [ 1 ] Whereas an oscilloscope or waveform monitor normally displays a plot of signal vs. time, a vectorscope displays an X-Y plot of two signals, which can reveal details about the relationship between these two signals. Vectorscopes are highly similar in operation to oscilloscopes operated in X-Y mode; however those used in video applications have specialized graticules, and accept standard television or video signals as input ( demodulating and demultiplexing the two components to be analyzed internally).
In video applications, a vectorscope supplements a waveform monitor for the purpose of measuring and testing television signals, regardless of format ( NTSC , PAL , SECAM or any number of digital television standards). While a waveform monitor allows a broadcast technician to measure the overall characteristics of a video signal, a vectorscope is used to visualize chrominance , which is encoded into the video signal as a subcarrier of specific frequencies . [ 2 ] The vectorscope locks exclusively to the chrominance subcarrier in the video signal (at 3.58 MHz for NTSC or 4.43 MHz for PAL) to drive its display. In digital applications, a vectorscope instead plots the Cb and Cr channels against each other (these are the two channels in digital formats which contain chroma information).
A vectorscope uses an overlaid circular reference display, or graticule , for visualizing chrominance signals, which is the best method of referring to the QAM scheme used to encode color into a video signal. The actual visual pattern that the incoming chrominance signal draws on the vectorscope is called the trace . Chrominance is measured using two methods—color saturation , encoded as the amplitude , or gain , of the subcarrier signal, and hue , encoded as the subcarrier's phase . The vectorscope's graticule roughly represents saturation as distance from the center of the circle, and hue as the angle, in standard position, around it. The graticule is also embellished with several elements corresponding to the various components of the standard color bars video test signal , including boxes around the circles for the colors in the main bars, and perpendicular lines corresponding to the U and V components of the chrominance signal (and additionally on an NTSC vectorscope, the I and Q components). NTSC vectorscopes have one set of boxes for the color bars, while their PAL counterparts have two sets of boxes, because the R-Y chrominance component in PAL reverses in phase on alternating lines. Another element in the graticule is a fine grid at 270° on the display (i.e. the -U position) used for measuring differential gain and phase .
Often two sets of bar targets are provided: one for color bars at 75% amplitude and one for color bars at 100% amplitude. The 100% bars represent the maximum amplitude (of the composite signal) that composite encoding allows for. 100% bars are not suitable for broadcast and are not broadcast-safe . 75% bars have reduced amplitude and are broadcast-safe.
Some vectorscope models have only one set of bar targets. The vectorscope can be set up for 75% or 100% bars by adjusting the gain so that the colorburst vector extends to the "75%" or "100%" marking on the graticule.
The reference signal used for the vectorscope's display is the colorburst that is transmitted before each line of video, which for NTSC is defined to have a phase of 180°, corresponding to the -U position on the graticule. The actual colorburst signal shows up on the vectorscope as a straight line pointing to the left from the center of the graticule. In the case of PAL, the colorburst phase alternates between 135° and 225°, resulting in two vectors pointing in the half-past-ten and half-past-seven positions on the graticule, respectively. In digital (and component analog) vectorscopes, colorburst doesn't exist; hence the phase relationship between the colorburst signal and the chroma subcarrier is simply not an issue. A vectorscope for SECAM uses a demodulator similar to the one found in a SECAM receiver to retrieve the U and V color signals since they are transmitted one at a time, namely the Thomson 8300 Vecamscope.
On older vectorscopes that use cathode-ray tubes (CRTs), the graticule was often a silk-screened overlay superimposed over the front surface of the screen. One notable exception was the Tektronix WFM601 series of instruments, which are combined waveform monitors and vectorscopes used to measure CCIR 601 television signals. The waveform-mode graticule of these instruments is implemented with a silkscreen, whereas the vectorscope graticule (consisting only of bar targets, as this family did not support composite video) was drawn on the CRT by the electron beam . Modern instruments have graticules drawn using computer graphics, and both graticule and trace are rendered on an external VGA monitor or an internal VGA-compatible LCD display.
Most modern waveform monitors include vectorscope functionality built in; and many allow the two modes to be displayed side-by-side. The combined device is typically referred to as a waveform monitor, and standalone vectorscopes are rapidly becoming obsolete. [ citation needed ]
In audio applications, a vectorscope is used to measure the difference between channels of stereo audio signals. One stereo channel drives the horizontal deflection of the display, and the other drives the vertical deflection. A monaural signal, consisting of identical left and right signals, results in a straight line with a gradient of +1. Any stereo separation is visible as a deviation from this line, creating a Lissajous figure . If a straight line appears with a gradient of −1, this indicates that the left and right channels are 180° out of phase.
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https://en.wikipedia.org/wiki/Vectorscope
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In crystallography , materials science and metallurgy , Vegard's law is an empirical finding ( heuristic approach) resembling the rule of mixtures . In 1921, Lars Vegard discovered that the lattice parameter of a solid solution of two constituents is approximately a weighted mean of the two constituents' lattice parameters at the same temperature: [ 1 ] [ 2 ]
e.g. , in the case of a mixed oxide of uranium and plutonium as used in the fabrication of MOX nuclear fuel :
Vegard's law assumes that both components A and B in their pure form ( i.e. , before mixing) have the same crystal structure . Here, a A (1- x ) B x is the lattice parameter of the solid solution, a A and a B are the lattice parameters of the pure constituents, and x is the molar fraction of B in the solid solution.
Vegard's law is seldom perfectly obeyed; often deviations from the linear behavior are observed. A detailed study of such deviations was conducted by King. [ 3 ] However, it is often used in practice to obtain rough estimates when experimental data are not available for the lattice parameter for the system of interest.
For systems known to approximately obey Vegard's law, the approximation may also be used to estimate the composition of a solution from knowledge of its lattice parameters, which are easily obtained from diffraction data. [ 4 ] For example, consider the semiconductor compound InP x As (1- x ) . A relation exists between the constituent elements and their associated lattice parameters, a , such that:
When variations in lattice parameters are very small across the entire composition range, Vegard's law becomes equivalent to Amagat's law .
In many binary semiconducting systems, the band gap in semiconductors is approximately a linear function of the lattice parameter. Therefore, if the lattice parameter of a semiconducting system follows Vegard's law, one can also write a linear relationship between the band gap and composition. Using InP x As (1- x ) as before, the band gap energy, E g {\displaystyle E_{g}} , can be written as:
Sometimes, the linear interpolation between the band gap energies is not accurate enough, and a second term to account for the curvature of the band gap energies as a function of composition is added. This curvature correction is characterized by the bowing parameter, b :
The following excerpt from Takashi Fujii (1960) [ 5 ] summarises well the limits of Vegard’s law in the context of mineralogy and also makes the link with the Gladstone–Dale equation :
In mineralogy, the tacit assumption for the linear correlation of the density and the chemical composition of a solid solution is twofold: one is an ideal solid solution and the other identical or nearly identical molar volumes of the components. …
Coefficients of thermal expansion and compressibilities of the ideal solid solution can be discussed in the same manner. But when the solid solution is ideal, the linear correlation of molar heat capacities and chemical composition is possible. The linear correlation of refractive index and chemical composition of an isotropic solid solution can be derived from the Gladstone–Dale equation , but it is required that the system must be ideal and the molar volumes of the components are equal or nearly equal. If the concept of the volume fraction is introduced, density, coefficient of thermal expansion , compressibility and refractive index can be correlated linearly with the volume fraction in an ideal system.“ [ 6 ]
When considering the empirical correlation of some physical properties and the chemical composition of solid compounds, other relationships, rules, or laws, also closely resembles Vegard's law, and in fact the more general rule of mixtures:
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https://en.wikipedia.org/wiki/Vegard's_law
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The Vegetable Production System ( Veggie ) is a plant growth system developed and used by NASA in space environments. The purpose of Veggie is to provide a self-sufficient and sustainable food source for astronauts as well as a means of recreation and relaxation through therapeutic gardening. [ 2 ] Veggie was designed in conjunction with ORBITEC and went operational aboard the International Space Station in 2014, with another Veggie module added to the ISS in 2017. [ 3 ]
The ‘Veggie’ vegetable production system was deployed to the ISS as an applied research platform for food production in space.. Among the goals of this project are to learn about how plants grow in a micro-gravity environment and to learn about how plants can efficiently be grown for crew use in space. [ 2 ] Veggie was designed to be low maintenance, using low power and having a low launch mass. Thus, Veggie provides a minorly regulated environment with minimal control over the atmosphere and temperature of the module. The successor to the Veggie project is the Advanced Plant Habitat (APH), which was delivered to the International Space Station in 2017. [ 4 ] [ 5 ]
In 2016 the Veggie-3 experiments (VEG-03 A,B,C,D) grew "outredgeous" red romaine lettuce, Tokyo Bekana Chinese cabbage, Mizuna mustard and Waldmann's Green lettuce . [ 3 ] [ 6 ]
A Veggie module weighs less than 8 kg (18 lb) and uses 90 watts. [ 7 ] It consists of three parts: a lighting system, a bellows enclosure, and a reservoir. [ 8 ] The lighting system regulates the amount and intensity of light plants receive, the bellows enclosure keeps the environment inside the unit separate from its surroundings, and the reservoir connects to plant pillows where the seeds grow.
Veggie's lighting system consists of three different types of coloreds LEDs : red, blue, and green. Each color corresponds to a different light intensity that the plants will receive. [ 2 ] Although the lighting system can be reconfigured, the following table shows the default settings and their corresponding intensities in micromoles per second per square meter. [ 2 ]
In addition to this lighting system, Veggie also uses opaque bellows to obstruct external sources of light. [ 2 ]
The bellows enclosure controls the flow and pressure of air within the container. The bellows are made from a fluorinated polymer and connected to the lighting system at its top and a baseplate at its bottom. Power and cooling is provided to the hardware that powers the bellows by ExPRESS Racks . [ 2 ] Although the bellows regulate air flow and air pressure, temperature and humidity are left controlled by the surrounding environment of the Veggie module. [ 8 ]
The reservoir of the Veggie module contains and provides water to the plant pillows in which plants grow. The plant pillows contain all other material such as fertilizer and seeds for the plant to grow. Seeds are oriented inside the sticky plant pillow so that their roots will grow downwards into the substrate provided by the plant pillow and that their stems will grow upwards outside of the plant pillow. [ 9 ]
The following plants have been grown using the Vegetable Production System:
In 2010, Desert Research and Technology Studies (Desert RATS) performed operational tests of the Vegetable Production System with lettuce. The three lettuce cultivars that were initially planted yielded positive results, growing and being consumed in 14 days. The Desert RATS team reported uniformly positive psychological results from the test crew. [ 15 ] No substantial information has been released as of yet on the differences between the nutritional values of space-grown plants and earth-grown plants.
As of August 2015, the Veggie system has succeeded in growing edible plants on the ISS. [ 9 ] Further, NASA has announced plans to launch a more advanced plant growth system named Advanced Plant Habitat in 2017. [ 2 ] No results have been recorded on the psychological benefits of the Veggie system in space.
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https://en.wikipedia.org/wiki/Vegetable_Production_System
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Vegetation is an assemblage of plant species and the ground cover they provide. [ 2 ] It is a general term, without specific reference to particular taxa , life forms, structure, spatial extent, or any other specific botanical or geographic characteristics. It is broader than the term flora which refers to species composition . Perhaps the closest synonym is plant community , but vegetation can, and often does, refer to a wider range of spatial scales than that term does, including scales as large as the global. Primeval redwood forests , coastal mangrove stands, sphagnum bogs , desert soil crusts , roadside weed patches, wheat fields, cultivated gardens and lawns; all are encompassed by the term vegetation .
The vegetation type is defined by characteristic dominant species, or a common aspect of the assemblage, such as an elevation range or environmental commonality. [ 3 ] The contemporary use of vegetation approximates that of ecologist Frederic Clements' term earth cover , an expression still used by the Bureau of Land Management .
The distinction between vegetation (the general appearance of a community) and flora (the taxonomic composition of a community) was first made by Jules Thurmann (1849). Prior to this, the two terms (vegetation and flora) were used indiscriminately, [ 4 ] [ 5 ] and still are in some contexts. Augustin de Candolle (1820) also made a similar distinction but he used the terms "station" ( habitat type) and "habitation" ( botanical region ). [ 6 ] [ 7 ] Later, the concept of vegetation would influence the usage of the term biome with the inclusion of the animal element. [ 8 ]
Other concepts similar to vegetation are " physiognomy of vegetation" ( Humboldt , 1805, 1807) and "formation" ( Grisebach , 1838, derived from " Vegetationsform ", Martius , 1824). [ 5 ] [ 9 ] [ 10 ] [ 11 ] [ 12 ]
Departing from Linnean taxonomy , Humboldt established a new science, dividing plant geography between taxonomists who studied plants as taxa and geographers who studied plants as vegetation. [ 13 ] The physiognomic approach in the study of vegetation is common among biogeographers working on vegetation on a world scale, or when there is a lack of taxonomic knowledge of someplace (e.g., in the tropics, where biodiversity is commonly high). [ 14 ]
The concept of " vegetation type " is more ambiguous. The definition of a specific vegetation type may include not only physiognomy but also floristic and habitat aspects. [ 15 ] [ 16 ] Furthermore, the phytosociological approach in the study of vegetation relies upon a fundamental unit, the plant association , which is defined upon flora. [ 17 ]
An influential, clear and simple classification scheme for types of vegetation was produced by Wagner & von Sydow (1888). [ 18 ] [ 19 ] Other important works with a physiognomic approach includes Grisebach (1872), Warming (1895, 1909), Schimper (1898), Tansley and Chipp (1926), Rübel (1930), Burtt Davy (1938), Beard (1944, 1955), André Aubréville (1956, 1957), Trochain (1955, 1957), Küchler (1967), Ellenberg and Mueller-Dombois (1967) (see vegetation classification ).
There are many approaches for the classification of vegetation (physiognomy, flora, ecology, etc.). [ 20 ] [ 21 ] [ 22 ] [ 23 ] Much of the work on vegetation classification comes from European and North American ecologists, and they have fundamentally different approaches. In North America, vegetation types are based on a combination of the following criteria: climate pattern, plant habit , phenology and/or growth form, and dominant species. In the current US standard (adopted by the Federal Geographic Data Committee (FGDC), and originally developed by UNESCO and The Nature Conservancy ), the classification is hierarchical and incorporates the non-floristic criteria into the upper (most general) five levels and limited floristic criteria only into the lower (most specific) two levels. In Europe, classification often relies much more heavily, sometimes entirely, on floristic (species) composition alone, without explicit reference to climate, phenology or growth forms. It often emphasizes indicator or diagnostic species which may distinguish one classification from another.
In the FGDC standard, the hierarchy levels, from most general to most specific, are: system, class, subclass, group, formation, alliance, and association . The lowest level, or association, is thus the most precisely defined, and incorporates the names of the dominant one to three (usually two) species of a type. An example of a vegetation type defined at the level of class might be " Forest, canopy cover > 60% "; at the level of a formation as " Winter-rain, broad-leaved, evergreen, sclerophyllous, closed-canopy forest "; at the level of alliance as " Arbutus menziesii forest"; and at the level of association as " Arbutus menziesii-Lithocarpus dense flora forest", referring to Pacific madrone-tanoak forests which occur in California and Oregon, US. In practice, the levels of the alliance and/or an association are the most often used, particularly in vegetation mapping, just as the Latin binomial is most often used in discussing particular species in taxonomy and in general communication.
Like all biological systems, plant communities are temporally and spatially dynamic; they change at all possible scales. Dynamism in vegetation is defined primarily as changes in species composition and structure.
Temporally, many processes or events can cause change, but for the sake of simplicity, they can be categorized roughly as abrupt or gradual. Abrupt changes are generally referred to as disturbances ; these include things like wildfires , high winds , landslides , floods , avalanches and the like. Their causes are usually external ( exogenous ) to the community—they are natural processes occurring (mostly) independently of the natural processes of the community (such as germination, growth, death, etc.). Such events can change vegetation structure and composition very quickly and for long periods, and they can do so over large areas. Very few ecosystems are without some disturbance as a regular and recurring part of the long-term system dynamic. Fire and wind disturbances are prevalent throughout many vegetation types worldwide. Fire is particularly potent because of its ability to destroy not only living plants but also the seeds, spores, and living meristems representing the potential next generation, and because of fire's impact on fauna populations, soil characteristics and other ecosystem elements and processes (for further discussion of this topic see fire ecology ).
Temporal change at a slower pace is ubiquitous; it comprises the ecological succession field. Succession is the relatively gradual structure and taxonomic composition change that arises as the vegetation modifies various environmental variables over time, including light, water, and nutrient levels. These modifications change the suite of species most adapted to grow, survive, and reproduce in an area, causing floristic changes. These floristic changes contribute to structural changes inherent in plant growth even in the absence of species changes (especially where plants have a large maximum size, i.e., trees), causing slow and broadly predictable changes in the vegetation. Succession can be interrupted at any time by disturbance, setting the system back to a previous state or off on another trajectory altogether. Because of this, successional processes may or may not lead to some static, final state . Moreover, accurately predicting the characteristics of such a state, even if it does arise, is not always possible. In short, vegetative communities are subject to many variables that set limits on future conditions' predictability.
Generally, the larger an area under consideration, the more likely the vegetation will be heterogeneous. Two main factors are at work. First, the temporal dynamics of disturbance and succession are increasingly unlikely to be in synchrony across any area as the size of that area increases. Different areas will be at various developmental stages due to other local histories, particularly their times since the last significant disturbance. This fact interacts with inherent environmental variability (e.g., in soils, climate, topography, etc.), also a function of area. Environmental variability constrains the suite of species that can occupy a given area, and the two factors interact to create a mosaic of vegetation conditions across the landscape. Only in agricultural or horticultural systems does vegetation ever approach perfect uniformity. There is always heterogeneity in natural systems, although its scale and intensity will vary widely.
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Vegetation classification is the process of classifying and mapping the vegetation over an area of the Earth's surface. Vegetation classification is often performed by state based agencies as part of land use, resource and environmental management. Many different methods of vegetation classification have been used. In general, there has been a shift from structural classification used by forestry for the mapping of timber resources, to floristic community mapping for biodiversity management . Whereas older forestry-based schemes considered factors such as height, species and density of the woody canopy, floristic community mapping shifts the emphasis onto ecological factors such as climate, soil type and floristic associations. Classification mapping is usually now done using geographic information systems (GIS) software.
Following, some important classification schemes.
Although this scheme is in fact of a climate classification , it has a deep relationship with vegetation studies:
Wagner & von Sydow (1888) scheme: Vegetationsgürtel (vegetation belts): [ 1 ]
Warming (1895, 1909) oecological classes: [ 2 ] [ 3 ]
Warming's types of formations:
Schimper (1898, 1903) climatic chief formation [ clarification needed ] types: [ 4 ]
Schimper formation types across the zones and regions
Formation-types: [ 5 ] [ 6 ]
Ellenberg and Mueller-Dombois (1967) scheme:
A vegetation classification with six main criteria ("hierarchical attributes", with exemplified categories applicable mainly to Neotropical region): [ 8 ] [ 9 ]
Other important schemes: Grisebach (1872), Tansley and Chipp (1926), Rübel (1930), Burtt Davy (1938), Beard (1944, 1955), André Aubréville (1956, 1957), Trochain (1955, 1957), Dansereau (1958), Küchler (1967), Webb and Tracey (1975). [ 10 ] [ 11 ] [ 12 ] [ 13 ] [ 14 ] [ 15 ] [ 16 ] [ 17 ] [ 18 ] [ 19 ] [ 20 ] [ 21 ] [ 22 ]
In the sixties, A. W. Kuchler coordinated an extensive review of vegetation maps from all the continents, compiling the terminology used for the types of vegetation. [ 23 ]
The Braun-Blanquet method focuses on the composition of plant species within a community. It examines which species grow together, looking at patterns and differences in species groups across different areas. This method uses data collected from specific plots to compare the plant communities and understand how these patterns are influenced by environmental factors. [ 24 ]
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A vegetation index ( VI ) is a spectral imaging transformation of two or more image bands designed to enhance the contribution of vegetation properties and allow reliable spatial and temporal inter-comparisons of terrestrial photosynthetic activity and canopy structural variations. [ 2 ] [ 3 ]
There are many VIs, with many being functionally equivalent. Many of the indices make use of the inverse relationship between red and near-infrared reflectance associated with healthy green vegetation. Since the 1960s scientists have used satellite remote sensing to monitor fluctuation in vegetation at the Earth's surface. Measurements of vegetation attributes include leaf area index (LAI), percent green cover, chlorophyll content, green biomass and absorbed photosynthetically active radiation (APAR).
VIs have been historically classified based on a range of attributes, including the number of spectral bands (2 or greater than 2); the method of calculations (ratio or orthogonal), depending on the required objective; or by their historical development (classified as first generation VIs or second generation VIs). [ 4 ] For the sake of comparison of the effectiveness of different VIs, Lyon, Yuan et al. (1998) [ 5 ] classified 7 VIs based on their computation methods (Subtraction, Division or Rational Transform). Due to advances in hyperspectral remote sensing technology, high-resolution reflectance spectrums are now available, which can be used with traditional multispectral VIs. In addition, VIs have been developed to be used specifically with hyperspectral data, such as the use of Narrow Band Vegetation Indices.
Vegetation indices have been used to:
With the advent of hyperspectral data, vegetation index have been developed specifically for hyperspectral data.
With the emergence of machine learning , certain algorithms can be used to determine vegetation indices from data. This allows to take into account all spectral bands and to discover hidden parameters that can be useful to strengthen these vegetation indices. Thus, they can be more robust against light variations, shadows or even uncalibrated images if these artifacts exist in the training data.
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Vegetative reproduction (also known as vegetative propagation , vegetative multiplication or cloning ) is a form of asexual reproduction occurring in plants in which a new plant grows from a fragment or cutting of the parent plant or specialized reproductive structures, which are sometimes called vegetative propagules . [ 1 ] [ 2 ] [ 3 ]
Many plants naturally reproduce this way, but it can also be induced artificially . Horticulturists have developed asexual propagation techniques that use vegetative propagules to replicate plants. Success rates and difficulty of propagation vary greatly. Monocotyledons typically lack a vascular cambium , making them more challenging to propagate.
Plant propagation is the process of plant reproduction of a species or cultivar, and it can be sexual or asexual. It can happen through the use of vegetative parts of the plants, such as leaves , stems , and roots to produce new plants or through growth from specialized vegetative plant parts. [ 4 ]
While many plants reproduce by vegetative reproduction, they rarely exclusively use that method to reproduce. Vegetative reproduction is not evolutionary advantageous; it does not allow for genetic diversity and could lead plants to accumulate deleterious mutations. [ 5 ] Vegetative reproduction is favored when it allows plants to produce more offspring per unit of resource than reproduction through seed production. [ 6 ] In general, juveniles of a plant are easier to propagate vegetatively. [ 7 ]
Although most plants normally reproduce sexually, many can reproduce vegetatively, or can be induced to do so via hormonal treatments. This is because meristematic cells capable of cellular differentiation are present in many plant tissues.
Vegetative propagation is usually considered a cloning method. [ 8 ] However, root cuttings of thornless blackberries ( Rubus fruticosus ) will revert to thorny type because the adventitious shoot develops from a cell that is genetically thorny. Thornless blackberry is a chimera, with the epidermal layers genetically thornless but the tissue beneath it genetically thorny. [ 9 ]
Grafting is often not a complete cloning method because seedlings are used as rootstocks. In that case, only the top of the plant is clonal. In some crops, particularly apples, the rootstocks are vegetatively propagated so the entire graft can be clonal if the scion and rootstock are both clones. Apomixis (including apospory and diplospory) is a type of reproduction that does not involve fertilization. In flowering plants, unfertilized seeds are produced, or plantlets that grow instead of flowers. Hawkweed ( Hieracium ), dandelion ( Taraxacum ), some citrus ( Citrus ) and many grasses such as Kentucky bluegrass ( Poa pratensis ) all use this form of asexual reproduction. Bulbils are sometimes formed instead of the flowers of garlic.
Meristem tissue makes the process of asexual reproduction possible. It is normally found in stems, leaves, and tips of stems and roots and consists of undifferentiated cells that are constantly dividing allowing for plant growth and give rise to plant tissue systems. The meristem tissue's ability to continuously divide allows for vegetative propagation to occur. [ 10 ]
Another important ability that allows for vegetative propagation is the ability to develop adventitious roots which arise from other vegetative parts of the plants such as the stem or leaves. These roots allow for the development of new plants from body parts from other plants. [ 11 ]
There are several advantages of vegetative reproduction, mainly that the produced offspring are clones of their parent plants. If a plant has favorable traits, it can continue to pass down its advantageous genetic information to its offspring. It can be economically beneficial for commercial growers to clone a certain plant to ensure consistency throughout their crops. [ 12 ] Vegetative propagation also allows plants to avoid the costly and complex process of producing sexual reproduction organs such as flowers and the subsequent seeds and fruits . [ 13 ] Developing an ace cultivar is extremely difficult, so, once farmers develop the desired traits in, for example, a lily, they use grafting and budding to ensure the consistency of the new cultivar and its successful production on a commercial level. However, as can be seen in many variegated plants, this does not always apply, because many plants actually are chimeras and cuttings might reflect the attributes of only one or some of the parent cell lines. Vegetative propagation also allows plants to circumvent the immature seedling phase and reach the mature phase faster. [ 14 ] In nature, that increases the chances for a plant to successfully reach maturity, and, commercially, it saves farmers a lot of time and money as it allows for faster crop overturn. [ 15 ]
Vegetative reproduction offers research advantages in several areas of biology and has practical usage when it comes to afforestation . The most common use made of vegetative propagation by forest geneticists and tree breeders has been to move genes from selected trees to some convenient location, usually designated a gene bank , clone bank , clone-holding orchard, or seed orchard where their genes can be recombined in pedigreed offspring. [ 15 ]
Some analyses suggest that vegetative reproduction is a characteristic which makes a plant species more likely to become invasive. Since vegetative reproduction is often faster than sexual reproduction, it "quickly increases populations and may contribute to recovery following disturbance" (such as fires and floods). [ 16 ]
A major disadvantage of vegetative propagation is that it prevents species genetic diversity which can lead to reductions in crop yields . [ 17 ] [ 18 ] The plants are genetically identical and are all, therefore, susceptible to pathogenic plant viruses , bacteria and fungi that can wipe out entire crops. [ 19 ]
Natural vegetative propagation is mostly a process found in herbaceous and woody perennial plants, and typically involves structural modifications of the stem , although any horizontal, underground part of a plant (whether stem, leaf, or root ) can contribute to vegetative reproduction of a plant. Most plant species that survive and significantly expand by vegetative reproduction would be perennial almost by definition, since specialized organs of vegetative reproduction, like seeds of annuals, serve to survive seasonally harsh conditions. A plant that persists in a location through vegetative reproduction of individuals over a long period of time constitutes a clonal colony .
In a sense, this process is not one of reproduction but one of survival and expansion of biomass of the individual. When an individual organism increases in size via cell multiplication and remains intact, the process is called "vegetative growth". However, in vegetative reproduction, the new plants that result are new individuals in almost every respect except genetic. Of considerable interest is how this process appears to reset the aging clock . [ 20 ]
As previously mentioned, plants vegetatively propagate both artificially and naturally. Most common methods of natural vegetative reproduction involve the development of a new plant from specialized structures of a mature plant. In addition to adventitious roots , roots that arise from plant structures other than the root, such as stems or leaves, modified stems , leaves and roots play an important role in plants' ability to naturally propagate. The most common modified stems, leaves and roots that allow for vegetative propagation are: [ 21 ]
Also known as stolons , runners are modified stems that, unlike rhizomes, grow from existing stems just below the soil surface. As they are propagated, the buds on the modified stems produce roots and stems. Those buds are more separated than the ones found on the rhizome. [ 22 ]
Examples of plants that use runners are strawberries and currants .
Bulbs are inflated parts of the stem within which lie the central shoots of new plants. They are typically underground and are surrounded by plump and layered leaves that provide nutrients to the new plant. [ 23 ]
Examples of plants that use bulbs are shallots , lilies and tulips .
Tubers develop from either the stem or the root. Stem tubers grow from rhizomes or runners that swell from storing nutrients while root tubers propagate from roots that are modified to store nutrients and get too large and produce a new plant. [ 22 ]
Examples of stem tubers are potatoes and yams and examples of root tubers are sweet potatoes and dahlias .
Corms are solid enlarged underground stems that store nutrients in their fleshy and solid stem tissue and are surrounded by papery leaves. Corms differ from bulbs in that their centers consists of solid tissue while bulbs consist of layered leaves. [ 24 ]
Examples of plants that use corms are gladiolus and taro .
Also known as root sprouts , suckers are plant stems that arise from buds on the base of the parent plant's stems or roots. [ 25 ]
Examples of plants that use suckers are apple , elm , and banana trees .
Plantlets are miniature structures that arise from meristem in leaf margins that eventually develops roots and drop from the leaves they grew on. [ 26 ]
An example of a plant that uses plantlets is the Bryophyllum daigremontianum (syn. Kalanchoe daigremontianum ), which is also known as mother of thousands for its many plantlets.
Keikis are additional offshoots which develop on vegetative stems or flower stalks of several orchids genera . [ 14 ]
Examples of plants that use keikis are the Phalaenopsis , Epidendrum , and Dendrobium genera of orchids .
Apomixis is the process of asexual reproduction through seed, in the absence of meiosis and fertilization, generating clonal progeny of maternal origin. [ 27 ]
Vegetative propagation of particular cultivars that have desirable characteristics is very common practice. It is used by farmers and horticulturalists to produce better crops with desirable qualities. The most common methods of artificial vegetative propagation are: [ 21 ]
A cutting is a part of the plant, usually a stem or a leaf, is cut off and planted. Adventitious roots grow from cuttings and a new plant eventually develops. Usually those cuttings are treated with hormones before being planted to induce growth. [ 28 ]
Grafting involves attaching a scion , or a desired cutting, to the stem of another plant called stock that remains rooted in the ground. Eventually both tissue systems become grafted or integrated and a plant with the characteristics of the grafted plant develops, [ 29 ] e.g. mango, guava, etc.
Layering is a process which includes the bending of plant branches or stems so that they touch the ground and are covered with soil. Adventitious roots develop from the underground part of the plant, which is known as the layer. This method of vegetative reproduction also occurs naturally. Another similar method, air layering , involved the scraping and replanting of tree branches which develop into trees. Examples are Jasmine and Bougainvillea. [ 30 ]
Suckers grow and form a dense compact mat that is attached to the parent plant. Too many suckers can lead to smaller crop size, so excess suckers are pruned , and mature suckers are transplanted to a new area where they develop into new plants. [ 31 ]
In tissue culture , plant cells are taken from various parts of the plant and are cultured and nurtured in a sterilized medium. The mass of developed tissue, known as the callus , is then cultured in a hormone-ladened medium and eventually develops into plantlets which are then planted and eventually develop into grown plants. [ 12 ] [ 32 ]
An offset is the lower part of a single culm with the rhizome axis basal to it and its roots. Planting of these is the most convenient way of propagating bamboo.
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A Vegetative Treatment System (VTS) is a combination of treatment steps for managing runoff . It treats runoff by settling, infiltrating , and nutrient usage. Individual components of a VTS include, a settling structure, an outlet structure, a distribution system, and a Vegetative Treatment Area (VTA). All these components when used together are considered to be a Vegetative Treatment System.
A Vegetative Treatment System (VTS) is a new alternative treatment option for treating the runoff from an animal feeding operation in an effort to protect water quality in South Dakota (SD). A VTS consists of a sediment basin to settle the solids from the feedlot, and uses controlled release of the liquids to a vegetated treatment area (VTA).
The VTA area is commonly confused with vegetative buffer (or filter) strips. A buffer strip is a narrow strip of vegetation (usually 30–60 feet wide) between cropland or a water source, such as a river , lake , or stream . In contrast, a VTA is a specifically sized area of perennial vegetation to which runoff from a barnyard or feedlot is applied uniformly. The VTA utilizes the water holding capacity of the soil to store the runoff water until the nutrients and water can be used by the vegetation. Therefore, the application of the runoff to the VTA must be at a rate to prevent deep percolation below the root zone, and not allow the flow to extend past the end of the VTA.
A VTS can be an economical alternative to runoff retention (holding) ponds for controlling runoff from an open lot feeding production system ( feedlots ). A Vegetative Treatment Area (VTA) is an area of perennial vegetation, such as a grass or a forage. The VTA is used to treat runoff from a feedlot or barnyard. It treats runoff by settling, infiltration, and nutrient use. Runoff passes through buffers with some “filtering” of pollutants, but no attempt is made to control solids or flow. A VTS, however, collects runoff from a barnyard or feedlot, separates the solids from the liquids, and uniformly distributes the liquid over the vegetated area. Little or no runoff should leave a VTA.
Runoff is first collected from an open lot or barnyard area in a sediment settling structure, usually a basin. Such basins are very effective for removing most solids. The runoff then flows into a VTA where the soil treats and stores the runoff. Once the runoff is in the soil, natural processes allow plants to use the nutrients. The general idea behind VTS technology is that the plants will take up the nutrients contained in the runoff and that natural processes will eliminate undesirable components such as pathogens. There are many different types of VTA’s such as level, infiltration basins, sloped, sprinkler, dual and multiple systems, etc.
A Vegetative Treatment System can be used to manage runoff from open lots of both AFOs and CAFOs . VTS systems for large CAFOs can be permitted under the National Pollutant Discharge Elimination System ( NPDES ) in the US.
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A vehicle (from Latin vehiculum ) [ 1 ] is a machine designed for self- propulsion , usually to transport people, cargo , or both. The term "vehicle" typically refers to land vehicles such as human-powered vehicles (e.g. bicycles , tricycles , velomobiles ), animal-powered transports (e.g. horse-drawn carriages / wagons , ox carts , dog sleds ), motor vehicles (e.g. motorcycles , cars , trucks , buses , mobility scooters ) and railed vehicles ( trains , trams and monorails ), but more broadly also includes cable transport ( cable cars and elevators ), watercraft ( ships , boats and underwater vehicles ), amphibious vehicles (e.g. screw-propelled vehicles , hovercraft , seaplanes ), aircraft ( airplanes , helicopters , gliders and aerostats ) and space vehicles ( spacecraft , spaceplanes and launch vehicles ). [ 2 ]
This article primarily concerns the more ubiquitous land vehicles, which can be broadly classified by the type of contact interface with the ground : wheels , tracks , rails or skis , as well as the non-contact technologies such as maglev . ISO 3833-1977 is the international standard for road vehicle types, terms and definitions. [ 3 ]
It is estimated by historians that boats have been used since prehistory ; rock paintings depicting boats, dated from around 50,000 to 15,000 BC, were found in Australia . [ 4 ] The oldest boats found by archaeological excavation are logboats , with the oldest logboat found, the Pesse canoe found in a bog in the Netherlands, being carbon dated to 8040–7510 BC, making it 9,500–10,000 years old, [ 5 ] [ 6 ] [ 7 ] [ 8 ] A 7,000 year-old seagoing boat made from reeds and tar has been found in Kuwait. [ 9 ] Boats were used between 4000 -3000 BC in Sumer , [ 10 ] ancient Egypt [ 11 ] and in the Indian Ocean. [ 10 ]
There is evidence of camel pulled wheeled vehicles about 4000–3000 BC. [ 12 ] The earliest evidence of a wagonway , a predecessor of the railway, found so far was the 6 to 8.5 km (4 to 5 mi) long Diolkos wagonway, which transported boats across the Isthmus of Corinth in Greece since around 600 BC. [ 13 ] [ 14 ] Wheeled vehicles pulled by men and animals ran in grooves in limestone , which provided the track element, preventing the wagons from leaving the intended route. [ 14 ]
In 200 CE, Ma Jun built a south-pointing chariot , a vehicle with an early form of guidance system. [ 15 ] The stagecoach , a four-wheeled vehicle drawn by horses, originated in 13th century England. [ 16 ]
Railways began reappearing in Europe after the Dark Ages . The earliest known record of a railway in Europe from this period is a stained-glass window in the Minster of Freiburg im Breisgau dating from around 1350. [ 17 ] In 1515, Cardinal Matthäus Lang wrote a description of the Reisszug , a funicular railway at the Hohensalzburg Fortress in Austria. The line originally used wooden rails and a hemp haulage rope and was operated by human or animal power, through a treadwheel . [ 18 ] [ 19 ] 1769: Nicolas-Joseph Cugnot is often credited with building the first self-propelled mechanical vehicle or automobile in 1769. [ 20 ]
In Russia, in the 1780s, Ivan Kulibin developed a human-pedalled, three-wheeled carriage with modern features such as a flywheel , brake , gear box and bearings ; however, it was not developed further. [ 21 ]
In 1783, the Montgolfier brothers developed the first balloon vehicle.
In 1801, Richard Trevithick built and demonstrated his Puffing Devil road locomotive, which many believe was the first demonstration of a steam-powered road vehicle, though it could not maintain sufficient steam pressure for long periods and was of little practical use.
In 1817, The Laufmaschine ("running machine"), invented by the German Baron Karl von Drais , became the first human means of transport to make use of the two-wheeler principle . It is regarded as the forerunner of the modern bicycle (and motorcycle). [ 22 ] In 1885, Karl Benz built (and subsequently patented) the Benz Patent-Motorwagen , the first automobile, powered by his own four-stroke cycle gasoline engine .
In 1885, Otto Lilienthal began experimental gliding and achieved the first sustained, controlled, reproducible flights.
In 1903, the Wright brothers flew the Wright Flyer , the first controlled, powered aircraft, in Kitty Hawk, North Carolina .
In 1907, Gyroplane No.I became the first tethered rotorcraft to fly. The same year, the Cornu helicopter became the first rotorcraft to achieve free flight. [ 23 ]
In 1928, Opel initiated the Opel-RAK program, the first large-scale rocket program. The Opel RAK.1 became the first rocket car ; the following year, it also became the first rocket-powered aircraft .
In 1961, the Soviet space program 's Vostok 1 carried Yuri Gagarin into space.
In 1969, NASA 's Apollo 11 achieved the first Moon landing .
In 2010, the number of motor vehicles in operation worldwide surpassed 1 billion, roughly one for every seven people. [ 24 ]
There are over 1 billion bicycles in use worldwide. [ 25 ] In 2002 there were an estimated 590 million cars and 205 million motorcycles in service in the world. [ 26 ] [ 27 ] At least 500 million Chinese Flying Pigeon bicycles have been made, more than any other single model of vehicle. [ 28 ] [ 29 ] The most-produced model of motor vehicle is the Honda Super Cub motorcycle, having sold 60 million units in 2008. [ 30 ] [ 31 ] The most-produced car model is the Toyota Corolla , with at least 35 million made by 2010. [ 32 ] [ 33 ] The most common fixed-wing airplane is the Cessna 172 , with about 44,000 having been made as of 2017. [ 34 ] [ 35 ] The Soviet Mil Mi-8 , at 17,000, is the most-produced helicopter. [ 36 ] The top commercial jet airliner is the Boeing 737 , at about 10,000 in 2018. [ 37 ] [ 38 ] [ 39 ] At around 14,000 for both, the most produced trams are the KTM-5 and Tatra T3 . [ 40 ] The most common trolleybus is ZiU-9 .
Locomotion consists of a means that allows displacement with little opposition, a power source to provide the required kinetic energy and a means to control the motion, such as a brake and steering system. By far, most vehicles use wheels which employ the principle of rolling to enable displacement with very little rolling friction .
It is essential that a vehicle have a source of energy to drive it. Energy can be extracted from external sources, as in the cases of a sailboat , a solar-powered car , or an electric streetcar that uses overhead lines. Energy can also be stored, provided it can be converted on demand and the storing medium's energy density and power density are sufficient to meet the vehicle's needs.
Human power is a simple source of energy that requires nothing more than humans. Despite the fact that humans cannot exceed 500 W (0.67 hp) for meaningful amounts of time, [ 41 ] the land speed record for human-powered vehicles (unpaced) is 133 km/h (83 mph), as of 2009 on a recumbent bicycle . [ 42 ]
The energy source used to power vehicles is fuel . External combustion engines can use almost anything that burns as fuel, whilst internal combustion engines and rocket engines are designed to burn a specific fuel, typically gasoline, diesel or ethanol . Food is the fuel used to power non-motor vehicles such as cycles, rickshaws and other pedestrian-controlled vehicles.
Another common medium for storing energy is batteries , which have the advantages of being responsive, useful in a wide range of power levels, environmentally friendly, efficient, simple to install, and easy to maintain. Batteries also facilitate the use of electric motors, which have their own advantages. On the other hand, batteries have low energy densities, short service life, poor performance at extreme temperatures, long charging times, and difficulties with disposal (although they can usually be recycled). Like fuel, batteries store chemical energy and can cause burns and poisoning in event of an accident. [ 43 ] Batteries also lose effectiveness with time. [ 44 ] The issue of charge time can be resolved by swapping discharged batteries with charged ones; [ 45 ] however, this incurs additional hardware costs and may be impractical for larger batteries. Moreover, there must be standard batteries for battery swapping to work at a gas station. Fuel cells are similar to batteries in that they convert from chemical to electrical energy, but have their own advantages and disadvantages.
Electrified rails and overhead cables are a common source of electrical energy on subways, railways, trams, and trolleybuses. Solar energy is a more modern development, and several solar vehicles have been successfully built and tested, including Helios , a solar-powered aircraft.
Nuclear power is a more exclusive form of energy storage, currently limited to large ships and submarines, mostly military. Nuclear energy can be released by a nuclear reactor , nuclear battery , or repeatedly detonating nuclear bombs . There have been two experiments with nuclear-powered aircraft, the Tupolev Tu-119 and the Convair X-6 .
Mechanical strain is another method of storing energy, whereby an elastic band or metal spring is deformed and releases energy as it is allowed to return to its ground state. Systems employing elastic materials suffer from hysteresis , and metal springs are too dense to be useful in many cases. [ clarification needed ]
Flywheels store energy in a spinning mass. Because a light and fast rotor is energetically favorable, flywheels can pose a significant safety hazard. Moreover, flywheels leak energy fairly quickly and affect a vehicle's steering through the gyroscopic effect . They have been used experimentally in gyrobuses .
Wind energy is used by sailboats and land yachts as the primary source of energy. It is very cheap and fairly easy to use, the main issues being dependence on weather and upwind performance. Balloons also rely on the wind to move horizontally. Aircraft flying in the jet stream may get a boost from high altitude winds.
Compressed gas is currently an experimental method of storing energy. In this case, compressed gas is simply stored in a tank and released when necessary. Like elastics, they have hysteresis losses when gas heats up during compression.
Gravitational potential energy is a form of energy used in gliders, skis, bobsleds and numerous other vehicles that go down hill. Regenerative braking is an example of capturing kinetic energy where the brakes of a vehicle are augmented with a generator or other means of extracting energy. [ 46 ]
When needed, the energy is taken from the source and consumed by one or more motors or engines. Sometimes there is an intermediate medium, such as the batteries of a diesel submarine. [ 47 ]
Most motor vehicles have internal combustion engines . They are fairly cheap, easy to maintain, reliable, safe and small. Since these engines burn fuel, they have long ranges but pollute the environment. A related engine is the external combustion engine . An example of this is the steam engine. Aside from fuel, steam engines also need water, making them impractical for some purposes. Steam engines also need time to warm up, whereas IC engines can usually run right after being started, although this may not be recommended in cold conditions. Steam engines burning coal release sulfur into the air, causing harmful acid rain . [ 48 ]
While intermittent internal combustion engines were once the primary means of aircraft propulsion, they have been largely superseded by continuous internal combustion engines, such as gas turbines . Turbine engines are light and, particularly when used on aircraft, efficient. [ citation needed ] On the other hand, they cost more and require careful maintenance. They can also be damaged by ingesting foreign objects, and they produce a hot exhaust. Trains using turbines are called gas turbine-electric locomotives . Examples of surface vehicles using turbines are M1 Abrams , MTT Turbine SUPERBIKE and the Millennium . Pulse jet engines are similar in many ways to turbojets but have almost no moving parts. For this reason, they were very appealing to vehicle designers in the past; however, their noise, heat, and inefficiency have led to their abandonment. A historical example of the use of a pulse jet was the V-1 flying bomb . Pulse jets are still occasionally used in amateur experiments. With the advent of modern technology, the pulse detonation engine has become practical and was successfully tested on a Rutan VariEze . While the pulse detonation engine is much more efficient than the pulse jet and even turbine engines, it still suffers from extreme noise and vibration levels. Ramjets also have few moving parts, but they only work at high speed, so their use is restricted to tip jet helicopters and high speed aircraft such as the Lockheed SR-71 Blackbird . [ 49 ] [ 50 ]
Rocket engines are primarily used on rockets, rocket sleds and experimental aircraft. Rocket engines are extremely powerful. The heaviest vehicle ever to leave the ground, the Saturn V rocket, was powered by five F-1 rocket engines generating a combined 180 million horsepower [ 51 ] (134.2 gigawatt). Rocket engines also have no need to "push off" anything, a fact that the New York Times denied in error . Rocket engines can be particularly simple, sometimes consisting of nothing more than a catalyst, as in the case of a hydrogen peroxide rocket. [ 52 ] This makes them an attractive option for vehicles such as jet packs. Despite their simplicity, rocket engines are often dangerous and susceptible to explosions. The fuel they run off may be flammable, poisonous, corrosive or cryogenic. They also suffer from poor efficiency. For these reasons, rocket engines are only used when absolutely necessary. [ citation needed ]
Electric motors are used in electric vehicles such as electric bicycles , electric scooters, small boats, subways, trains , trolleybuses , trams and experimental aircraft . Electric motors can be very efficient: over 90% efficiency is common. [ 53 ] Electric motors can also be built to be powerful, reliable, low-maintenance and of any size. Electric motors can deliver a range of speeds and torques without necessarily using a gearbox (although it may be more economical to use one). Electric motors are limited in their use chiefly by the difficulty of supplying electricity. [ citation needed ]
Compressed gas motors have been used on some vehicles experimentally. They are simple, efficient, safe, cheap, reliable and operate in a variety of conditions. One of the difficulties met when using gas motors is the cooling effect of expanding gas. These engines are limited by how quickly they absorb heat from their surroundings. [ 54 ] The cooling effect can, however, double as air conditioning. Compressed gas motors also lose effectiveness with falling gas pressure. [ citation needed ]
Ion thrusters are used on some satellites and spacecraft. They are only effective in a vacuum, which limits their use to spaceborne vehicles. Ion thrusters run primarily off electricity, but they also need a propellant such as caesium , or, more recently xenon . [ 55 ] [ 56 ] Ion thrusters can achieve extremely high speeds and use little propellant; however, they are power-hungry. [ 57 ]
The mechanical energy that motors and engines produce must be converted to work by wheels, propellers, nozzles, or similar means.
Aside from converting mechanical energy into motion, wheels allow a vehicle to roll along a surface and, with the exception of railed vehicles, to be steered. [ 58 ] Wheels are ancient technology, with specimens being discovered from over 5000 years ago. [ 59 ] Wheels are used in a plethora of vehicles, including motor vehicles, armoured personnel carriers , amphibious vehicles, airplanes, trains, skateboards and wheelbarrows.
Nozzles are used in conjunction with almost all reaction engines. [ 60 ] Vehicles using nozzles include jet aircraft, rockets, and personal watercraft . While most nozzles take the shape of a cone or bell , [ 60 ] some unorthodox designs have been created such as the aerospike . Some nozzles are intangible, such as the electromagnetic field nozzle of a vectored ion thruster. [ 61 ]
Continuous track is sometimes used instead of wheels to power land vehicles. Continuous track has the advantages of a larger contact area, easy repairs on small damage, and high maneuverability. [ 62 ] Examples of vehicles using continuous tracks are tanks, snowmobiles and excavators. Two continuous tracks used together allow for steering. The largest land vehicle in the world, [ 63 ] the Bagger 293 , is propelled by continuous tracks.
Propellers (as well as screws, fans and rotors) are used to move through a fluid. Propellers have been used as toys since ancient times; however, it was Leonardo da Vinci who devised what was one of the earliest propeller driven vehicles, the "aerial-screw". [ 64 ] In 1661, Toogood & Hays adopted the screw for use as a ship propeller. [ 65 ] Since then, the propeller has been tested on many terrestrial vehicles, including the Schienenzeppelin train and numerous cars. [ 66 ] In modern times, propellers are most prevalent on watercraft and aircraft, as well as some amphibious vehicles such as hovercraft and ground-effect vehicles . Intuitively, propellers cannot work in space as there is no working fluid; however, some sources have suggested that since space is never empty , a propeller could be made to work in space. [ 67 ]
Similarly to propeller vehicles, some vehicles use wings for propulsion. Sailboats and sailplanes are propelled by the forward component of lift generated by their sails/wings. [ 68 ] [ 69 ] Ornithopters also produce thrust aerodynamically. Ornithopters with large rounded leading edges produce lift by leading-edge suction forces. [ 70 ] Research at the University of Toronto Institute for Aerospace Studies [ 71 ] lead to a flight with an actual ornithopter on July 31, 2010.
Paddle wheels are used on some older watercraft and their reconstructions. These ships were known as paddle steamers . Because paddle wheels simply push against the water, their design and construction is very simple. The oldest such ship in scheduled service is the Skibladner . [ 72 ] Many pedalo boats also use paddle wheels for propulsion.
Screw-propelled vehicles are propelled by auger -like cylinders fitted with helical flanges. Because they can produce thrust on both land and water, they are commonly used on all-terrain vehicles. The ZiL-2906 was a Soviet-designed screw-propelled vehicle designed to retrieve cosmonauts from the Siberian wilderness. [ 73 ]
All or almost all of the useful energy produced by the engine is usually dissipated as friction; so minimizing frictional losses is very important in many vehicles. The main sources of friction are rolling friction and fluid drag (air drag or water drag).
Wheels have low bearing friction, and pneumatic tires give low rolling friction. Steel wheels on steel tracks are lower still. [ 74 ]
Aerodynamic drag can be reduced by streamlined design features.
Friction is desirable and important in supplying traction to facilitate motion on land. Most land vehicles rely on friction for accelerating, decelerating and changing direction. Sudden reductions in traction can cause loss of control and accidents.
Most vehicles, with the notable exception of railed vehicles, have at least one steering mechanism. Wheeled vehicles steer by angling their front [ 75 ] or rear [ 76 ] wheels. The B-52 Stratofortress has a special arrangement in which all four main wheels can be angled. [ citation needed ] Skids can also be used to steer by angling them, as in the case of a snowmobile . Ships, boats, submarines, dirigibles and aeroplanes usually have a rudder for steering. On an airplane, ailerons are used to bank the airplane for directional control, sometimes assisted by the rudder.
With no power applied, most vehicles come to a stop due to friction . But it is often required to stop a vehicle faster than by friction alone, so almost all vehicles are equipped with a braking system. Wheeled vehicles are typically equipped with friction brakes, which use the friction between brake pads (stators) and brake rotors to slow the vehicle. [ 46 ] Many airplanes have high-performance versions of the same system in their landing gear for use on the ground. A Boeing 757 brake, for example, has 3 stators and 4 rotors. [ 77 ] The Space Shuttle also uses frictional brakes on its wheels. [ 78 ] As well as frictional brakes, hybrid and electric cars, trolleybuses and electric bicycles can also use regenerative brakes to recycle some of the vehicle's potential energy. [ 46 ] High-speed trains sometimes use frictionless Eddy-current brakes ; however, widespread application of the technology has been limited by overheating and interference issues. [ 79 ]
Aside from landing gear brakes, most large aircraft have other ways of decelerating. In aircraft, air brakes are aerodynamic surfaces that provide braking force by increasing the frontal cross section, thus increasing the increasing the aerodynamic drag of the aircraft. These are usually implemented as flaps that oppose air flow when extended and are flush with the aircraft when retracted. Reverse thrust is also used in many aeroplane engines. Propeller aircraft achieve reverse thrust by reversing the pitch of the propellers, while jet aircraft do so by redirecting their engine exhausts forward. [ 80 ] On aircraft carriers , arresting gears are used to stop an aircraft. Pilots may even apply full forward throttle on touchdown, in case the arresting gear does not catch and a go around is needed. [ 81 ]
Parachutes are used to slow down vehicles travelling very fast. Parachutes have been used in land, air and space vehicles such as the ThrustSSC , Eurofighter Typhoon and Apollo Command Module . Some older Soviet passenger jets had braking parachutes for emergency landings. [ 82 ] Boats use similar devices called sea anchors to maintain stability in rough seas.
To further increase the rate of deceleration or where the brakes have failed, several mechanisms can be used to stop a vehicle. Cars and rolling stock usually have hand brakes that, while designed to secure an already parked vehicle, can provide limited braking should the primary brakes fail. A secondary procedure called forward-slip is sometimes used to slow airplanes by flying at an angle, causing more drag.
Motor vehicle and trailer categories are defined according to the following international classification: [ 83 ]
In the European Union the classifications for vehicle types are defined by: [ 84 ]
European Community is based on the Community's WVTA (whole vehicle type-approval) system. Under this system, manufacturers can obtain certification for a vehicle type in one Member State if it meets the EC technical requirements and then market it EU-wide with no need for further tests. Total technical harmonization already has been achieved in three vehicle categories (passenger cars, motorcycles, and tractors) and soon will extend to other vehicle categories ( coaches and utility vehicles ). It is essential that European car manufacturers be ensured access to as large a market as possible.
While the Community type-approval system allows manufacturers to fully benefit fully from internal market opportunities, worldwide technical harmonization in the context of the United Nations Economic Commission for Europe ( UNECE ) offers a market beyond European borders.
In many cases, it is unlawful to operate a vehicle without a license or certification. The least strict form of regulation usually limits what passengers the driver may carry or prohibits them completely (e.g., a Canadian ultralight license without endorsements). [ 87 ] The next level of licensing may allow passengers, but without any form of compensation or payment. A private driver's license usually has these conditions. Commercial licenses that allow the transport of passengers and cargo are more tightly regulated. The most strict form of licensing is generally reserved for school buses, hazardous materials transports and emergency vehicles.
The driver of a motor vehicle is typically required to hold a valid driver's license while driving on public lands, whereas the pilot of an aircraft must have a license at all times, regardless of where in the jurisdiction the aircraft is flying.
Vehicles are often required to be registered. Registration may be for purely legal reasons, for insurance reasons, or to help law enforcement recover stolen vehicles. The Toronto Police Service , for example, offers free and optional bicycle registration online. [ 88 ] On motor vehicles, registration often takes the form of a vehicle registration plate , which makes it easy to identify a vehicle. In Russia , trucks and buses have their licence plate numbers repeated in large black letters on the back. [ citation needed ] On aircraft, a similar system is used, where a tail number is painted on various surfaces. Like motor vehicles and aircraft, watercraft also have registration numbers in most jurisdictions; however, the vessel name is still the primary means of identification as has been the case since ancient times. For this reason, duplicate registration names are generally rejected. In Canada , boats with an engine power of 10 hp (7.5 kW) or greater require registration, [ 89 ] leading to the ubiquitous "9.9 hp (7.4 kW)" engine.
Registration may be conditional on the vehicle being approved for use on public highways, as in the case of the UK [ 90 ] and Ontario. [ 91 ] Many U.S. states also have requirements for vehicles operating on public highways. [ 92 ] Aircraft have more stringent requirements, as they pose a high risk of damage to people and property in the event of an accident. In the U.S., the FAA requires aircraft to have an airworthiness certificate . [ 93 ] [ 94 ] Because U.S. aircraft must be flown for some time before they are certified, [ 95 ] there is a provision for an experimental airworthiness certificate. [ 96 ] FAA experimental aircraft are restricted in operation, including no overflights of populated areas, in busy airspace, or with unessential passengers. [ 95 ] Materials and parts used in FAA certified aircraft must meet the criteria set forth by the technical standard orders . [ 97 ]
In many jurisdictions, the operator of a vehicle is legally obligated to carry safety equipment with or on them. Common examples include seat belts in cars, helmets on motorcycles and bicycles, fire extinguishers on boats, buses and airplanes, and life jackets on boats and commercial aircraft. Passenger aircraft carry a great deal of safety equipment, including inflatable slides, rafts, oxygen masks, oxygen tanks, life jackets, satellite beacons and first aid kits. Some equipment, such as life jackets has led to debate regarding their usefulness. In the case of Ethiopian Airlines Flight 961 , the life jackets saved many people but also led to many deaths when passengers inflated their vests prematurely.
There are specific real-estate arrangements made to allow vehicles to travel from one place to another. The most common arrangements are public highways, where appropriately licensed vehicles can navigate without hindrance. These highways are on public land and are maintained by the government. Similarly, toll routes are open to the public after paying a toll. These routes and the land they rest on may be government-owned, privately owned or a combination of both. Some routes are privately owned but grant access to the public. These routes often have a warning sign stating that the government does not maintain them. An example of this are byways in England and Wales . In Scotland , land is open to unmotorized vehicles if it meets certain criteria . Public land is sometimes open to use by off-road vehicles . On U.S. public land , the Bureau of Land Management (BLM) decides where vehicles may be used.
Railways often pass over land not owned by the railway company. The right to this land is granted to the railway company through mechanisms such as easement . Watercraft are generally allowed to navigate public waters without restriction as long as they do not cause a disturbance. Passing through a lock , however, may require paying a toll.
Despite the common law tradition Cuius est solum, eius est usque ad coelum et ad inferos of owning all the air above one's property, the U.S. Supreme Court ruled that aircraft in the U.S. have the right to use air above someone else's property without their consent. While the same rule generally applies in all jurisdictions, some countries, such as Cuba and Russia, have taken advantage of air rights on a national level to earn money. [ 98 ] There are some areas that aircraft are barred from overflying. This is called prohibited airspace . Prohibited airspace is usually strictly enforced due to potential damage from espionage or attack. In the case of Korean Air Lines Flight 007 , the airliner entered prohibited airspace over Soviet territory and was shot down as it was leaving. [ citation needed ]
Several different metrics used to compare and evaluate the safety of different vehicles. The main three are deaths per billion passenger-journeys , deaths per billion passenger-hours and deaths per billion passenger-kilometers .
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https://en.wikipedia.org/wiki/Vehicle
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Vehicle-to-device (V2D) communication is a particular type of vehicular communication system that consists in the exchange of information between a vehicle and any electronic device that may be connected to the vehicle itself.
The ever-increasing tendency of developing mobile applications for our everyday use has ultimately entered also the automotive sector. Vehicle connectivity with mobile apps have the great potential to offer a better driving experience, by providing information regarding the surrounding vehicles and infrastructure and making the interaction between the car and its driver much simpler. [ 1 ] The fact that apps may significantly improve driving safety has attracted the attention of car users and caused a rise in the number of new apps developed specifically for the car industry. [ 2 ] This trend has such a great influence that now manufacturers are beginning to design cars taking care of their interaction with mobile phones. For example, starting from 2017 Volvo is going to sell keyless cars, thanks to an app that makes it possible to open and start the vehicle remotely. [ 3 ] Another sector that could coherently benefit from this technology is car sharing .
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https://en.wikipedia.org/wiki/Vehicle-to-device
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Vehicle-to-everything ( V2X ) describes wireless communication between a vehicle and any entity that may affect, or may be affected by, the vehicle. Sometimes called C-V2X , it is a vehicular communication system that is intended to improve road safety and traffic efficiency while reducing pollution and saving energy.
The automotive and communications industries, along with the U.S. government, [ 1 ] European Union [ 2 ] and South Korea [ 3 ] are actively promoting V2X and C-V2X as potentially life-saving, pollution-reducing technologies. The U.S. Department of Transport has said V2X technologies offer significant transportation safety and mobility benefits. [ 1 ] The U.S. NHTSA estimates a minimum of 13% reduction in traffic accidents if a V2V system were implemented, resulting in 439,000 fewer crashes per year. [ 4 ] V2X technology is already being used in Europe and China. [ 5 ]
There are two standards for dedicated V2X communications depending on the underlying wireless technology being used: (1) WLAN -based, and (2) cellular -based. V2X also incorporates various more specific types of communication including :
The history of working on vehicle-to-vehicle communication projects to increase safety, reduce accidents and driver assistance can be traced back to the 1970s with projects such as the US Electronic Road Guidance System (ERGS) and Japan's CACS. [ 7 ] Most milestones in the history of vehicle networks originate from the United States, Europe, and Japan. [ 7 ]
Standardization of WLAN-based V2X supersedes that of cellular-based V2X systems. IEEE first published the specification of WLAN-based V2X ( IEEE 802.11p ) in 2010. [ 8 ] It supports direct communication between vehicles (V2V) and between vehicles and infrastructure (V2I). This technology is referred to as Dedicated Short Range Communication ( DSRC ). DSRC uses the underlying radio communication provided by 802.11p.
In 2016, Toyota became the first automaker globally to introduce automobiles equipped with V2X. These vehicles use DSRC technology and are only for sale in Japan. In 2017, GM became the second automaker to introduce V2X. GM sells a Cadillac model in the United States that also is equipped with DSRC V2X.
In 2016, 3GPP published V2X specifications based on LTE as the underlying technology. It is generally referred to as "cellular V2X" (C-V2X) to differentiate itself from the 802.11p based V2X technology. In addition to the direct communication (V2V, V2I), C-V2X also supports wide area communication over a cellular network (V2N).
As of December 2017, a European automotive manufacturer has announced to deploy V2X technology based on 802.11p from 2019. [ 9 ] While some studies and analysis in 2017 [ 9 ] and 2018, [ 10 ] all performed by the 5G Automotive Association (5GAA) – the industry organisation supporting and developing the C-V2X technology – indicate that cellular-based C-V2X technology in direct communication mode is superior to 802.11p in multiple aspects, such as performance, communication range, and reliability, many of these claims are disputed, e.g. in a whitepaper published by NXP, [ 11 ] one of the companies active in the 802.11p based V2X technology,
but also published by peer-reviewed journals. [ 12 ]
This technology can be misused to remotely control the vehicle. The Police of the Czech Republic(2024) announced, in cooperation with universities, has developed a system for remote stopping of vehicles with reference to the fact that such a procedure is legal even under the current legislation. [ 13 ]
The original V2X communication uses WLAN technology and works directly between vehicles (V2V) as well as vehicles and traffic infrastructure (V2I), which form a vehicular ad-hoc network as two V2X senders come within each other's range. Hence it does not require any communication infrastructure for vehicles to communicate, which is key to assure safety in remote or little-developed areas. WLAN is particularly well-suited for V2X communication , due to its low latency. It transmits messages known as Cooperative Awareness Messages (CAM) or Basic Safety Message (BSM), and Decentralised Environmental Notification Messages (DENM). Other roadside infrastructure related messages are Signal Phase and Timing Message (SPAT), In Vehicle Information Message (IVI), and Service Request Message (SRM). The data volume of these messages is very low. The radio technology is part of the WLAN IEEE 802.11 family of standards and known in the US as Wireless Access in Vehicular Environments (WAVE) and in Europe as ITS-G5. [ 14 ] To complement the direct communication mode, vehicles can be equipped with traditional cellular communication technologies, supporting V2N based services. This extension with V2N was achieved in Europe under the C-ITS platform umbrella [ 15 ] with cellular systems and broadcast systems (TMC/DAB+).
More recent V2X communication uses cellular networks and is called cellular V2X (or C-V2X) to differentiate it from the WLAN-based V2X. There have been multiple industry organizations, such as the 5G Automotive Association (5GAA) promoting C-V2X due to its advantages over WLAN based V2X (without considering disadvantages at the same time). [ 16 ] C-V2X is initially defined as LTE in 3GPP Release 14 and is designed to operate in several modes:
In 3GPP Release 15, the V2X functionalities are expanded to support 5G . C-V2X includes support of both direct communication between vehicles (V2V) and traditional cellular-network based communication. Also, C-V2X provides a migration path to 5G based systems and services, which implies incompatibility and higher costs compared to 4G based solutions.
The direct communication between vehicle and other devices (V2V, V2I) uses so-called PC5 interface. PC5 refers to a reference point where the User Equipment (UE), i.e. mobile handset, directly communicates with another UE over the direct channel. In this case, the communication with the base station is not required. In system architectural level, proximity service (ProSe) is the feature that specifies the architecture of the direct communication between UEs. In 3GPP RAN specifications, "sidelink" is the terminology to refer to the direct communication over PC5. PC5 interface was originally defined to address the needs of mission-critical communication for public safety community (Public Safety-LTE, or PS-LTE) in release 13. The motivation of the mission-critical communication was to allow law enforcement agencies or emergency rescue to use the LTE communication even when the infrastructure is not available, such as natural disaster scenario. In release 14 onwards, the use of PC5 interface has been expanded to meet various market needs, such as communication involving wearable devices such as smartwatch . In C-V2X, PC5 interface is re-applied to the direct communication in V2V and V2I.
The Cellular V2X mode 4 communication relies on a distributed resource allocation scheme, namely sensing-based semipersistent scheduling which schedules radio resources in a stand-alone fashion in each user equipment (UE). [ 17 ] [ 18 ] [ 19 ]
In addition to the direct communication over PC5, C-V2X also allows the C-V2X device to use the cellular network connection in the traditional manner over Uu interface. Uu refers to the logical interface between the UE and the base station. This is generally referred to as vehicle-to-network (V2N). V2N is a unique use case to C-V2X and does not exist in 802.11p based V2X given that the latter supports direct communication only. However, similar to WLAN based V2X also in case of C-V2X, two communication radios are required to be able to communicate simultaneously via a PC5 interface with nearby stations and via the UU interface with the network.
While 3GPP defines the data transport features that enable V2X, it does not include V2X semantic content but proposes usage of ITS-G5 standards like CAM, DENM, BSM, etc. over 3GPP V2X data transport features. [ 20 ]
Through its instant communication, V2X enables road safety applications such as (non-exhaustive list):
In June 2024 the U.S. Department of Transportation announced that it is awarding $60 million in grants to advance connected and interoperable vehicle technologies under a program called "Saving Lives with Connectivity: Accelerating V2X Deployment program". [ 21 ] It said the grants to recipients in Arizona, Texas and Utah would serve as national models to accelerate and spur new deployments of V2X technologies. European standardisation body ETSI and SAE published standards on what they see as use cases. [ 22 ] [ 23 ] Early use cases focus on road safety and efficiency. [ 24 ] Organizations such as 3GPP and 5GAA continuously introduce and test new cases. The 5GAA has published several roadmaps [ 25 ] which highlight the technical potential and challenges of new use cases. Some use cases address high levels of automation. [ 7 ]
C-V2X offers further use cases including slippery road, roadworks and road hazard information to cars and trucks over hills, around curves and over longer distances than is possible with direct communications. Volvo, for example, has sold new cars that warn other Volvos of slippery roads ahead using C-V2X communications since 2016 in Denmark, and has announced plans to complement that with general accident-ahead warnings and offer the same functionality in other European markets over time. [ 26 ]
In the medium term, V2X is perceived as a key enabler for autonomous driving, assuming it would be allowed to intervene into the actual driving. In that case, vehicles would be able to join platoons the way HGVs do. With the advent of connected and autonomous mobility, V2X discussions are seen to play an important role, especially in the context of teleoperations for autonomous vehicles [ 27 ] and platooning [ 28 ] [ 29 ]
WLAN-based V2X communication is based on a set of standards drafted by the American Society for Testing and Materials (ASTM). The ASTM E 2213 series of standards looks at wireless communication for high-speed information exchange between vehicles themselves as well as road infrastructure. The first standard of this series was published 2002. Here the acronym Wireless Access in Vehicular Environments (WAVE) was first used for V2X communication.
From 2004 onwards the Institute of Electrical and Electronics Engineers (IEEE) started to work on wireless access for vehicles under the umbrella of their standards family IEEE 802.11 for Wireless Local Area Networks (WLAN). Their initial standard for wireless communication for vehicles is known as IEEE 802.11p and is based on the work done by the ASTM. Later on in 2012 IEEE 802.11p was incorporated in IEEE 802.11.
Around 2007 when IEEE 802.11p got stable, IEEE started to develop the 1609.x standards family standardising applications and a security framework [ 30 ] (IEEE uses the term WAVE), and soon after SAE started to specify standards for V2V communication applications. SAE uses the term DSRC for this technology (this is how the term was coined in the US). In parallel at ETSI the technical committee for Intelligent transportation system (ITS) was founded and started to produce standards for protocols and applications [ 31 ] (ETSI coined the term ITS-G5). All these standards are based on IEEE 802.11p technology.
Between 2012 and 2013, the Japanese Association of Radio Industries and Businesses (ARIB) specified, also based on IEEE 802.11, a V2V and V2I communication system in the 700 MHz frequency band. [ 32 ]
In 2015 ITU published as summary of all V2V and V2I standards that are worldwide in use, comprising the systems specified by ETSI, IEEE, ARIB, and TTA (Republic of Korea, Telecommunication Technology Association). [ 33 ]
3GPP started standardization work of cellular V2X (C-V2X) in Release 14 in 2014. It is based on LTE as the underlying technology. Specifications were published in 2017. Because this C-V2X functionalities are based on LTE, it is often referred to as LTE-V2X. The scope of functionalities supported by C-V2X includes both direct communication (V2V, V2I) as well as wide area cellular network communication (V2N).
In Release 15, 3GPP continued its C-V2X standardization to be based on 5G. Specifications are published in 2018 as Release 15 comes to completion. To indicate the underlying technology, the term 5G-V2X is often used in contrast to LTE-based V2X (LTE-V2X). Either case, C-V2X is the generic terminology that refers to the V2X technology using the cellular technology irrespective of the specific generation of technology.
In Release 16, 3GPP further enhances the C-V2X functionality. The work is currently in progress. In this way, C-V2X is inherently future-proof by supporting migration path to 5G.
Study and analysis were done [ 9 ] [ 10 ] to compare the effectiveness of direct communication technologies between LTE-V2X PC5 and 802.11p from the perspective of accident avoided and reduction in fatal and serious injuries. The study shows that LTE-V2X achieves higher level of accident avoidance and reduction in injury. [ 9 ] It also indicates LTE-V2X performs higher percentage of successful packet delivery and communication range. Another link-level and system-level simulation result indicates that, to achieve the same link performance for both line-of-sight (LOS) and non-line-of-sight (NLOS) scenarios, lower signal-to-noise-ratio (SNR) are achievable by LTE-V2X PC5 interface compared to IEEE 802.11p. [ 10 ]
Cellular-based V2X solution also leads to the possibility of further protecting other types of road users (e.g. pedestrian, cyclist) by having PC5 interface to be integrated into smartphones, effectively integrating those road users into the overall C-ITS solution. Vehicle-to-person (V2P) includes Vulnerable Road User (VRU) scenarios to detect pedestrians and cyclists to avoid accident and injuries involving those road users.
As both direct communication and wide area cellular network communication are defined in the same standard (3GPP), both modes of communication will likely be integrated into a single chipset. Commercialization of those chipsets further enhances economy of scale and leads to possibilities to wider range of business models and services using both types of communications.
In 1999 the US Federal Communications Commission (FCC) allocated 75 MHz in the spectrum of 5.850-5.925 GHz for intelligent transport systems. [ 34 ]
Since then the US Department of Transportation (USDOT) has been working with a range of stakeholders on V2X. In 2012 a pre-deployment project was implemented in Ann Arbor, Michigan. 2800 vehicles covering cars, motorcycles, buses and HGV of different brands took part using equipment by different manufacturers. [ 35 ] The US National Highway Traffic Safety Administration (NHTSA) saw this model deployment as proof that road safety could be improved and that WAVE standard technology was interoperable. In August 2014 NHTSA published a report arguing vehicle-to-vehicle technology was technically proven as ready for deployment. [ 36 ] On 20 August 2014 the NHTSA published an Advance Notice of Proposed Rulemaking (ANPRM) in the Federal Register, [ 37 ] arguing that the safety benefits of V2X communication could only be achieved if a significant part of the vehicles fleet was equipped. Because of the lack of an immediate benefit for early adopters, the NHTSA proposed a mandatory introduction. On 25 June 2015 the US House of Representatives held a hearing on the matter, [ 38 ] where again the NHTSA, as well as other stakeholders argued the case for V2X. [ 39 ]
On November 18, 2020, the FCC reallocated 45 MHz in the 5.850–5.895 GHz range to Wi-Fi , and the rest of the V2X band to C-V2X, citing the failure of DSRC to take off. [ 40 ] The advocacy organizations ITS America and American Association of State Highway and Transportation Officials sued the FCC, arguing that the decision harms users of DSRC; on August 12, 2022, a federal court permitted the reassignment to go ahead. [ 41 ]
To acquire EU-wide spectrum, radio applications require a harmonised standard, in case of ITS-G5 ETSI EN 302 571, [ 42 ] first published in 2008. A harmonised standard in turn requires an ETSI System Reference Document, here ETSI TR 101 788. [ 43 ] Commission Decision 2008/671/EC harmonises the use of the 5875 to 5905 MHz frequency band for transport safety ITS applications. [ 44 ] In 2010 the ITS Directive 2010/40/EU [ 45 ] was adopted. It aims to assure that ITS applications are interoperable and can operate across national borders, it defines priority areas for secondary legislation, which cover V2X and requires technologies to be mature. In 2014 the European Commission's industry stakeholder “C-ITS Deployment Platform” started working on a regulatory framework for V2X in the EU. [ 46 ] It identified key approaches to an EU-wide V2X security Public Key infrastructure (PKI) and data protection, as well as facilitating a mitigation standard [ 47 ] to prevent radio interference between ITS-G5 based V2X and road charging systems. The European Commission recognised ITS-G5 as the initial communication technology in its 5G Action Plan [ 48 ] and the accompanying explanatory document, [ 49 ] to form a communication environment consisting of ITS-G5 and cellular communication as envisioned by EU Member States. [ 50 ] Various pre-deployment projects exist at EU or EU Member State level, such as SCOOP@F, the Testfeld Telematik, the digital testbed Autobahn, the Rotterdam-Vienna ITS Corridor, Nordic Way, COMPASS4D or C-ROADS. [ 51 ] There exist real scenarios of implementation V2X standard as well. The first commercial project where V2X standard is used for Intersection movement assist use-case. It has been realized in Brno City / Czech Republic where 80 pcs of cross intersections are controlled by V2X communication standard from public transport vehicles of municipality Brno. [ 52 ]
Spectrum allocation for C-ITS in various countries is shown in the following table. Due to the standardization of V2X in 802.11p preceding C-V2X standardization in 3GPP , spectrum allocation was originally intended for the 802.11p based system. However, the regulations are technology neutral so that the deployment of C-V2X is not excluded.
In 2022, US Federal Courts told the FCC that it could reallocate 45 MHz of V2X spectrum to wireless and cellular carriers, citing years of no use by V2X constituents.
The deployment of V2X technology (either C-V2X or 802.11p based products) will occur gradually over time. New cars will be equipped with either of the two technologies starting around 2020 and its proportion on the road is expected to increase gradually. The Volkswagen Golf 8th generation was the first passenger car to be fitted with V2X technology powered by NXP technology. [ 53 ] In the meantime, existing (legacy) vehicles will continue to exist on the road. This implies that the V2X capable vehicles will need to co-exist with non-V2X (legacy) vehicles or with V2X vehicles of incompatible technology.
The main obstacles to its adoption are legal issues and the fact that, unless almost all vehicles adopt it, its effectiveness is limited. [ 54 ] British weekly The Economist argued in 2016 that autonomous driving is more driven by regulations than by technology. [ 55 ]
However, a 2017 study [ 9 ] indicated that there are benefits in reducing traffic accidents even during the transitional period in which the technology is being adopted in the market.
Many books and papers have been written in the topic:
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Vehicle Information and Communication System ( VICS ) is a technology used in Japan for delivering traffic and travel information to road vehicle drivers. It provides simple maps showing information about traffic jams, travel time, and road work - usually relevant to your location and usually incorporating infrared beacons.
It can be compared with the European TMC technology.
VICS is transmitted using:
It is an application of ITS .
The VICS information can be displayed on the car navigation unit at 3 levels:
Information transmitted generally includes traffic congestion data, data on availability of service areas (SA) and parking areas (PA), information on road works and traffic collisions .
Some advanced navigation units might utilize this data for route calculation (e.g., choosing a route to avoid congestion) or the driver might use his/her own discretion while using this information.
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Vehicle dynamics is the study of vehicle motion, e.g., how a vehicle's forward movement changes in response to driver inputs, propulsion system outputs, ambient conditions, air/surface/water conditions, etc.
Vehicle dynamics is a part of engineering primarily based on classical mechanics .
It may be applied for motorized vehicles (such as automobiles), bicycles and motorcycles , aircraft , and watercraft .
The aspects of a vehicle's design which affect the dynamics can be grouped into drivetrain and braking, suspension and steering, distribution of mass, aerodynamics and tires.
Some attributes relate to the geometry of the suspension , steering and chassis . These include:
Some attributes or aspects of vehicle dynamics are purely due to mass and its distribution. These include:
Some attributes or aspects of vehicle dynamics are purely aerodynamic . These include:
Some attributes or aspects of vehicle dynamics can be attributed directly to the tires . These include:
Some attributes or aspects of vehicle dynamics are purely dynamic . These include:
The dynamic behavior of vehicles can be analysed in several different ways. [ 1 ] This can be as straightforward as a simple spring mass system, through a three- degree of freedom (DoF) bicycle model, to a large degree of complexity using a multibody system simulation package such as MSC ADAMS or Modelica . As computers have gotten faster, and software user interfaces have improved, commercial packages such as CarSim have become widely used in industry for rapidly evaluating hundreds of test conditions much faster than real time. Vehicle models are often simulated with advanced controller designs provided as software in the loop (SIL) with controller design software such as Simulink , or with physical hardware in the loop (HIL).
Vehicle motions are largely due to the shear forces generated between the tires and road, and therefore the tire model is an essential part of the math model. In current vehicle simulator models, the tire model is the weakest and most difficult part to simulate. [ 2 ] The tire model must produce realistic shear forces during braking, acceleration, cornering, and combinations, on a range of surface conditions. Many models are in use. Most are semi-empirical, such as the Pacejka Magic Formula model.
Racing car games or simulators are also a form of vehicle dynamics simulation. In early versions many simplifications were necessary in order to get real-time performance with reasonable graphics. However, improvements in computer speed have combined with interest in realistic physics, leading to driving simulators that are used for vehicle engineering using detailed models such as CarSim.
It is important that the models should agree with real world test results, hence many of the following tests are correlated against results from instrumented test vehicles.
Techniques include:
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A vehicle frame , also historically known as its chassis , is the main supporting structure of a motor vehicle to which all other components are attached, comparable to the skeleton of an organism.
Until the 1930s, virtually every car had a structural frame separate from its body, known as body-on-frame construction. Both mass production of completed vehicles by a manufacturer using this method, epitomized by the Ford Model T , and supply of rolling chassis to coachbuilders for both mass production (as by Fisher Body in the United States) and to smaller firms (such as Hooper ) for bespoke bodies and interiors was practiced.
By the 1960s, unibody construction in passenger cars had become common, and the trend towards building unibody passenger cars continued over the ensuing decades. [ 1 ]
Nearly all trucks , buses, and most pickups continue to use a separate frame as their chassis.
The main functions of a frame in a motor vehicle are: [ 2 ]
Typically, the material used to construct vehicle chassis and frames include carbon steel for strength or aluminum alloys to achieve a more lightweight construction. In the case of a separate chassis, the frame is made up of structural elements called the rails or beams . These are ordinarily made of steel channel sections by folding, rolling, or pressing steel plate.
There are three main designs for these. If the material is folded twice, an open-ended cross-section, either C-shaped or hat-shaped (U-shaped), results.
"Boxed" frames contain closed chassis rails, either by welding them up or by using premanufactured metal tubing .
By far the most common, the C-channel rail has been used on nearly every type of vehicle at one time or another. [ citation needed ] It is made by taking a flat piece of steel (usually ranging in thickness from 1/8" to 3/16", but up to 1/2" or more in some heavy-duty trucks [ 3 ] [ 4 ] ) and rolling both sides over to form a C-shaped beam running the length of the vehicle. C-channel is typically more flexible than (fully) boxed of the same gauge.
Hat frames resemble a "U" and may be either right-side-up or inverted, with the open area facing down. They are not commonly used due to weakness and a propensity to rust. However, they can be found on 1936–1954 Chevrolet cars and some Studebakers .
Abandoned for a while, the hat frame regained popularity when companies started welding it to the bottom of unibody cars, effectively creating a boxed frame.
Originally, boxed frames were made by welding two matching C-rails together to form a rectangular tube. Modern techniques, however, use a process similar to making C-rails in that a piece of steel is bent into four sides and then welded where both ends meet.
In the 1960s, the boxed frames of conventional American cars were spot-welded in multiple places down the seam; when turned into NASCAR "stock car" racers, the box was continuously welded from end to end for extra strength.
While appearing at first glance as a simple form made of metal, frames encounter significant stress and are built accordingly. The first issue addressed is "beam height", or the height of the vertical side of a frame. The taller the frame, the better it can resist vertical flex when force is applied to the top of the frame. This is the reason semi-trucks have taller frame rails than other vehicles instead of just being thicker.
As looks, ride quality, and handling became more important to consumers, new shapes were incorporated into frames. The most visible of these are arches and kick-ups. Instead of running straight over both axles , arched frames sit lower—roughly level with their axles—and curve up over the axles and then back down on the other side for bumper placement. Kick-ups do the same thing without curving down on the other side and are more common on the front ends.
Another feature are the tapered rails that narrow vertically or horizontally in front of a vehicle's cabin. This is done mainly on trucks to save weight and slightly increase room for the engine since the front of the vehicle does not bear as much load as the back. Design developments include frames that use multiple shapes in the same frame rail. For example, some pickup trucks have a boxed frame in front of the cab, shorter, narrower rails underneath the cab, and regular C-rails under the bed.
On perimeter frames, the areas where the rails connect from front to center and center to rear are weak compared to regular frames, so that section is boxed in, creating what are called "torque boxes".
Named for its resemblance to a ladder, the ladder frame is one of the oldest, simplest, and most frequently used under-body, separate chassis/frame designs. It consists of two symmetrical beams, rails, or channels, running the length of the vehicle, connected by several transverse cross-members. Initially seen on almost all vehicles, the ladder frame was gradually phased out on cars in favor of perimeter frames and unitized body construction. It is now seen mainly on large trucks. This design offers good side impact resistance because of its continuous rails from front to rear, but poor resistance to torsion or warping if simple, perpendicular cross-members are used. The vehicle's overall height will be greater due to the floor pan sitting above the frame instead of inside it.
A backbone chassis is a type of automotive construction with chassis that is similar to the body-on-frame design. Instead of a relatively flat, ladder-like structure with two longitudinal, parallel frame rails, it consists of a central, strong tubular backbone (usually rectangular in cross-section) that carries the power-train and connects the front and rear suspension attachment structures. Although the backbone is frequently drawn upward into, and mostly above the floor of the vehicle, the body is still placed on or over (sometimes straddling) this structure from above.
An X-frame is built generally in the shape of the letter X, beginning in its simplest form with two frame rails parallel to one another in the engine compartment, crossing (or joining) in the middle, then returning to parallel at or after the rear axle. The purpose of the design was to allow the floor pan to be placed lower than had been possible sitting atop a full ladder frame. Centerline humps, however, provided for the power train and central crossmember, intruded into the cabin space.
The X-frame varied in stiffness depending on the gauge and proportion of its cross-section, but could be rigid when heavy enough.
It was widely used, as in the exclusive Mercedes-Benz 300 "Adenaeur" limousines, and for some full-sized GM cars of the late 1950s and early 1960s.
A shortcoming was weakness to side-impact, resulting in the addition of side rails (that still allowed a recessed cabin), spurring development of the perimeter frame. [ 5 ]
Similar to a ladder frame, but the middle sections of the frame rails sit outboard of the front and rear rails, routed around the passenger footwells, inside the rocker and sill panels. This allowed the floor pan to be lowered, especially the passenger footwells, lowering the passengers' seating height and thereby reducing both the roof-line and overall vehicle height, as well as the center of gravity, thus improving handling and road-holding in passenger cars.
This became the prevalent design for body-on-frame cars in the United States, but not in the rest of the world, until the unibody gained popularity. For example, Hudson introduced this construction on their third generation Commodore models in 1948. This frame type allowed for annual model changes , and lower cars, introduced in the 1950s to increase sales – without costly structural changes.
The Ford Panther platform , discontinued in 2011, was one of the last perimeter frame passenger car platforms in the United States. [ 1 ] The fourth to seventh generation Chevrolet Corvette used a perimeter frame integrated with an internal skeleton that serves as a clamshell.
In addition to a lowered roof, the perimeter frame allows lower seating positions when that is desirable, and offers better safety in the event of a side impact. However, the design lacks stiffness because the transition areas from front to center and center to rear reduce beam and torsional resistance and is used in combination with torque boxes and soft suspension settings.
This is a modification of the perimeter frame, or of the backbone frame, in which the passenger compartment floor, and sometimes the luggage compartment floor, have been integrated into the frame as loadbearing parts for strength and rigidity. The sheet metal used to assemble the components needs to be stamped with ridges and hollows to give it strength.
Platform chassis were used on several successful European cars, most notably the Volkswagen Beetle , where it was called "body-on-pan" construction. Another German example are the Mercedes-Benz "Ponton" cars of the 1950s and 1960s, [ 6 ] where it was called a "frame floor" in English-language advertisements.
The French Renault 4 , of which over eight million were made, also used a platform frame. The frame of the Citroën 2CV used a minimal interpretation of a platform chassis under its body.
Originally known as a "tubular frame", the space frame (also "spaceframe") utilizes tubular steel, alloy, or carbon fibre to create a load-bearing three-dimensional skeleton, to which the suspension, engine, and body panels are attached. As the body panels have limited or no structural function, geometry is used to maximize rigidity and minimize weight, frequently employing triangles where all the forces in each strut are either tensile or compressive. The lack of bending forces allows members to be kept to a minimum weight and cross-section.
The first true spaceframe chassis were designed and produced in the 1930s by Buckminster Fuller and William Bushnell Stout , who understood the theory supporting them from either architecture or aircraft design, resulting in the bus-like Dymaxion and Stout Scarab . [ 7 ] Maximizing space while minimizing weight were the goals.
With its high strength-to-weight ratio, the space frame was adapted to automobile racing following World War II. The 1951 Jaguar C-Type racing sports car utilized a lightweight, multi-tubular, triangulated frame over which an aerodynamic aluminum body was crafted. The form saw mass production with the 1954 introduction of the Mercedes-Benz 300 SL "Gullwing" sports car, the fastest road-going automobile of its day. The car's exceptionally high sills made conventional doors impractical, spawning the model's iconic gullwing doors .
In 1994, the Audi A8 was the first mass-market car with an aluminium chassis, made feasible by integrating an aluminium space-frame into the bodywork. Audi A8 models have since used this construction method co-developed with Alcoa , and marketed as the Audi Space Frame . [ 8 ]
A tubular frame that is not load-bearing is not a true space frame. The Italian term Superleggera (meaning 'super-light') was trademarked for lightweight sports-car body construction that still requires its own chassis, and thus only resembles a space-frame chassis in general look and construction method. It utilizes a geodesic -like network of narrow tubes running under the body, up the fenders and over the radiator, cowl, and roof, and under the rear window, to provide form and attachment points for a sheetmetal skin, typically aluminum for weight-savings, as rigidity is not a consideration.
The terms "unibody" and "unit-body" are short for "unitized body", "unitary construction", or alternatively (fully) integrated body and frame/chassis. It is defined as: [ 9 ]
A type of body/frame construction in which the body of the vehicle, its floor pan and chassis form a single structure. Such a design is generally lighter and more rigid than a vehicle having a separate body and frame.
Vehicle structure has shifted from the traditional body-on-frame architecture to the lighter unitized/integrated body structure that is now used for most cars. [ 10 ]
Integral frame and body construction requires more than simply welding an unstressed body to a conventional frame. In a fully integrated body structure, the entire car is a load-carrying unit that handles all the loads experienced by the vehicle – forces from driving and cargo loads. Integral-type bodies for wheeled vehicles are typically manufactured by welding preformed metal panels and other components together, by forming or casting whole sections as one piece, or by combining these techniques. Although this is sometimes also referred to as a monocoque structure, because the car's outer skin and panels are made load-bearing, there are still ribs, bulkheads, and box sections to reinforce the body, making the description semi-monocoque more appropriate.
The first attempt to develop such a design technique was on the 1922 Lancia Lambda to provide structural stiffness and a lower body height for its torpedo car body. [ 11 ] The Lambda had an open layout with unstressed roof, which made it less of a monocoque shell and more like a bowl. One thousand were produced. [ 12 ]
A key role in developing the unitary body was played by the American firm the Budd Company, now ThyssenKrupp Budd . [ 12 ] Budd supplied pressed-steel bodywork, fitted to separate frames, to automakers Dodge , Ford , Buick , and the French company, Citroën .
In 1930, Joseph Ledwinka , an engineer with Budd, designed an automobile prototype with a full unitary construction. [ 13 ]
Citroën purchased this fully unitary body design for the Citroën Traction Avant . This high-volume, mass-production car was introduced in 1934 and sold 760,000 units over the next 23 years of production. [ 12 ] This application was the first iteration of the modern structural integration of body and chassis, using spot welded deeply stamped steel sheets into a structural cage, including sills, pillars, and roof beams. [ 11 ] In addition to a unitary body with no separate frame, the Traction Avant also featured other innovations such as front-wheel drive . The result was a low-slung vehicle with an open, flat-floored interior. [ 14 ]
For the Chrysler Airflow (1934–1937), Budd supplied a variation – three main sections from the Airflow's body were welded into what Chrysler called a bridge-truss construction. Unfortunately, this method was not ideal because the panel fits were poor. [ 12 ] To convince a skeptical public of the strength of unibody, both Citroën and Chrysler created advertising films showing cars surviving after being pushed off a cliff. [ 12 ]
Opel was the second European and the first German car manufacturer to produce a car with a unibody structure – production of the compact Olympia started in 1935. A larger Kapitän went into production in 1938, although its front longitudinal beams were stamped separately and then attached to the main body. It was so successful that the Soviet post-war mass produced GAZ-M20 Pobeda of 1946 copied unibody structure from the Opel Kapitän. [ 15 ] Later Soviet limousine GAZ-12 ZIM of 1950 introduced unibody design to automobiles with a wheelbase as long as 3.2 m (126 in). [ 16 ]
The streamlined 1936 Lincoln-Zephyr with conventional front-engine, rear-wheel-drive layout utilized a unibody structure. [ 17 ] By 1941, unit construction was no longer a new idea for cars, "but it was unheard of in the [American] low-price field [and] Nash wanted a bigger share of that market." [ 18 ] [ 19 ] The single unit-body construction of the Nash 600 provided weight savings and Nash's Chairman and CEO, George W. Mason was convinced "that unibody was the wave of the future." [ 20 ] [ 21 ]
Since then, more cars were redesigned to the unibody structure, which is now "considered standard in the industry". [ 21 ] By 1960, the unitized body design was used by Detroit's Big Three on their compact cars ( Ford Falcon , Plymouth Valiant , and Chevrolet Corvair ). After Nash merged with Hudson Motors to form American Motors Corporation , its Rambler-badged automobiles continued exclusively building variations of the unibody.
Although the 1934 Chrysler Airflow had a weaker-than-usual frame and body framework welded to the chassis to provide stiffness, in 1960, Chrysler moved from body-on-frame construction to a unit-body design for most of its cars. [ 22 ]
Most of the American-manufactured unibody automobiles used torque boxes in their vehicle design to reduce vibrations and chassis flex, except for the Chevy II , which had a bolt-on front apron (erroneously referred to as a subframe).
The unibody is now the preferred construction for mass-market automobiles. This design provides weight savings, improved space utilization, and ease of manufacture. Acceptance grew dramatically in the wake of the two energy crises of the 1970s and that of the 2000s in which compact SUVs using a truck platform (primarily the USA market) were subjected to CAFE standards after 2005 (by the late 2000s truck-based compact SUVs were phased out and replaced with crossovers). An additional advantage of a strong-bodied car lies in the improved crash protection for its passengers.
American Motors (with its partner Renault ) during the late 1970s incorporated unibody construction when designing the Jeep Cherokee (XJ) platform using the manufacturing principles (unisides, floorplan with integrated frame rails and crumple zones, and roof panel) used in its passenger cars, such as the Hornets and all-wheel-drive Eagles for a new type of frame called the "Uniframe [...] a robust stamped steel frame welded to a strong unit-body structure, giving the strength of a conventional heavy frame with the weight advantages of Unibody construction." [ 23 ] This design was also used with the XJC concept developed by American Motors before its absorption by Chrysler, which later became the Jeep Grand Cherokee (ZJ) . The design is still used in modern-day sport utility vehicles such as the Jeep Grand Cherokee and Land Rover Defender . This design is also used in large vans such as Ford Transit , VW Crafter and Mercedes Sprinter .
A subframe is a distinct structural frame component, to reinforce or complement a particular section of a vehicle's structure. Typically attached to a unibody or a monocoque, the rigid subframe can handle great forces from the engine and drive train. It can transfer them evenly to a wide area of relatively thin sheet metal of a unitized body shell. Subframes are often found at the front or rear end of cars and are used to attach the suspension to the vehicle. A subframe may also contain the engine and transmission . It normally has pressed or box steel construction but may be tubular and/or other material.
Examples of passenger car use include the 1967–1981 GM F platform , the numerous years and models built on the GM X platform (1962) , GM's M/L platform vans ( Chevrolet Astro /GMC Safari, which included an all-wheel drive variant), and the unibody AMC Pacer that incorporated a front subframe to isolate the passenger compartment from the engine, suspension, and steering loads. [ 24 ] [ 25 ]
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A Vehicular ad hoc network ( VANET ) is a proposed type of mobile ad hoc network (MANET) involving road vehicles. [ 1 ] VANETs were first proposed [ 2 ] in 2001 as " car-to-car ad-hoc mobile communication and networking" applications, where networks could be formed and information could be relayed among cars. It has been shown that vehicle-to-vehicle and vehicle-to-roadside communications architectures could co-exist in VANETs to provide road safety , navigation, and other roadside services.
VANETs could be a key part of the intelligent transportation systems (ITS) framework. Sometimes, VANETs are referred to as Intelligent Transportation Networks. [ 3 ] They could evolve into a broader " Internet of vehicles ". [ 4 ] which itself could evolve into an "Internet of autonomous vehicles". [ 5 ]
While, in the early 2000s, VANETs were seen as a mere one-to-one application of MANET principles, they have since then developed into a field of research in their own right. By 2015, [ 6 ] : 3 the term VANET became mostly synonymous with the more generic term inter-vehicle communication ( IVC ), although the focus remains on the aspect of spontaneous networking, much less on the use of infrastructure like Road Side Units (RSUs) or cellular networks.
VANETs are in development and are not in use by commercially available vehicles. [ 7 ]
VANETs could support a wide range of applications – from simple one hop information dissemination of, e.g., cooperative awareness messages (CAMs) to multi-hop dissemination of messages over vast distances.
Most of the principles of mobile ad hoc networks (MANETs) apply to VANETs, but the details differ. [ 8 ] Rather than moving at random, vehicles tend to move in an organized fashion. The interactions with roadside equipment can likewise be characterized fairly accurately. And finally, most vehicles are restricted in their range of motion, for example by being constrained to follow a paved highway.
Potential applications of VANETs include: [ 6 ] : 56
VANETs could use any wireless networking technology as their basis. The most prominent are short-range radio technologies are WLAN and DSRC . In addition, cellular technologies or LTE and 5G can be used for VANETs.
Prior to the implementation of VANETs on the roads, realistic computer simulations of VANETs using a combination of Urban Mobility simulation [ 13 ] and Network simulation are thought to be necessary. Typically open source simulator like SUMO [ 14 ] (which handles road traffic simulation) is combined with a network simulator like TETCOS NetSim , [ 15 ] or NS-2 to study the performance of VANETs.
Further simulations could also be done for communication channel modeling that captures the complexities of wireless network for VANETs. [ 16 ]
Major standardization of VANET protocol stacks is taking place in the U.S., in Europe, and in Japan, corresponding to these regions' dominance in the automotive industry . [ 6 ] : 5
In the U.S., the IEEE 1609 WAVE Wireless Access in Vehicular Environments protocol stack builds on IEEE 802.11p WLAN operating on seven reserved channels in the 5.9 GHz frequency band.
The WAVE protocol stack is designed to provide multi-channel operation (even for vehicles equipped with only a single radio), security, and lightweight application layer protocols.
Within the IEEE Communications Society , there is a Technical Subcommittee on Vehicular Networks & Telematics Applications (VNTA). The charter of this committee is to actively promote technical activities in the field of vehicular networks, V2V, V2R and V2I communications, standards, communications-enabled road and vehicle safety, real-time traffic monitoring , intersection management technologies, future telematics applications, and ITS -based services.
In the US, the systems could use a region of the 5.9 GHz band set aside by the United States Congress, the unlicensed frequency also used by Wi-Fi . The US V2V standard, commonly known as WAVE ("Wireless Access for Vehicular Environments"), builds upon the lower-level IEEE 802.11p standard, as early as 2004.
The European Commission Decision 2008/671/EC harmonises the use of the 5 875-5 905 MHz frequency band for transport safety ITS applications. [ 17 ] In Europe V2V is standardised as ETSI ITS, [ 18 ] a standard also based on IEEE 802.11p . C-ITS, cooperative ITS, is also a term used in EU policy making, closely linked to ITS-G5 and V2V.
V2V is also known as VANET (vehicular ad hoc network). It is a variation of MANET ( Mobile ad hoc network ), with the emphasis being now the node is the vehicle. In 2001, it was mentioned in a publication [ 19 ] that ad hoc networks can be formed by cars and such networks can help overcome blind spots, avoid accidents, etc. The infrastructure also participates in such systems, then referred to as V2X (vehicle-to-everything). Over the years, there have been considerable research and projects in this area, applying VANETs for a variety of applications, ranging from safety to
navigation and law enforcement.
In 1999 the US Federal Communications Commission (FCC) allocated 75 MHz in the spectrum of 5.850-5.925 GHz for intelligent transport systems.
As of 2016, V2V is under threat from cable television and other tech firms that want to take away a big chunk of the radio spectrum currently reserved for it and use those frequencies for high-speed internet service. V2V's current share of spectrum was set aside by the government in 1999. The auto industry is trying to retain all it can saying that it desperately needs the spectrum for V2V. The Federal Communications Commission has taken the side of the tech companies with the National Traffic Safety Board supporting the position of the auto industry. Internet service providers who want the spectrum claim that self-driving cars will make extensive use of V2V unnecessary. The auto industry said it is willing to share the spectrum if V2V service is not slowed or disrupted; the FCC plans to test several sharing schemes. [ 20 ]
Research in VANETs started as early as 2000, in universities and research labs, having evolved from researchers working on wireless ad hoc networks. Many have worked on media access protocols, routing, warning message dissemination, and VANET application scenarios. V2V is currently in active development by General Motors , which demonstrated the system in 2006 using Cadillac vehicles. Other automakers working on V2V include Toyota , [ 21 ] BMW , Daimler , Honda , Audi , Volvo and the Car-to-Car communication consortium. [ 22 ]
Since then, the United States Department of Transportation (USDOT) has been working with a range of stakeholders on V2X . In 2012, a pre-deployment project was implemented in Ann Arbor , Michigan. 2800 vehicles covering cars, motorcycles, buses and HGV of different brands took part using equipment by different manufacturers. [ 23 ] The US National Highway Traffic Safety Administration (NHTSA) saw this model deployment as proof that road safety could be improved and that WAVE standard technology was interoperable. In August 2014, NHTSA published a report arguing vehicle-to-vehicle technology was technically proven as ready for deployment. [ 24 ] In April 2014 it was reported that U.S. regulators were close to approving V2V standards for the U.S. market. [ 25 ] On 20 August 2014 the NHTSA published an Advance Notice of Proposed Rulemaking (ANPRM) in the Federal Register, [ 26 ] arguing that the safety benefits of V2X communication could only be achieved, if a significant part of the vehicles fleet was equipped. Because of the lacking immediate benefit for early adopters the NHTSA proposed a mandatory introduction. On 25 June 2015, the US House of Representatives held a hearing on the matter, where again the NHTSA, as well as other stakeholders argued the case for V2X . [ 27 ]
In the EU the ITS Directive 2010/40/EU [ 28 ] was adopted in 2010. It aims to assure that ITS applications are interoperable and can operate across national borders, it defines priority areas for secondary legislation, which cover V2X and requires technologies to be mature. In 2014 the European Commission's industry stakeholder "C-ITS Deployment Platform" started working on a regulatory framework for V2X in the EU. [ 29 ] It identified key approaches to an EU-wide V2X security Public Key infrastructure (PKI) and data protection, as well as facilitating a mitigation standard [ 30 ] to prevent radio interference between ITS-G5 based V2X and CEN DSRC-based road charging systems. The European Commission recognised ITS-G5 as the initial communication technology in its 5G Action Plan [ 31 ] and the accompanying explanatory document, [ 32 ] to form a communication environment consisting of ITS-G5 and cellular communication as envisioned by EU Member States. [ 33 ] Various pre-deployment projects exist at EU or EU Member State level, such as SCOOP@F, the Testfeld Telematik, the digital testbed Autobahn, the Rotterdam-Vienna ITS Corridor, Nordic Way, COMPASS4D or C-ROADS. [ 34 ] Further projects are under preparation.
While using VANET in urban scenarios there are some aspects that are important to take in count. The first one is the analysis of the idle time [ 35 ] and the choosing of a routing protocol that satisfy the specifications of our network. [ 36 ] The other one is to try to minimize the data download time by choosing the right network architecture after analyzing the urban scenario where we want to implement it. [ 37 ]
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https://en.wikipedia.org/wiki/Vehicular_ad_hoc_network
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Vehicular automation is the use of technology to assist or replace the operator of a vehicle such as a car, truck, aircraft, rocket, military vehicle, or boat. [ 2 ] [ 3 ] [ 4 ] [ 5 ] [ 6 ] Assisted vehicles are semi-autonomous , whereas vehicles that can travel without a human operator are autonomous . [ 3 ] The degree of autonomy may be subject to various constraints such as conditions. Autonomy is enabled by advanced driver-assistance systems (ADAS) of varying capacity.
Related technology includes advanced software, maps, vehicle changes, and support outside the vehicle.
Autonomy presents varying issues for road travel, air travel, and marine travel. Roads present the greatest complexity given the unpredictability of the driving environment, including diverse road designs, driving conditions, traffic, obstacles, and geographical/cultural differences. [ 7 ]
Autonomy implies that the vehicle is responsible for all perception, monitoring, and control functions. [ 8 ]
The Society of Automotive Engineers (SAE) classifies road vehicle autonomy in six levels: [ 9 ] [ 10 ]
Level 0 refers, for instance, to vehicles without adaptive cruise control . Level 1 and 2 refer to vehicles where one part of the driving task is performed by the ADAS under the responsibility/liability of the driver.
From level 3, the driver can transfer the driving task to the vehicle, but the driver must assume control when the ADAS reaches its limits. For instance an automated traffic jam pilot can drive in a traffic jam , but otherwise passes control to the driver. Level 5 refers to a vehicle that can handle any situation. [ 11 ]
Autonomous systems typically rely on machine learning software to operate. [ 12 ] Systems must perceive the driving environment, control the vehicle, monitor vehicle status, and provide detailed driving directions.
The perception system is responsible for observing the environment. It must identify everything that could affect the trip, including other vehicles, pedestrians, cyclists, their movements, road conditions, obstacles, and other issues. [ 13 ] Various makers use cameras, radar , lidar , sonar , and microphones that can collaboratively minimize errors. [ 13 ] [ 14 ]
Navigation systems are a necessary element in autonomous vehicles. The Global Positioning System (GPS) is used for navigation by air and water vehicles, and by land vehicles as well, particularly for off-road navigation.
For road vehicles, two approaches are prominent. One is to use maps that hold data about lanes and intersections, relying on the vehicle's perception system to fill in the details. The other is to use highly detailed maps that reduce the scope of realtime decision-making, but require significant maintenance as the environment evolves. [ 12 ] Some systems crowdsource their map updates, using the vehicles themselves to update the map to reflect changes such as construction or traffic that is then used by the entire vehicle fleet. [ 15 ]
Another potential source of information is the environment itself. Traffic data may be supplied by roadside monitoring systems and used to route vehicles to best use a limited road system. [ 16 ] Additionally, modern GNSS enhancement technologies, such as real-time kinematic (RTK) and precise point positioning (PPP), enhance the accuracy of vehicle positioning to sub-meter level precision, which is crucial for autonomous navigation and decision-making. [ 17 ]
Automated vehicles in European Union legislation refer specifically to road vehicles (car, truck, or bus). [ 18 ] For those vehicles, a specific difference is legally defined between advanced driver-assistance system and autonomous/automated vehicles, based on liability differences.
AAA Foundation for Traffic Safety tested two automatic emergency braking systems: some designed to prevent crashes and others that aim to make a crash less severe. The test looked at popular models like the 2016 Volvo XC90 , Subaru Legacy , Lincoln MKX , Honda Civic , and Volkswagen Passat . Researchers tested how well each system stopped when approaching moving and nonmoving targets. It found that systems capable of preventing crashes reduced vehicle speeds by twice that of the systems designed to mitigate crash severity. When the two test vehicles traveled within 30 mph of each other, even those designed to simply lessen crash severity avoided crashes 60 percent of the time. [ 19 ]
The SAfe Road TRains for the Environment (Sartre) project's goal was to enable platooning, in which a line of cars and trucks (a "train") follow a human-driven vehicle. Trains were predicted to provide comfort and allow the following vehicles to travel safely to a destination. Human drivers encountering a train could join and delegate driving to the human driver. [ 20 ]
Self-driving Uber vehicles were tested in Pittsburgh, Pennsylvania. The tests were paused after an autonomous car killed a woman in Arizona. [ 21 ] [ 22 ] Automated busses have been tested in California. [ 23 ] In San Diego, California, an automated bus test used magnetic markers. The longitudinal control of automated truck platoons used millimeter wave radio and radar. Waymo and Tesla have conducted tests. Tesla FSD allows drivers to enter a destination and let the car take over.
Ford offers Blue Cruise, technology that allows geofenced cars to drive autonomously. [ 24 ]
Drivers are directed to stay attentive and safety warnings are implemented to alert the driver when corrective action is needed. [ 25 ] Tesla, Incorporated has one recorded incident that resulted in a fatality involving the automated driving system in the Tesla Model S. [ 26 ] The accident report reveals the accident was a result of the driver being inattentive and the autopilot system not recognizing the obstruction ahead. [ 26 ] Tesla has also had multiple instances where the vehicle crashed into a garage door. According to the book "The Driver in the Driverless Car: How Your Technology Choices Create the Future" a Tesla performed an update overnight automatically. The morning after the update the driver used his app to "summon" his car, it crashed into his garage door.
Another flaw with automated driving systems is that in situations where unpredictable events such as weather or the driving behavior of others may cause fatal accidents due to sensors that monitor the surroundings of the vehicle not being able to provide corrective action. [ 25 ]
To overcome some of the challenges for automated driving systems, novel methodologies based on virtual testing, traffic flow simulation and digital prototypes have been proposed, [ 27 ] especially when novel algorithms based on Artificial Intelligence approaches are employed which require extensive training and validation data sets.
The implementation of automated driving systems poses the possibility of changing built environments in urban areas, such as the expansion of suburban areas due to the increased ease of mobility. [ 28 ]
Around 2015, several self-driving car companies including Nissan and Toyota promised self-driving cars by 2020. However, the predictions turned out to be far too optimistic. [ 29 ]
There are still many obstacles in developing fully autonomous Level 5 vehicles, which is the ability to operate in any conditions. Currently, companies are focused on Level 4 automation, which is able to operate under certain environmental circumstances. [ 29 ]
There is still debate about what an autonomous vehicle should look like. For example, whether to incorporate lidar to autonomous driving systems is still being argued. Some researchers have come up with algorithms using camera-only data that achieve the performance that rival those of lidar. On the other hand, camera-only data sometimes draw inaccurate bounding boxes, and thus lead to poor predictions. This is due to the nature of superficial information that stereo cameras provide, whereas incorporating lidar gives autonomous vehicles precise distance to each point on the vehicle. [ 29 ]
These features require numerous sensors, many of which rely on micro-electro-mechanical systems (MEMS) to maintain a small size, high efficiency, and low cost. Foremost among MEMS sensors in vehicles are accelerometers and gyroscopes to measure acceleration around multiple orthogonal axes—critical to detecting and controlling the vehicle's motion.
One critical step to achieve the implementation of autonomous vehicles is the acceptance by the general public. It provides guidelines for the automobile industry to improve their design and technology. Studies have shown that many people believe that using autonomous vehicles is safer, which underlines the necessity for the automobile companies to assure that autonomous vehicles improve safety benefits. The TAM research model breaks down important factors that affect the consumer's acceptance into: usefulness, ease to use, trust, and social influence. [ 31 ]
Real-time testing of autonomous vehicles is an inevitable part of the process. At the same time, vehicular automation regulators are faced with challenges to protect public safety and yet allow autonomous vehicle companies to test their products. Groups representing autonomous vehicle companies are resisting most regulations, whereas groups representing vulnerable road users and traffic safety are pushing for regulatory barriers. To improve traffic safety, the regulators are encouraged to find a middle ground that protects the public from immature technology while allowing autonomous vehicle companies to test the implementation of their systems. [ 32 ] There have also been proposals to adopt the aviation automation safety regulatory knowledge into the discussions of safe implementation of autonomous vehicles, due to the experience that has been gained over the decades by the aviation sector on safety topics. [ 33 ]
In some countries, specific laws and regulations apply to road traffic motor vehicles (such as cars, bus and trucks) while other laws and regulations apply to other ground vehicles such as tram, train or automated guided vehicles making them to operate in different environments and conditions.
An automated driving system is defined in a proposed amendment to Article 1 of the Vienna Convention on Road Traffic :
(ab) " Automated driving system " refers to a vehicle system that uses both hardware and
software to exercise dynamic control of a vehicle on a sustained basis.
(ac) "Dynamic control" refers to carrying out all the real-time operational and tactical functions required to move the vehicle. This includes controlling the vehicle's lateral and longitudinal motion, monitoring the road environment, responding to events in the road traffic environment, and planning and signalling for manoeuvres. [ 34 ]
This amendment will enter into force on 14 July 2022, unless it is rejected before 13 January 2022. [ 35 ]
An automated driving feature must be described sufficiently clearly so that it is distinguished from an assisted driving feature.
There are two clear states – a vehicle is either assisted with a driver being supported by technology or automated where the technology is effectively and safely replacing the driver.
Ground vehicles employing automation and teleoperation include shipyard gantries, mining trucks, bomb-disposal robots, robotic insects, and driverless tractors .
There are many autonomous and semi-autonomous ground vehicles being made for the purpose of transporting passengers. One such example is the free-ranging on grid ( FROG ) technology which consists of autonomous vehicles, a magnetic track and a supervisory system. The FROG system is deployed for industrial purposes in factory sites and has been in use since 1999 on the ParkShuttle , a PRT -style public transport system in the city of Capelle aan den IJssel to connect the Rivium business park with the neighboring city of Rotterdam (where the route terminates at the Kralingse Zoom metro station). The system experienced a crash in 2005 [ 37 ] that proved to be caused by a human error. [ 38 ]
Applications for automation in ground vehicles include the following:
Research is ongoing and prototypes of autonomous ground vehicles exist.
Extensive automation for cars focuses on either introducing robotic cars or modifying modern car designs to be semi-autonomous.
Semi-autonomous designs could be implemented sooner as they rely less on technology that is still at the forefront of research. An example is the dual mode monorail. Groups such as RUF (Denmark) and TriTrack (USA) are working on projects consisting of specialized private cars that are driven manually on normal roads but also that dock onto a monorail/guideway along which they are driven autonomously.
As a method of automating cars without extensively modifying the cars as much as a robotic car , Automated highway systems (AHS) aims to construct lanes on highways that would be equipped with, for example, magnets to guide the vehicles. Automation vehicles have auto-brakes named as Auto Vehicles Braking System (AVBS). Highway computers would manage the traffic and direct the cars to avoid crashes.
In 2006, The European Commission has established a smart car development program called the Intelligent Car Flagship Initiative . [ 39 ] The goals of that program include:
There are further uses for automation in relation to cars. These include:
Singapore also announced a set of provisional national standards on January 31, 2019, to guide the autonomous vehicle industry. The standards, known as Technical Reference 68 (TR68), will promote the safe deployment of fully driverless vehicles in Singapore, according to a joint press release by Enterprise Singapore (ESG), Land Transport Authority (LTA), Standards Development Organisation and Singapore Standards Council (SSC). [ 42 ]
Since 1999, the 12-seat/10-standing ParkShuttle has been operating on an 1.8 kilometres (1.1 mi) exclusive right of way in the city of Capelle aan den IJssel in The Netherlands. The system uses small magnets in the road surface to allow the vehicle to determine its position. The use of shared autonomous vehicles was trialed around 2012 in a hospital car park in Portugal. [ 43 ] From 2012 to 2016, the European Union funded CityMobil2 project examined the use of shared autonomous vehicles and passenger experience including short term trials in seven cities. This project led to the development of the EasyMile EZ10. [ 44 ]
In the 2010s, self-driving shuttle became able to run in mixed traffic without the need for embedded guidance markers. [ 45 ] So far the focus has been on low speed, 20 miles per hour (32 km/h), with short, fixed routes for the "last mile" of journeys. This means issues of collision avoidance and safety are significantly less challenging than those for automated cars, which seek to match the performance of conventional vehicles. Many trials have been undertaken, mainly on quiet roads with little traffic or on public pathways or private roadways and specialised test sites. [ citation needed ] The capacity of different models varies significantly, between 6-seats and 20-seats. (Above this size there are conventional buses that have driverless technology installed.)
In December 2016, the Jacksonville Transportation Authority has announced its intention to replace the Jacksonville Skyway monorail with driverless vehicles that would run on the existing elevated superstructure as well as continue onto ordinary roads. [ 46 ] The project has since been named the "Ultimate Urban Circulator" or "U2C" and testing has been carried out on shuttles from six different manufacturers. The cost of the project is estimated at $379 million. [ 47 ]
In January 2017, it was announced the ParkShuttle system in the Netherlands will be renewed and expanded including extending the route network beyond the exclusive right of way so vehicles will run in mixed traffic on ordinary roads. [ 48 ] The plans were delayed and the extension into mixed traffic was expected in 2021. [ 49 ]
In July 2018, Baidu stated it had built 100 of its 8-seat Apolong model, with plans for commercial sales. [ 50 ] As of July 2021, they had not gone into volume production.
In August 2020, it was reported there were 25 autonomous shuttle manufacturers, [ 51 ] including the 2GetThere , Local Motors , Navya , Baidu , Easymile , Toyota and Ohmio.
In December 2020, Toyota showcased its 20-passenger "e-Palette" vehicle, which is due to be used at the 2021 Tokyo Olympic Games . [ 52 ] Toyota announced it intends to have the vehicle available for commercial applications before 2025. [ 53 ]
In January 2021, Navya released an investor report which predicted global autonomous shuttle sales will reach 12,600 units by 2025, with a market value of EUR 1.7 billion. [ 54 ]
In June 2021, Chinese maker Yutong claimed to have delivered 100 models of its 10-seat Xiaoyu 2.0 autonomous bus for use in Zhengzhou . Testing has been carried out in a number of cities since 2019 with trials open to the public planned for July 2021. [ 55 ]
Self-driving shuttles are already in use on some private roads, such as at the Yutong factory in Zhengzhou where they are used to transport workers between buildings of the world's largest bus factory. [ 56 ]
A large number of trials have been conducted since 2016, with most involving only one vehicle on a short route for a short period of time and with an onboard conductor. The purpose of the trials has been to both provide technical data and to familiarize the public with the driverless technology. A 2021 survey of over 100 shuttle experiments across Europe concluded that low speed – 15–20 kilometres per hour (9.3–12.4 mph) – was the major barrier to implementation of autonomous shuttle buses. The current cost of the vehicles at €280,000 and the need for onboard attendants were also issues. [ 57 ]
In March 2023, "ZEN drive Pilot" became the first legally approved Level 4 Automatic operation device under the amended "Road Traffic Act" of 2023. [ 103 ]
Vehicle names are in quotes
Autonomous buses are proposed as well as self driving cars and trucks. Grade 2 level automated minibuses were trialed for a few weeks in Stockholm. [ 109 ] [ 110 ] China has a small fleet of self-driving public buses in the tech district of Shenzhen, Guangdong. [ 111 ]
The first autonomous bus trial in the United Kingdom commenced in mid-2019, with an Alexander Dennis Enviro200 MMC single-decker bus modified with autonomous software from Fusion Processing able to operate in driverless mode within Stagecoach Manchester 's Sharston bus depot, performing tasks such as driving to the washing station, refuelling point and then parking at a dedicated parking space in the depot. [ 112 ] Passenger-carrying driverless bus trials in Scotland commenced in January 2023, with a fleet of five identical vehicles to the Manchester trial used on a 14 miles (23 km) Stagecoach Fife park-and-ride route across the Forth Road Bridge , from the north bank of the Forth to Edinburgh Park station . [ 113 ] [ 114 ]
Another autonomous trial in Oxfordshire , England, which uses a battery electric Fiat Ducato minibus on a circular service to Milton Park , operated by FirstBus with support from Fusion Processing, Oxfordshire County Council and the University of the West of England , entered full passenger service also in January 2023. The trial route is planned to be extended to Didcot Parkway railway station following the acquisition of a larger single-decker by the end of 2023. [ 115 ] [ 116 ]
In July 2020 in Japan, AIST Human-Centered Mobility Research Center with Nippon Koei and Isuzu started a series of demonstration tests for mid-sized buses, Isuzu "Erga Mio" with autonomous driving systems, in five areas; Ōtsu city in Shiga prefecture , Sanda city in Hyōgo Prefecture and other three areas in sequence. [ 117 ] [ 118 ] [ 119 ]
In October 2023, Imagry , an Israeli AI startup, introduced its mapless autonomous driving solution at Busworld Europe, leveraging a real-time image recognition system and a spatial deep convolutional neural network (DCNN) to mimic human driving behavior. [ 120 ]
The concept for autonomous vehicles has been applied for commercial uses, such as autonomous or nearly autonomous trucks .
Companies such as Suncor Energy , a Canadian energy company, and Rio Tinto Group were among the first to replace human-operated trucks with driverless commercial trucks run by computers. [ 121 ] In April 2016, trucks from major manufacturers including Volvo and the Daimler Company completed a week of autonomous driving across Europe, organized by the Dutch, in an effort to get self-driving trucks on the road. With developments in self-driving trucks progressing, U.S. self-driving truck sales is expected to reach 60,000 by 2035 according to a report released by IHS Incorporated in June 2016. [ 122 ]
As reported in June 1995 in Popular Science magazine, self-driving trucks were being developed for combat convoys, whereby only the lead truck would be driven by a human and the following trucks would rely on satellite, an inertial guidance system and ground-speed sensors. [ 123 ] Caterpillar Incorporated made early developments in 2013 with the Robotics Institute at Carnegie Mellon University to improve efficiency and reduce cost at various mining and construction sites. [ 124 ]
In Europe, the Safe Road Trains for the Environment is such an approach.
From PWC's Strategy & Report, [ 125 ] self driving trucks will be the source of concern around how this technology will impact around 3 million truck drivers in the US, as well as 4 million employees in support of the trucking economy in gas stations, restaurants, bars and hotels. At the same time, some companies like Starsky, are aiming for Level 3 Autonomy, which would see the driver playing a control role around the truck's environment. The company's project, remote truck driving, would give truck drivers a greater work-life balance, enabling them to avoid long periods away from their home. This would however provoke a potential mismatch between the driver's skills with the technological redefinition of the job.
Companies that buy driverless trucks could massively cut costs: human drivers would no longer be required, companies' liabilities due to truck accidents would diminish, and productivity would increase (as the driverless truck doesn't need to rest). The usage of self driving trucks will go hand in hand with the use of real-time data to optimize both efficiency and productivity of the service delivered, as a way to tackle traffic congestion for example. Driverless trucks could enable new business models that would see deliveries shift from day time to night time or time slots in which traffic is less heavily dense.
Several self-balancing autonomous motorcycles were demonstrated in 2017 and 2018 from BMW, Honda and Yamaha. [ 135 ] [ 136 ] [ 137 ]
The concept for autonomous vehicles has also been applied for commercial uses, like for autonomous trains. The world's first driverless urban transit system is the Port Island Line in Kobe , Japan, opened in 1981. [ 141 ] The first self-driving train in the UK was launched in London on the Thameslink route. [ 142 ]
An example of an automated train network is the Docklands Light Railway in London .
Also see List of automated train systems .
In 2018 the first autonomous trams in Potsdam were trialed. [ 143 ]
An automated guided vehicle or automatic guided vehicle (AGV) is a mobile robot that follows markers or wires in the floor, or uses vision, magnets, or lasers for navigation. They are most often used in industrial applications to move materials around a manufacturing facility or warehouse. Application of the automatic guided vehicle had broadened during the late 20th century.
Aircraft have received much attention for automation, especially for navigation. A system capable of autonomously navigating a vehicle (especially aircraft) is known as autopilot .
Various industries such as packages and food have experimented with delivery drones. Traditional and new transportation companies are competing in the market. For example, UPS Flight Forward , Alphabet Wing, and Amazon Prime Air are all developing delivery drones. [ 144 ] Zipline , an American medical drone delivery company, has the largest active drone delivery operations in the world, and its drones are capable of Level 4 autonomy. [ 145 ]
However, even if technology seems to allow for those solutions to function correctly as various tests of various companies show, the main throwback to the market launch and use of such drones is inevitably the legislation in place and regulatory agencies have to decide on the framework they wish to take to draft regulation. This process is in different phases across the world as each country will tackle the topic independently. For example, Iceland's government and departments of transport, aviation, police have already started issuing licenses for drone operations. It has a permissive approach and together with Costa Rica, Italy, the UAE, Sweden and Norway, has a fairly unrestricted legislation on commercial drone use. Those countries are characterized by a body of regulation that may give operational guidelines or require licensing, registration and insurance. [ 146 ]
On the other side, other countries have decided to ban, either directly (outright ban) or indirectly (effective ban), the use of commercial drones. The RAND Corporation thus notes the difference between countries forbidding drones and those that have a formal process for commercial drone licensing, but requirements are either impossible to meet or licenses do not appear to have been approved. In the US, United Parcel Service is the only delivery service with the Part 135 Standard certification that is required to use drones to deliver to real customers. [ 144 ]
However, most countries seem to be struggling on the integration of drones for commercial uses into their aviation regulatory frameworks. Thus, constraints are placed on the use of those drones such as that they must be operating within the visual line of sight (VLOS) of the pilot and thus limiting their potential range. This would be the case of the Netherlands and Belgium. Most countries let pilots operate outside the VLOS but is subject to restrictions and pilot ratings, which would be the case of the US.
The general trend is that legislation is moving fast and laws are constantly being reevaluated. Countries are moving towards a more permissive approach but the industry still lacks infrastructures to ensure the success of such a transition. To provide safety and efficiency, specialized training courses, pilot exams (type of UAV and flying conditions) as well as liability management measures regarding insurances may need to be developed.
There is a sense of urgency related to this innovation as competition is high and companies lobby to integrate them rapidly in their products and services offerings. Since June 2017, the US Senate legislation reauthorized the Federal Aviation Administration and the Department of Transportation to create a carrier certificate allowing for package deliveries by drones. [ 147 ]
Autonomous boats can provide security, perform research, or conduct hazardous or repetitive tasks (such as guiding a large ship into a harbor or transporting cargo).
Sea Machines offers an autonomous system for workboats. While it requires a human operator to oversee its actions, the system takes care of many active domain perception and navigation duties that normally a few members of the crew would have to do. They use AI to have situational awareness for different ships within the route. They use camera, lidar, and proprietary software to inform the operator of its status. [ 148 ] [ 149 ]
Buffalo Automation , a team formed from the University of Buffalo, creates technology for semi-autonomous features for boats. They started by creating navigation assist technologies for freighters called AutoMate, which is like having another very experienced “first mate” that will look out for the ship. [ 150 ] The system helps navigate difficult waterways. [ 149 ] [ 151 ]
This Massachusetts based company has led the forefront of unmanned sailing drones. The Datamarans are autonomously sailing to collect ocean data. They are created to enable large payload packages. Due to the automated system and their solar panels, they are able to navigate for longer periods of time. Their technologies on advanced metocean surveys, collect “wind velocity profiles with altitude, water current, conductivity, temperature profiles with depth, hi-resolution bathymetry, sub-bottom profiling, [and] magnetometer measurements”. [ 152 ] [ 149 ]
The autonomous vessel called Mayflower is expected to be the first large ship that makes an unmanned transatlantic journey. [ 153 ]
This autonomous unmanned vessel uses both solar and wind energy to navigate. [ 154 ]
Sea Hunter is an autonomous unmanned surface vehicle (USV) launched in 2016 as part of the DARPA Anti-Submarine Warfare Continuous Trail Unmanned Vessel ( ACTUV ) program.
Underwater vehicles have been a focus for automation for tasks such as pipeline inspection and underwater mapping.
This four-legged robot was created to be able to navigate through many different terrain outdoors and indoors. It can walk on its own without colliding into anything. It uses many different sensors, including 360-degree vision cameras and gyroscopes. It is able to keep its balance even when pushed over. This vehicle, while it is not intended to be ridden, can carry heavy loads for construction workers or military personnel through rough terrain. [ 155 ]
The British Highway Code states that:
By self-driving vehicles , we mean those listed as automated vehicles by the Secretary of State for Transport under the Automated and Electric Vehicles Act 2018.
The UK considers the way to update its British Highway Code for automated code:
Automated vehicles can perform all the tasks involved in driving, in at least some situations. They differ from vehicles fitted with assisted driving features (like cruise control and lane-keeping assistance ), which carry out some tasks, but where the driver is still responsible for driving. If you are driving a vehicle with assisted driving features, you MUST stay in control of the vehicle.
If the vehicle is designed to require you to resume driving after being prompted to, while the vehicle is driving itself, you MUST remain in a position to be able to take control. For example, you should not move out of the driving seat. You should not be so distracted that you cannot take back control when prompted by the vehicle.
Through the autonomy level, it is shown that the higher the level of autonomy, the less control humans have on their vehicles (highest level of autonomy needing zero human interventions). One concerns regarding the development of vehicular automation is related to the end-users’ trust in the technology that controls automated vehicles. [ 157 ] According to a nationally conducted survey made by Kelley Blue Book (KBB) in 2016, it was shown that the majority of people would choose to have a certain level of control behind their own vehicle rather than having the vehicle operate in Level 5 autonomy, or in other words, complete autonomy. [ 158 ] According to half of the respondents, the idea of safety in an autonomous vehicle diminishes as the level of autonomy increases. [ 158 ] This distrust of autonomous driving systems proved to be unchanged throughout the years when a nationwide survey conducted by AAA Foundation for Traffic and Safety (AAAFTS) in 2019 showed the same outcome as the survey KBB did in 2016. AAAFTS survey showed that even though people have a certain level of trust in automated vehicles, most people also have doubts and distrust towards the technology used in autonomous vehicles, with most distrust in Level 5 autonomous vehicles. [ 159 ] It is shown by AAAFTS’ survey that people's trust in autonomous driving systems increased when their level of understanding increased. [ 159 ]
The possibility of autonomous vehicle's technology to experience malfunctions is also one of the causes of user's distrust in autonomous driving systems. [ 157 ] It is the concern that most respondents voted for in the AAAFTS survey. [ 159 ] Even though autonomous vehicles are made to improve traffic safety by minimizing crashes and their severity, [ 159 ] they still caused fatalities. At least 113 autonomous vehicle related accidents have occurred until 2018. [ 160 ] In 2015, Google declared that their automated vehicles experienced at least 272 failures, and drivers had to intervene around 13 times to prevent fatalities. [ 161 ] Furthermore, other automated vehicles’ manufacturers also reported automated vehicles’ failures, including the Uber car incident. [ 161 ] A self-driving Uber car accident in 2018 is an example of autonomous vehicle accidents that are also listed among self-driving car fatalities. A report made by the National Transportation Safety Board (NTSB) showed that the self-driving Uber car was unable to identify the victim in a sufficient amount of time for the vehicle to slow down and avoid crashing into the victim. [ 162 ]
Another concern related to vehicle automation is its ethical issues. In reality, autonomous vehicles can encounter inevitable traffic accidents. In such situations, many risks and calculations need to be made in order to minimize the amount of damage the accident could cause. [ 163 ] When a human driver encounters an inevitable accident, the driver will take a spontaneous action based on ethical and moral logic. However, when a driver has no control over the vehicle (Level 5 autonomy), the system of an autonomous vehicle needs to make that quick decision. [ 163 ] Unlike humans, autonomous vehicles can only make decisions based on what it is programmed to do. [ 163 ] However, the situation and circumstances of accidents differ from one another, and any one decision might not be the best decision for certain accidents. Based on two research studies in 2019, [ 164 ] [ 165 ] the implementation of fully automated vehicles in traffic where semi-automated and non-automated vehicles are still present might lead to complications. [ 164 ] Some flaws that still need consideration include the structure of liability, distribution of responsibilities, [ 165 ] efficiency in decision making, and the performance of autonomous vehicles with its diverse surroundings. [ 164 ] Still, researchers Steven Umbrello and Roman V. Yampolskiy propose that the value sensitive design approach is one method that can be used to design autonomous vehicles to avoid some of these ethical issues and design for human values. [ 166 ]
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https://en.wikipedia.org/wiki/Vehicular_automation
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Vehicular communication systems are computer networks in which vehicles and roadside units are the communicating nodes , providing each other with information, such as safety warnings and traffic information. They can be effective in avoiding accidents and traffic congestion. Both types of nodes are dedicated short-range communications (DSRC) devices. DSRC works in 5.9 GHz band with bandwidth of 75 MHz and approximate range of 300 metres (980 ft). [ 1 ] Vehicular communications is usually developed as a part of intelligent transportation systems (ITS).
The beginnings of vehicular communications go back to the 1970s. Work began on projects such as Electronic Route Guidance System (ERGS) and CACS in the United States and Japan respectively. [ 2 ] While the term Inter-Vehicle Communications (IVC) began to circulate in the early 1980s. [ 3 ] Various media were used before the standardization activities began, such as lasers, infrared, and radio waves.
The PATH project in the United States between 1986 and 1997 was an important breakthrough in vehicular communications projects. [ 4 ] Projects related to vehicular communications in Europe were launched with the PROMETHEUS project between 1986 and 1995. [ 5 ] Numerous subsequent projects have been implemented all over the world such as the Advanced Safety Vehicle (ASV) program, [ 6 ] CHAUFFEUR I and II, [ 7 ] FleetNet, [ 8 ] CarTALK 2000, [ 9 ] etc.
In the early 2000s, the term Vehicular Ad Hoc Network (VANET) was introduced as an application of the principles of Mobile Ad-Hoc Networks (MANETs) to the vehicular field. The terms VANET and IVC do not differ and are used interchangeably to refer to communications between vehicles with or without reliance on roadside infrastructure, although some have argued that IVC refers to direct V2V connections only. [ 10 ] Many projects have appeared in EU, Japan, USA and other parts of the world for example, ETC, [ 11 ] SAFESPOT, [ 12 ] PReVENT, [ 13 ] COMeSafety, [ 14 ] NoW, [ 15 ] IVI. [ 16 ]
Several terms have been used to refer to vehicular communications. These acronyms differ from each other either in historical context, technology used, standard, or country ( vehicle telematics , DSRC , WAVE, [ 17 ] VANET , IoV , 802.11p , ITS-G5, [ 18 ] V2X ). Currently, cellular based on 3GPP-Release 16 [ 19 ] and WiFi based on IEEE 802.11p have proven to be potential communication technologies enabling connected vehicles. However, this does not negate that other technologies for example, VLC , ZigBee , WiMAX , microwave , mmWave are still a vehicular communication research area. [ 20 ]
Many organizations and governmental agencies are concerned with issuing standards and regulation for vehicular communication ( ASTM , IEEE , ETSI , SAE , 3GPP , ARIB , TTC , TTA, [ 21 ] CCSA , ITU , 5GAA , ITS America , ERTICO, ITS Asia-Pacific [ 22 ] ). 3GPP is working on standards and specifications for cellular-based V2X communications, [ 23 ] while IEEE is working through the study group Next Generation V2X (NGV) on the issuance of the standard 802.11bd. [ 24 ]
The main motivation for vehicular communication systems is safety and eliminating the excessive cost of traffic collisions. According to the World Health Organization (WHO), road accidents annually cause approximately 1.2 million deaths worldwide; one fourth of all deaths caused by injury. Also about 50 million persons are injured in traffic accidents. Road death was the ninth-leading cause of death in 1990. [ 25 ] A study from the American Automobile Association (AAA) concluded that car crashes cost the United States $300 billion per year. [ 26 ] It can be used for automated traffic intersection control. [ 1 ]
However the deaths caused by car crashes are in principle avoidable. The U.S. Department of Transportation states that 21,000 of the annual 43,000 road accident deaths in the US are caused by roadway departures and intersection-related incidents. [ 27 ] This number can be significantly lowered by deploying local warning systems through vehicular communications. Departing vehicles can inform other vehicles that they intend to depart the highway and arriving cars at intersections can send warning messages to other cars traversing that intersection. They can also notify when they intend to change lanes or if there is a traffic jam. [ 28 ] According to a 2010 study by the US National Highway Traffic Safety Administration , vehicular communication systems could help avoid up to 79% of all traffic accidents. [ 29 ] Studies show that in Western Europe a mere 5 km/h decrease in average vehicle speeds could result in 25% decrease in deaths. [ 30 ]
Over the years, there have been considerable research and projects in this area, applying VANETs for a variety of applications, ranging from safety to navigation and law enforcement. In December 2016, the US Department of Transportation proposed draft rules that would gradually make V2V communication capabilities to be mandatory for light-duty vehicles. [ 31 ] The technology is not completely specified, so critics have argued that manufacturers "could not take what’s in this document and know what their responsibility will be under the Federal Motor Vehicle Safety Standards". [ 31 ] PKI (public key infrastructure) is the current security system being used in V2V communications. [ 32 ]
V2V is under threat from cable television and other tech firms that want to take away a big chunk of the radio spectrum currently reserved for it and use those frequencies for high-speed internet service. In the USA , V2V's current share of the radio spectrum was set aside by the government in 1999, but has gone unused. The automotive industry is trying to retain all it can, saying that it desperately needs the spectrum for V2V. The Federal Communications Commission (FCC) has taken the side of the tech companies, with the National Transportation Safety Board supporting the position of the automotive industry. Internet service providers (who want to use the spectrum) claim that autonomous cars will render V2V communication unnecessary. The US automotive industry has said that it is willing to share the spectrum if V2V service is not slowed or disrupted; and the FCC plans to test several sharing schemes. [ 33 ]
With governments in different locales supporting incompatible spectra for V2V communication, vehicle manufacturers may be discouraged from adopting the technology for some markets. In Australia for instance, there is no spectrum reserved for V2V communication, so vehicles would suffer interference from non-vehicle communications. [ 34 ] The spectra reserved for V2V communications in some locales are as follows:
In 2012, computer scientists at the University of Texas in Austin began developing smart intersections designed for automated cars. The intersections will have no traffic lights and no stop signs, instead of using computer programs that will communicate directly with each car on the road. [ 36 ] In the case of autonomous vehicles, it is essential for them to connect with other 'devices' in order to function most effectively. Autonomous vehicles are equipped with communication systems that allow them to communicate with other autonomous vehicles and roadside units to provide them, amongst other things, with information about road work or traffic congestion. In addition, scientists believe that the future will have computer programs that connect and manage each individual autonomous vehicle as it navigates through an intersection. [ 36 ] These types of characteristics drive and further develop the ability of autonomous vehicles to understand and cooperate with other products and services (such as intersection computer systems) in the autonomous vehicles market. Eventually, this can lead to more autonomous vehicles using the network because the information has been validated through the usage of other autonomous vehicles. Such movements will strengthen the value of the network and are called network externalities.
In 2017, Researchers from Arizona State University developed a 1/10 scale intersection and proposed an intersection management technique called Crossroads. It was shown that Crossroads is very resilient to network delay of both V2I communication and Worst-case Execution time of the intersection manager. [ 37 ] In 2018, a robust approach was introduced which is resilient to both model mismatch and external disturbances such as wind and bumps. [ 38 ]
In November 2019, an applications of Cellular V2X (Cellular Vehicle-to-Everything) based on 5G were demonstrated on open city streets and a test track in Turin . [ 39 ] V2V equipped cars broadcast a message to following vehicles in the case of sudden braking to notify them timely of the potentially dangerous situation. Other applications demonstrated use cases such as; alerting drivers about a crossing pedestrian. [ 40 ]
Intelligent Transportation Society of America (ITSA) aims to improve cooperation among public and private sector organizations. ITSA summarizes its mission statement as "vision zero" meaning its goal is to reduce the fatal accidents and delays as much as possible.
Many universities are pursuing research and development of vehicular ad hoc networks. For example, University of California , Berkeley is participating in California Partners for Advanced Transit and Highways (PATH). [ 4 ]
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Velocimetry is the measurement of the velocity of fluids . This is a task often taken for granted, and involves far more complex processes than one might expect. It is often used to solve fluid dynamics problems, study fluid networks, in industrial and process control applications, as well as in the creation of new kinds of fluid flow sensors . Methods of velocimetry include particle image velocimetry and particle tracking velocimetry , Molecular tagging velocimetry , laser-based interferometry , ultrasonic Doppler methods, Doppler sensors, and new signal processing methodologies.
In general, velocity measurements are made in the Lagrangian or Eulerian frames of reference (see Lagrangian and Eulerian coordinates ). Lagrangian methods assign a velocity to a volume of fluid at a given time, whereas Eulerian methods assign a velocity to a volume of the measurement domain at a given time. A classic example of the distinction is particle tracking velocimetry, where the idea is to find the velocity of individual flow tracer particles (Lagrangian) and particle image velocimetry, where the objective is to find the average velocity within a sub-region of the field of view (Eulerian). [ 1 ]
Velocimetry can be traced back to the days of Leonardo da Vinci , who would float grass seeds on a flow and sketch the resulting trajectories of the seeds that he observed (a Lagrangian measurement). [ 2 ] Eventually da Vinci's flow visualizations were used in his cardio vascular studies, attempting to learn more about blood flow throughout the human body. [ 3 ]
Methods similar to da Vinci's were carried out for close to four hundred years due to technological limitations. One other notable study comes from Felix Savart in 1833. Using a stroboscopic instrument, he sketched water jet impacts. [ 3 ]
In the late 19th century a huge breakthrough was made in these technologies when it became possible to take photographs of flow patterns. One notable instance of this is Ludwig Mach using particles unresolvable by the naked eye to visualize streamlines. [ 4 ] Another notable contribution occurred in the 20th century by Étienne-Jules Marey who used photographic techniques to introduce the concept of the smoke box. This model allowed both for the directions of the flow to be tracked but also the speed, as streamlines closer together indicated faster flow. [ 3 ]
More recently, high speed cameras and digital technology has revolutionized the field. allowing for the possibility of many more techniques and rendering of flow fields in three dimensions. [ 3 ]
Today the basic ideas established by Leonardo are the same; the flow must be seeded with particles that can be observed by the method of choice. The seeding particles depend on many factors including the fluid, the sensing method, the size of the measurement domain, and sometimes the expected accelerations in the flow. [ 5 ] If the flow contains particles that can be measured naturally, seeding the flow is unnecessary. [ 6 ]
Spatial reconstruction of fluid streamtubes using long exposure imaging of tracer can be applied for streamlines imaging velocimetry, high resolution frame rate free velocimetry of stationary flows. [ 7 ] Temporal integration of velocimetric information can be used to totalize fluid flow. For measuring velocity and length on moving surfaces, laser surface velocimeters are used. [ 8 ]
The fluid generally limits the particle selection according to its specific gravity; the particles should ideally be of the same density as the fluid. This is especially important in flows with a high acceleration (for example, high-speed flow through a 90-degree pipe elbow). [ 9 ] Heavier fluids like water and oil are thus very attractive to velocimetry, whereas air ads a challenge in most techniques that it is rarely possible to find particles of the same density as air.
Still, even large-field measurement techniques like PIV have been performed successfully in air. [ 10 ] Particles used for seeding can be both liquid droplets or solid particles. Solid particles being preferred when high particle concentrations are necessary. [ 9 ] For point measurements like laser Doppler velocimetry , particles in the nanometre diameter range, such as those in cigarette smoke, are sufficient to perform a measurement. [ 6 ]
In water and oil there are a variety of inexpensive industrial beads that can be used, such as silver-coated hollow glass spheres manufactured to be conductive powders (tens of micrometres diameter range) or other beads used as reflectors and texturing agents in paints and coatings. [ 11 ] The particles need not be spherical; in many cases titanium dioxide particles can be used. [ 12 ]
PIV has been used in research for controlling aircraft noise. This noise is created by the high speed mixing of hot jet exhaust with the ambient temperature of the environment. PIV has been used to model this behavior. [ 13 ]
Additionally, Doppler velocimetry enables noninvasive techniques of determining whether fetuses are the proper size at a given term of pregnancy. [ 14 ]
Velocimetry has also been applied to medical images in order to obtain regional measurements of blood flow and tissue motion. Initially, standard PIV (single plane illumination) was adapted to work with x-ray images (full volume illumination), enabling the measurement of opaque flows such as blood flow. This was then extended to investigate the regional 2D motion of lung tissue, and was found to be a sensitive indicator of regional lung disease. [ 15 ]
Velocimetry was also expanded to 3D regional measurements blood flow and tissue motion with a new technique – computed tomographic x-ray velocimetry – which uses information contained within the PIV cross-correlation to extract 3D measurements from 2D image sequences. [ 16 ] Specifically, computed tomographic x-ray velocimetry generates a model solution, compares the cross-correlations of the model to the cross-correlation from the 2D image sequence, and iterates the model solution until the difference between the model cross-correlations and the image sequence cross-correlations are minimised. This technique is being used as a non invasive method to quantify functional performance of the lungs. It is being used in a clinical setting, [ 17 ] and is being utilised in clinical trails conducted by institutions including Duke University , [ 18 ] Vanderbilt University Medical Center [ 19 ] and Oregon Health Science University [ 20 ]
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In relativistic physics , a velocity-addition formula is an equation that specifies how to combine the velocities of objects in a way that is consistent with the requirement that no object's speed can exceed the speed of light . Such formulas apply to successive Lorentz transformations , so they also relate different frames . Accompanying velocity addition is a kinematic effect known as Thomas precession , whereby successive non-collinear Lorentz boosts become equivalent to the composition of a rotation of the coordinate system and a boost.
Standard applications of velocity-addition formulas include the Doppler shift , Doppler navigation , the aberration of light , and the dragging of light in moving water observed in the 1851 Fizeau experiment . [ 1 ]
The notation employs u as velocity of a body within a Lorentz frame S , and v as velocity of a second frame S ′ , as measured in S , and u ′ as the transformed velocity of the body within the second frame.
The speed of light in a fluid is slower than the speed of light in vacuum, and it changes if the fluid is moving along with the light. In 1851, Fizeau measured the speed of light in a fluid moving parallel to the light using an interferometer . Fizeau's results were not in accord with the then-prevalent theories. Fizeau experimentally correctly determined the zeroth term of an expansion of the relativistically correct addition law in terms of V / c as is described below. Fizeau's result led physicists to accept the empirical validity of the rather unsatisfactory theory by Fresnel that a fluid moving with respect to the stationary aether partially drags light with it, i.e. the speed is c / n + (1 − 1 / n 2 ) V instead of c / n + V , where c is the speed of light in the aether, n is the refractive index of the fluid, and V is the speed of the fluid with respect to the aether.
The aberration of light, of which the easiest explanation is the relativistic velocity addition formula, together with Fizeau's result, triggered the development of theories like Lorentz aether theory of electromagnetism in 1892. In 1905 Albert Einstein , with the advent of special relativity , derived the standard configuration formula ( V in the x -direction ) for the addition of relativistic velocities. [ 2 ] The issues involving aether were, gradually over the years, settled in favor of special relativity.
It was observed by Galileo that a person on a uniformly moving ship has the impression of being at rest and sees a heavy body falling vertically downward. [ 3 ] This observation is now regarded as the first clear statement of the principle of mechanical relativity. Galileo saw that from the point of view of a person standing on the shore, the motion of falling downwards on the ship would be combined with, or added to, the forward motion of the ship. [ 4 ] In terms of velocities, it can be said that the velocity of the falling body relative to the shore equals the velocity of that body relative to ship plus the velocity of the ship relative to the shore.
In general for three objects A (e.g. Galileo on the shore), B (e.g. ship), C (e.g. falling body on ship) the velocity vector u {\displaystyle \mathbf {u} } of C relative to A (velocity of falling object as Galileo sees it) is the sum of the velocity u ′ {\displaystyle \mathbf {u'} } of C relative to B (velocity of falling object relative to ship) plus the velocity v of B relative to A (ship's velocity away from the shore). The addition here is the vector addition of vector algebra and the resulting velocity is usually represented in the form
u = v + u ′ . {\displaystyle \mathbf {u} =\mathbf {v} +\mathbf {u'} .}
The cosmos of Galileo consists of absolute space and time and the addition of velocities corresponds to composition of Galilean transformations . The relativity principle is called Galilean relativity . It is obeyed by Newtonian mechanics .
According to the theory of special relativity , the frame of the ship has a different clock rate and distance measure, and the notion of simultaneity in the direction of motion is altered, so the addition law for velocities is changed. This change is not noticeable at low velocities but as the velocity increases towards the speed of light it becomes important. The addition law is also called a composition law for velocities . For collinear motions, the speed of the object, u ′ {\displaystyle u'} , e.g. a cannonball fired horizontally out to sea, as measured from the ship, moving at speed v {\displaystyle v} , would be measured by someone standing on the shore and watching the whole scene through a telescope as [ 5 ] u = v + u ′ 1 + ( v u ′ / c 2 ) . {\displaystyle u={v+u' \over 1+(vu'/c^{2})}.} The composition formula can take an algebraically equivalent form, which can be easily derived by using only the principle of constancy of the speed of light, [ 6 ] c − u c + u = ( c − u ′ c + u ′ ) ( c − v c + v ) . {\displaystyle {c-u \over c+u}=\left({c-u' \over c+u'}\right)\left({c-v \over c+v}\right).} The cosmos of special relativity consists of Minkowski spacetime and the addition of velocities corresponds to composition of Lorentz transformations . In the special theory of relativity Newtonian mechanics is modified into relativistic mechanics .
The formulas for boosts in the standard configuration follow most straightforwardly from taking differentials of the inverse Lorentz boost in standard configuration. [ 7 ] [ 8 ] If the primed frame is travelling with speed v {\displaystyle v} with Lorentz factor γ v = 1 / 1 − v 2 / c 2 {\textstyle \gamma _{_{v}}=1/{\sqrt {1-v^{2}/c^{2}}}} in the positive x -direction relative to the unprimed frame, then the differentials are
d x = γ v ( d x ′ + v d t ′ ) , d y = d y ′ , d z = d z ′ , d t = γ v ( d t ′ + v c 2 d x ′ ) . {\displaystyle dx=\gamma _{_{v}}(dx'+vdt'),\quad dy=dy',\quad dz=dz',\quad dt=\gamma _{_{v}}\left(dt'+{\frac {v}{c^{2}}}dx'\right).}
Divide the first three equations by the fourth,
d x d t = γ v ( d x ′ + v d t ′ ) γ v ( d t ′ + v c 2 d x ′ ) , d y d t = d y ′ γ v ( d t ′ + v c 2 d x ′ ) , d z d t = d z ′ γ v ( d t ′ + v c 2 d x ′ ) , {\displaystyle {\frac {dx}{dt}}={\frac {\gamma _{_{v}}(dx'+vdt')}{\gamma _{_{v}}(dt'+{\frac {v}{c^{2}}}dx')}},\quad {\frac {dy}{dt}}={\frac {dy'}{\gamma _{_{v}}(dt'+{\frac {v}{c^{2}}}dx')}},\quad {\frac {dz}{dt}}={\frac {dz'}{\gamma _{_{v}}(dt'+{\frac {v}{c^{2}}}dx')}},}
or
u x = d x d t = d x ′ d t ′ + v ( 1 + v c 2 d x ′ d t ′ ) , u y = d y d t = d y ′ d t ′ γ v ( 1 + v c 2 d x ′ d t ′ ) , u z = d z d t = d z ′ d t ′ γ v ( 1 + v c 2 d x ′ d t ′ ) , {\displaystyle u_{x}={\frac {dx}{dt}}={\frac {{\frac {dx'}{dt'}}+v}{(1+{\frac {v}{c^{2}}}{\frac {dx'}{dt'}})}},\quad u_{y}={\frac {dy}{dt}}={\frac {\frac {dy'}{dt'}}{\gamma _{_{v}}\ (1+{\frac {v}{c^{2}}}{\frac {dx'}{dt'}})}},\quad u_{z}={\frac {dz}{dt}}={\frac {\frac {dz'}{dt'}}{\gamma _{_{v}}\ (1+{\frac {v}{c^{2}}}{\frac {dx'}{dt'}})}},}
which is
u x = u x ′ + v 1 + v c 2 u x ′ , u x ′ = u x − v 1 − v c 2 u x , {\displaystyle u_{x}={\frac {u_{x}'+v}{1+{\frac {v}{c^{2}}}u_{x}'}},\quad u_{x}'={\frac {u_{x}-v}{1-{\frac {v}{c^{2}}}u_{x}}},} u y = u y ′ 1 − v 2 c 2 1 + v c 2 u x ′ , u y ′ = u y 1 − v 2 c 2 1 − v c 2 u x , {\displaystyle u_{y}={\frac {u_{y}'{\sqrt {1-{\frac {v^{2}}{c^{2}}}}}}{1+{\frac {v}{c^{2}}}u_{x}'}},\quad u_{y}'={\frac {u_{y}{\sqrt {1-{\frac {v^{2}}{c^{2}}}}}}{1-{\frac {v}{c^{2}}}u_{x}}},} u z = u z ′ 1 − v 2 c 2 1 + v c 2 u x ′ , u z ′ = u z 1 − v 2 c 2 1 − v c 2 u x , {\displaystyle u_{z}={\frac {u_{z}'{\sqrt {1-{\frac {v^{2}}{c^{2}}}}}}{1+{\frac {v}{c^{2}}}u_{x}'}},\quad u_{z}'={\frac {u_{z}{\sqrt {1-{\frac {v^{2}}{c^{2}}}}}}{1-{\frac {v}{c^{2}}}u_{x}}},}
in which expressions for the primed velocities were obtained using the standard recipe by replacing v by – v and swapping primed and unprimed coordinates. If coordinates are chosen so that all velocities lie in a (common) x – y plane, then velocities may be expressed as u x = u cos θ , u y = u sin θ , u x ′ = u ′ cos θ ′ , u y ′ = u ′ sin θ ′ , {\displaystyle u_{x}=u\cos \theta ,u_{y}=u\sin \theta ,\quad u_{x}'=u'\cos \theta ',\quad u_{y}'=u'\sin \theta ',} (see polar coordinates ) and one finds [ 2 ] [ 9 ]
u = u ′ 2 + v 2 + 2 v u ′ cos θ ′ − ( v u ′ sin θ ′ c ) 2 1 + v c 2 u ′ cos θ ′ , {\displaystyle u={\frac {\sqrt {u'^{2}+v^{2}+2vu'\cos \theta '-\left({\frac {vu'\sin \theta '}{c}}\right)^{2}}}{1+{\frac {v}{c^{2}}}u'\cos \theta '}},} tan θ = u y u x = 1 − v 2 c 2 u y ′ u x ′ + v = 1 − v 2 c 2 u ′ sin θ ′ u ′ cos θ ′ + v . {\displaystyle \tan \theta ={\frac {u_{y}}{u_{x}}}={\frac {{\sqrt {1-{\frac {v^{2}}{c^{2}}}}}u_{y}'}{u_{x}'+v}}={\frac {{\sqrt {1-{\frac {v^{2}}{c^{2}}}}}u'\sin \theta '}{u'\cos \theta '+v}}.}
u = u x 2 + u y 2 = ( u x ′ + v ) 2 + ( 1 − v 2 c 2 ) u y ′ 2 1 + v c 2 u x ′ = u x ′ 2 + v 2 + 2 u x ′ v + ( 1 − v 2 c 2 ) u y ′ 2 1 + v c 2 u x ′ = u ′ 2 cos 2 θ ′ + v 2 + 2 v u ′ cos θ ′ + u ′ 2 sin 2 θ ′ − v 2 c 2 u ′ 2 sin 2 θ ′ 1 + v c 2 u x ′ = u ′ 2 + v 2 + 2 v u ′ cos θ ′ − ( v u ′ sin θ ′ c ) 2 1 + v c 2 u ′ cos θ ′ {\displaystyle {\begin{aligned}u&={\sqrt {u_{x}^{2}+u_{y}^{2}}}={\frac {\sqrt {(u_{x}'+v)^{2}+(1-{\frac {v^{2}}{c^{2}}})u_{y}'^{2}}}{1+{\frac {v}{c^{2}}}u_{x}'}}={\frac {\sqrt {u_{x}'^{2}+v^{2}+2u_{x}'v+(1-{\frac {v^{2}}{c^{2}}})u_{y}'^{2}}}{1+{\frac {v}{c^{2}}}u_{x}'}}\\&={\frac {\sqrt {u'^{2}\cos ^{2}\theta '+v^{2}+2vu'\cos \theta '+u'^{2}\sin ^{2}\theta '-{\frac {v^{2}}{c^{2}}}u'^{2}\sin ^{2}\theta '}}{1+{\frac {v}{c^{2}}}u_{x}'}}\\&={\frac {\sqrt {u'^{2}+v^{2}+2vu'\cos \theta '-({\frac {vu'\sin \theta '}{c}})^{2}}}{1+{\frac {v}{c^{2}}}u'\cos \theta '}}\end{aligned}}}
The proof as given is highly formal. There are other more involved proofs that may be more enlightening, such as the one below.
Since a relativistic transformation rotates space and time into each other much as geometric rotations in the plane rotate the x - and y -axes, it is convenient to use the same units for space and time, otherwise a unit conversion factor appears throughout relativistic formulae, being the speed of light . In a system where lengths and times are measured in the same units, the speed of light is dimensionless and equal to 1 . A velocity is then expressed as fraction of the speed of light.
To find the relativistic transformation law, it is useful to introduce the four-velocities V = ( V 0 , V 1 , 0, 0) , which is the motion of the ship away from the shore, as measured from the shore, and U′ = ( U′ 0 , U′ 1 , U′ 2 , U′ 3 ) which is the motion of the fly away from the ship, as measured from the ship. The four-velocity is defined to be a four-vector with relativistic length equal to 1 , future-directed and tangent to the world line of the object in spacetime. Here, V 0 corresponds to the time component and V 1 to the x component of the ship's velocity as seen from the shore. It is convenient to take the x -axis to be the direction of motion of the ship away from the shore, and the y -axis so that the x – y plane is the plane spanned by the motion of the ship and the fly. This results in several components of the velocities being zero: V 2 = V 3 = U′ 3 = 0
The ordinary velocity is the ratio of the rate at which the space coordinates are increasing to the rate at which the time coordinate is increasing:
v = ( v 1 , v 2 , v 3 ) = ( V 1 / V 0 , 0 , 0 ) , u ′ = ( u 1 ′ , u 2 ′ , u 3 ′ ) = ( U 1 ′ / U 0 ′ , U 2 ′ / U 0 ′ , 0 ) {\displaystyle {\begin{aligned}\mathbf {v} &=(v_{1},v_{2},v_{3})=(V_{1}/V_{0},0,0),\\\mathbf {u} '&=(u'_{1},u'_{2},u'_{3})=(U'_{1}/U'_{0},U'_{2}/U'_{0},0)\end{aligned}}}
Since the relativistic length of V is 1 , V 0 2 − V 1 2 = 1 , {\displaystyle V_{0}^{2}-V_{1}^{2}=1,} so V 0 = 1 / 1 − v 1 2 = γ , V 1 = v 1 / 1 − v 1 2 = v 1 γ . {\displaystyle V_{0}=1/{\sqrt {1-v_{1}^{2}}}\ =\gamma ,\quad V_{1}=v_{1}/{\sqrt {1-v_{1}^{2}}}=v_{1}\gamma .}
The Lorentz transformation matrix that converts velocities measured in the ship frame to the shore frame is the inverse of the transformation described on the Lorentz transformation page, so the minus signs that appear there must be inverted here:
( γ v 1 γ 0 0 v 1 γ γ 0 0 0 0 1 0 0 0 0 1 ) {\displaystyle {\begin{pmatrix}\gamma &v_{1}\gamma &0&0\\v_{1}\gamma &\gamma &0&0\\0&0&1&0\\0&0&0&1\end{pmatrix}}}
This matrix rotates the pure time-axis vector (1, 0, 0, 0) to ( V 0 , V 1 , 0, 0) , and all its columns are relativistically orthogonal to one another, so it defines a Lorentz transformation.
If a fly is moving with four-velocity U′ in the ship frame, and it is boosted by multiplying by the matrix above, the new four-velocity in the shore frame is U = ( U 0 , U 1 , U 2 , U 3 ) , U 0 = V 0 U 0 ′ + V 1 U 1 ′ , U 1 = V 1 U 0 ′ + V 0 U 1 ′ , U 2 = U 2 ′ , U 3 = U 3 ′ . {\displaystyle {\begin{aligned}U_{0}&=V_{0}U'_{0}+V_{1}U'_{1},\\U_{1}&=V_{1}U'_{0}+V_{0}U'_{1},\\U_{2}&=U'_{2},\\U_{3}&=U'_{3}.\end{aligned}}}
Dividing by the time component U 0 and substituting for the components of the four-vectors U′ and V in terms of the components of the three-vectors u′ and v gives the relativistic composition law as
u 1 = v 1 + u 1 ′ 1 + v 1 u 1 ′ , u 2 = u 2 ′ ( 1 + v 1 u 1 ′ ) 1 V 0 = u 2 ′ 1 + v 1 u 1 ′ 1 − v 1 2 , u 3 = 0 {\displaystyle {\begin{aligned}u_{1}&={v_{1}+u'_{1} \over 1+v_{1}u'_{1}},\\u_{2}&={u'_{2} \over (1+v_{1}u'_{1})}{1 \over V_{0}}={u'_{2} \over 1+v_{1}u'_{1}}{\sqrt {1-v_{1}^{2}}},\\u_{3}&=0\end{aligned}}}
The form of the relativistic composition law can be understood as an effect of the failure of simultaneity at a distance. For the parallel component, the time dilation decreases the speed, the length contraction increases it, and the two effects cancel out. The failure of simultaneity means that the fly is changing slices of simultaneity as the projection of u′ onto v . Since this effect is entirely due to the time slicing, the same factor multiplies the perpendicular component, but for the perpendicular component there is no length contraction, so the time dilation multiplies by a factor of 1 / V 0 = √ (1 − v 1 2 ) .
Starting from the expression in coordinates for v parallel to the x -axis , expressions for the perpendicular and parallel components can be cast in vector form as follows, a trick which also works for Lorentz transformations of other 3d physical quantities originally in set up standard configuration. Introduce the velocity vector u in the unprimed frame and u ′ in the primed frame, and split them into components parallel (∥) and perpendicular (⊥) to the relative velocity vector v (see hide box below) thus u = u ∥ + u ⊥ , u ′ = u ∥ ′ + u ⊥ ′ , {\displaystyle \mathbf {u} =\mathbf {u} _{\parallel }+\mathbf {u} _{\perp },\quad \mathbf {u} '=\mathbf {u} '_{\parallel }+\mathbf {u} '_{\perp },} then with the usual Cartesian standard basis vectors e x , e y , e z , set the velocity in the unprimed frame to be u ∥ = u x e x , u ⊥ = u y e y + u z e z , v = v e x , {\displaystyle \mathbf {u} _{\parallel }=u_{x}\mathbf {e} _{x},\quad \mathbf {u} _{\perp }=u_{y}\mathbf {e} _{y}+u_{z}\mathbf {e} _{z},\quad \mathbf {v} =v\mathbf {e} _{x},} which gives, using the results for the standard configuration, u ∥ = u ∥ ′ + v 1 + v ⋅ u ∥ ′ c 2 , u ⊥ = 1 − v 2 c 2 u ⊥ ′ 1 + v ⋅ u ∥ ′ c 2 . {\displaystyle \mathbf {u} _{\parallel }={\frac {\mathbf {u} _{\parallel }'+\mathbf {v} }{1+{\frac {\mathbf {v} \cdot \mathbf {u} _{\parallel }'}{c^{2}}}}},\quad \mathbf {u} _{\perp }={\frac {{\sqrt {1-{\frac {v^{2}}{c^{2}}}}}\mathbf {u} _{\perp }'}{1+{\frac {\mathbf {v} \cdot \mathbf {u} _{\parallel }'}{c^{2}}}}}.} where · is the dot product . Since these are vector equations, they still have the same form for v in any direction. The only difference from the coordinate expressions is that the above expressions refers to vectors , not components.
One obtains u = u ∥ + u ⊥ = 1 1 + v ⋅ u ′ c 2 [ α v u ′ + v + ( 1 − α v ) ( v ⋅ u ′ ) v 2 v ] ≡ v ⊕ u ′ , {\displaystyle \mathbf {u} =\mathbf {u} _{\parallel }+\mathbf {u} _{\perp }={\frac {1}{1+{\frac {\mathbf {v} \cdot \mathbf {u} '}{c^{2}}}}}\left[\alpha _{v}\mathbf {u} '+\mathbf {v} +(1-\alpha _{v}){\frac {(\mathbf {v} \cdot \mathbf {u} ')}{v^{2}}}\mathbf {v} \right]\equiv \mathbf {v} \oplus \mathbf {u} ',} where α v = 1/ γ v is the reciprocal of the Lorentz factor . The ordering of operands in the definition is chosen to coincide with that of the standard configuration from which the formula is derived.
u ∥ ′ + v 1 + v ⋅ u ′ c 2 + α v u ⊥ ′ 1 + v ⋅ u ′ c 2 = v + v ⋅ u ′ v 2 v 1 + v ⋅ u ′ c 2 + α v u ′ − α v v ⋅ u ′ v 2 v 1 + v ⋅ u ′ c 2 = 1 + v ⋅ u ′ v 2 ( 1 − α v ) 1 + v ⋅ u ′ c 2 v + α v 1 1 + v ⋅ u ′ c 2 u ′ = 1 1 + v ⋅ u ′ c 2 v + α v 1 1 + v ⋅ u ′ c 2 u ′ + 1 1 + v ⋅ u ′ c 2 v ⋅ u ′ v 2 ( 1 − α v ) v = 1 1 + v ⋅ u ′ c 2 v + α v 1 1 + v ⋅ u ′ c 2 u ′ + 1 c 2 1 1 + v ⋅ u ′ c 2 v ⋅ u ′ v 2 / c 2 ( 1 − α v ) v = 1 1 + v ⋅ u ′ c 2 v + α v 1 1 + v ⋅ u ′ c 2 u ′ + 1 c 2 1 1 + v ⋅ u ′ c 2 v ⋅ u ′ ( 1 − α v ) ( 1 + α v ) ( 1 − α v ) v = 1 1 + v ⋅ u ′ c 2 [ α v u ′ + v + ( 1 − α v ) ( v ⋅ u ′ ) v 2 v ] . {\displaystyle {\begin{aligned}{\frac {\mathbf {u} '_{\parallel }+\mathbf {v} }{1+{\frac {\mathbf {v} \cdot \mathbf {u} '}{c^{2}}}}}+{\frac {\alpha _{v}\mathbf {u} '_{\perp }}{1+{\frac {\mathbf {v} \cdot \mathbf {u} '}{c^{2}}}}}&={\frac {\mathbf {v} +{\frac {\mathbf {v} \cdot \mathbf {u} '}{v^{2}}}\mathbf {v} }{1+{\frac {\mathbf {v} \cdot \mathbf {u} '}{c^{2}}}}}+{\frac {\alpha _{v}\mathbf {u} '-\alpha _{v}{\frac {\mathbf {v} \cdot \mathbf {u} '}{v^{2}}}\mathbf {v} }{1+{\frac {\mathbf {v} \cdot \mathbf {u} '}{c^{2}}}}}\\&={\frac {1+{\frac {\mathbf {v} \cdot \mathbf {u} '}{v^{2}}}(1-\alpha _{v})}{1+{\frac {\mathbf {v} \cdot \mathbf {u} '}{c^{2}}}}}\mathbf {v} +\alpha _{v}{\frac {1}{1+{\frac {\mathbf {v} \cdot \mathbf {u} '}{c^{2}}}}}\mathbf {u} '\\&={\frac {1}{1+{\frac {\mathbf {v} \cdot \mathbf {u} '}{c^{2}}}}}\mathbf {v} +\alpha _{v}{\frac {1}{1+{\frac {\mathbf {v} \cdot \mathbf {u} '}{c^{2}}}}}\mathbf {u} '+{\frac {1}{1+{\frac {\mathbf {v} \cdot \mathbf {u} '}{c^{2}}}}}{\frac {\mathbf {v} \cdot \mathbf {u} '}{v^{2}}}(1-\alpha _{v})\mathbf {v} \\&={\frac {1}{1+{\frac {\mathbf {v} \cdot \mathbf {u} '}{c^{2}}}}}\mathbf {v} +\alpha _{v}{\frac {1}{1+{\frac {\mathbf {v} \cdot \mathbf {u} '}{c^{2}}}}}\mathbf {u} '+{\frac {1}{c^{2}}}{\frac {1}{1+{\frac {\mathbf {v} \cdot \mathbf {u} '}{c^{2}}}}}{\frac {\mathbf {v} \cdot \mathbf {u} '}{v^{2}/c^{2}}}(1-\alpha _{v})\mathbf {v} \\&={\frac {1}{1+{\frac {\mathbf {v} \cdot \mathbf {u} '}{c^{2}}}}}\mathbf {v} +\alpha _{v}{\frac {1}{1+{\frac {\mathbf {v} \cdot \mathbf {u} '}{c^{2}}}}}\mathbf {u} '+{\frac {1}{c^{2}}}{\frac {1}{1+{\frac {\mathbf {v} \cdot \mathbf {u} '}{c^{2}}}}}{\frac {\mathbf {v} \cdot \mathbf {u} '}{(1-\alpha _{v})(1+\alpha _{v})}}(1-\alpha _{v})\mathbf {v} \\&={\frac {1}{1+{\frac {\mathbf {v} \cdot \mathbf {u} '}{c^{2}}}}}\left[\alpha _{v}\mathbf {u} '+\mathbf {v} +(1-\alpha _{v}){\frac {(\mathbf {v} \cdot \mathbf {u} ')}{v^{2}}}\mathbf {v} \right].\end{aligned}}}
Either the parallel or the perpendicular component for each vector needs to be found, since the other component will be eliminated by substitution of the full vectors.
The parallel component of u ′ can be found by projecting the full vector into the direction of the relative motion u ∥ ′ = v ⋅ u ′ v 2 v , {\displaystyle \mathbf {u} '_{\parallel }={\frac {\mathbf {v} \cdot \mathbf {u} '}{v^{2}}}\mathbf {v} ,} and the perpendicular component of u ′ can be found by the geometric properties of the cross product (see figure above right), u ⊥ ′ = − v × ( v × u ′ ) v 2 . {\displaystyle \mathbf {u} '_{\perp }=-{\frac {\mathbf {v} \times (\mathbf {v} \times \mathbf {u} ')}{v^{2}}}.}
In each case, v / v is a unit vector in the direction of relative motion.
The expressions for u ∥ and u ⊥ can be found in the same way. Substituting the parallel component into u = u ∥ ′ + v 1 + v ⋅ u ∥ ′ c 2 + 1 − v 2 c 2 ( u ′ − u ∥ ′ ) 1 + v ⋅ u ∥ ′ c 2 , {\displaystyle \mathbf {u} ={\frac {\mathbf {u} _{\parallel }'+\mathbf {v} }{1+{\frac {\mathbf {v} \cdot \mathbf {u} _{\parallel }'}{c^{2}}}}}+{\frac {{\sqrt {1-{\frac {v^{2}}{c^{2}}}}}(\mathbf {u} '-\mathbf {u} _{\parallel }')}{1+{\frac {\mathbf {v} \cdot \mathbf {u} _{\parallel }'}{c^{2}}}}},}
results in the above equation. [ 10 ]
Using an identity in α v {\displaystyle \alpha _{v}} and γ v {\displaystyle \gamma _{v}} , [ 11 ] [ nb 1 ]
v ⊕ u ′ ≡ u = 1 1 + u ′ ⋅ v c 2 [ v + u ′ γ v + 1 c 2 γ v 1 + γ v ( u ′ ⋅ v ) v ] = 1 1 + u ′ ⋅ v c 2 [ v + u ′ + 1 c 2 γ v 1 + γ v v × ( v × u ′ ) ] , {\displaystyle {\begin{aligned}\mathbf {v} \oplus \mathbf {u} '\equiv \mathbf {u} &={\frac {1}{1+{\frac {\mathbf {u} '\cdot \mathbf {v} }{c^{2}}}}}\left[\mathbf {v} +{\frac {\mathbf {u} '}{\gamma _{v}}}+{\frac {1}{c^{2}}}{\frac {\gamma _{v}}{1+\gamma _{v}}}(\mathbf {u} '\cdot \mathbf {v} )\mathbf {v} \right]\\&={\frac {1}{1+{\frac {\mathbf {u} '\cdot \mathbf {v} }{c^{2}}}}}\left[\mathbf {v} +\mathbf {u} '+{\frac {1}{c^{2}}}{\frac {\gamma _{v}}{1+\gamma _{v}}}\mathbf {v} \times (\mathbf {v} \times \mathbf {u} ')\right],\end{aligned}}} and in the forwards (v positive, S → S') direction v ⊕ u ≡ u ′ = 1 1 − u ⋅ v c 2 [ u γ v − v + 1 c 2 γ v 1 + γ v ( u ⋅ v ) v ] = 1 1 − u ⋅ v c 2 [ u − v + 1 c 2 γ v 1 + γ v v × ( v × u ) ] {\displaystyle {\begin{aligned}\mathbf {v} \oplus \mathbf {u} \equiv \mathbf {u} '&={\frac {1}{1-{\frac {\mathbf {u} \cdot \mathbf {v} }{c^{2}}}}}\left[{\frac {\mathbf {u} }{\gamma _{v}}}-\mathbf {v} +{\frac {1}{c^{2}}}{\frac {\gamma _{v}}{1+\gamma _{v}}}(\mathbf {u} \cdot \mathbf {v} )\mathbf {v} \right]\\&={\frac {1}{1-{\frac {\mathbf {u} \cdot \mathbf {v} }{c^{2}}}}}\left[\mathbf {u} -\mathbf {v} +{\frac {1}{c^{2}}}{\frac {\gamma _{v}}{1+\gamma _{v}}}\mathbf {v} \times (\mathbf {v} \times \mathbf {u} )\right]\end{aligned}}}
where the last expression is by the standard vector analysis formula v × ( v × u ) = ( v ⋅ u ) v − ( v ⋅ v ) u . The first expression extends to any number of spatial dimensions, but the cross product is defined in three dimensions only. The objects A , B , C with B having velocity v relative to A and C having velocity u relative to A can be anything. In particular, they can be three frames, or they could be the laboratory, a decaying particle and one of the decay products of the decaying particle.
The relativistic addition of 3-velocities is non-linear , so in general ( λ v ) ⊕ ( λ u ) ≠ λ ( v ⊕ u ) , {\displaystyle (\lambda \mathbf {v} )\oplus (\lambda \mathbf {u} )\neq \lambda (\mathbf {v} \oplus \mathbf {u} ),} for real number λ , although it is true that ( − v ) ⊕ ( − u ) = − ( v ⊕ u ) , {\displaystyle (-\mathbf {v} )\oplus (-\mathbf {u} )=-(\mathbf {v} \oplus \mathbf {u} ),}
Also, due to the last terms, is in general neither commutative v ⊕ u ≠ u ⊕ v , {\displaystyle \mathbf {v} \oplus \mathbf {u} \neq \mathbf {u} \oplus \mathbf {v} ,} nor associative v ⊕ ( u ⊕ w ) ≠ ( v ⊕ u ) ⊕ w . {\displaystyle \mathbf {v} \oplus (\mathbf {u} \oplus \mathbf {w} )\neq (\mathbf {v} \oplus \mathbf {u} )\oplus \mathbf {w} .}
It deserves special mention that if u and v′ refer to velocities of pairwise parallel frames (primed parallel to unprimed and doubly primed parallel to primed), then, according to Einstein's velocity reciprocity principle, the unprimed frame moves with velocity − u relative to the primed frame, and the primed frame moves with velocity − v′ relative to the doubly primed frame hence (− v′ ⊕ − u ) is the velocity of the unprimed frame relative to the doubly primed frame, and one might expect to have u ⊕ v′ = −(− v′ ⊕ − u ) by naive application of the reciprocity principle. This does not hold, though the magnitudes are equal. The unprimed and doubly primed frames are not parallel, but related through a rotation. This is related to the phenomenon of Thomas precession , and is not dealt with further here.
The norms are given by [ 12 ] | u | 2 ≡ | v ⊕ u ′ | 2 = 1 ( 1 + v ⋅ u ′ c 2 ) 2 [ ( v + u ′ ) 2 − 1 c 2 ( v × u ′ ) 2 ] = | u ′ ⊕ v | 2 . {\displaystyle |\mathbf {u} |^{2}\equiv |\mathbf {v} \oplus \mathbf {u} '|^{2}={\frac {1}{\left(1+{\frac {\mathbf {v} \cdot \mathbf {u} '}{c^{2}}}\right)^{2}}}\left[\left(\mathbf {v} +\mathbf {u} '\right)^{2}-{\frac {1}{c^{2}}}\left(\mathbf {v} \times \mathbf {u} '\right)^{2}\right]=|\mathbf {u} '\oplus \mathbf {v} |^{2}.} and | u ′ | 2 ≡ | v ⊕ u | 2 = 1 ( 1 − v ⋅ u c 2 ) 2 [ ( u − v ) 2 − 1 c 2 ( v × u ) 2 ] = | u ⊕ v | 2 . {\displaystyle |\mathbf {u} '|^{2}\equiv |\mathbf {v} \oplus \mathbf {u} |^{2}={\frac {1}{\left(1-{\frac {\mathbf {v} \cdot \mathbf {u} }{c^{2}}}\right)^{2}}}\left[\left(\mathbf {u} -\mathbf {v} \right)^{2}-{\frac {1}{c^{2}}}\left(\mathbf {v} \times \mathbf {u} \right)^{2}\right]=|\mathbf {u} \oplus \mathbf {v} |^{2}.}
( 1 + v ⋅ u ′ c 2 ) 2 | v ⊕ u ′ | 2 = [ v + u ′ + 1 c 2 γ v 1 + γ v v × ( v × u ′ ) ] 2 = ( v + u ′ ) 2 + 2 1 c 2 γ v γ v + 1 [ ( v ⋅ u ′ ) 2 − ( v ⋅ v ) ( u ′ ⋅ u ′ ) ] + 1 c 4 ( γ v γ v + 1 ) 2 [ ( v ⋅ v ) 2 ( u ′ ⋅ u ′ ) − ( v ⋅ u ′ ) 2 ( v ⋅ v ) ] = ( v + u ′ ) 2 + 2 1 c 2 γ v γ v + 1 [ ( v ⋅ u ′ ) 2 − ( v ⋅ v ) ( u ′ ⋅ u ′ ) ] + v 2 c 4 ( γ v γ v + 1 ) 2 [ ( v ⋅ v ) ( u ′ ⋅ u ′ ) − ( v ⋅ u ′ ) 2 ] = ( v + u ′ ) 2 + 2 1 c 2 γ v γ v + 1 [ ( v ⋅ u ′ ) 2 − ( v ⋅ v ) ( u ′ ⋅ u ′ ) ] + ( 1 − α v ) ( 1 + α v ) c 2 ( γ v γ v + 1 ) 2 [ ( v ⋅ v ) ( u ′ ⋅ u ′ ) − ( v ⋅ u ′ ) 2 ] = ( v + u ′ ) 2 + 2 1 c 2 γ v γ v + 1 [ ( v ⋅ u ′ ) 2 − ( v ⋅ v ) ( u ′ ⋅ u ′ ) ] + ( γ v − 1 ) c 2 ( γ v + 1 ) [ ( v ⋅ v ) ( u ′ ⋅ u ′ ) − ( v ⋅ u ′ ) 2 ] = ( v + u ′ ) 2 + 2 1 c 2 γ v γ v + 1 [ ( v ⋅ u ′ ) 2 − ( v ⋅ v ) ( u ′ ⋅ u ′ ) ] + ( 1 − γ v ) c 2 ( γ v + 1 ) [ ( v ⋅ u ′ ) 2 − ( v ⋅ v ) ( u ′ ⋅ u ′ ) ] = ( v + u ′ ) 2 + 1 c 2 γ v + 1 γ v + 1 [ ( v ⋅ u ′ ) 2 − ( v ⋅ v ) ( u ′ ⋅ u ′ ) ] = ( v + u ′ ) 2 − 1 c 2 | v × u ′ | 2 {\displaystyle {\begin{aligned}&\left(1+{\frac {\mathbf {v} \cdot \mathbf {u} '}{c^{2}}}\right)^{2}|\mathbf {v} \oplus \mathbf {u} '|^{2}\\&=\left[\mathbf {v} +\mathbf {u} '+{\frac {1}{c^{2}}}{\frac {\gamma _{v}}{1+\gamma _{v}}}\mathbf {v} \times (\mathbf {v} \times \mathbf {u} ')\right]^{2}\\&=(\mathbf {v} +\mathbf {u} ')^{2}+2{\frac {1}{c^{2}}}{\frac {\gamma _{v}}{\gamma _{v}+1}}\left[(\mathbf {v} \cdot \mathbf {u} ')^{2}-(\mathbf {v} \cdot \mathbf {v} )(\mathbf {u} '\cdot \mathbf {u} ')\right]+{\frac {1}{c^{4}}}\left({\frac {\gamma _{v}}{\gamma _{v}+1}}\right)^{2}\left[(\mathbf {v} \cdot \mathbf {v} )^{2}(\mathbf {u} '\cdot \mathbf {u} ')-(\mathbf {v} \cdot \mathbf {u} ')^{2}(\mathbf {v} \cdot \mathbf {v} )\right]\\&=(\mathbf {v} +\mathbf {u} ')^{2}+2{\frac {1}{c^{2}}}{\frac {\gamma _{v}}{\gamma _{v}+1}}\left[(\mathbf {v} \cdot \mathbf {u} ')^{2}-(\mathbf {v} \cdot \mathbf {v} )(\mathbf {u} '\cdot \mathbf {u} ')\right]+{\frac {v^{2}}{c^{4}}}\left({\frac {\gamma _{v}}{\gamma _{v}+1}}\right)^{2}\left[(\mathbf {v} \cdot \mathbf {v} )(\mathbf {u} '\cdot \mathbf {u} ')-(\mathbf {v} \cdot \mathbf {u} ')^{2}\right]\\&=(\mathbf {v} +\mathbf {u} ')^{2}+2{\frac {1}{c^{2}}}{\frac {\gamma _{v}}{\gamma _{v}+1}}\left[(\mathbf {v} \cdot \mathbf {u} ')^{2}-(\mathbf {v} \cdot \mathbf {v} )(\mathbf {u} '\cdot \mathbf {u} ')\right]+{\frac {(1-\alpha _{v})(1+\alpha _{v})}{c^{2}}}\left({\frac {\gamma _{v}}{\gamma _{v}+1}}\right)^{2}\left[(\mathbf {v} \cdot \mathbf {v} )(\mathbf {u} '\cdot \mathbf {u} ')-(\mathbf {v} \cdot \mathbf {u} ')^{2}\right]\\&=(\mathbf {v} +\mathbf {u} ')^{2}+2{\frac {1}{c^{2}}}{\frac {\gamma _{v}}{\gamma _{v}+1}}\left[(\mathbf {v} \cdot \mathbf {u} ')^{2}-(\mathbf {v} \cdot \mathbf {v} )(\mathbf {u} '\cdot \mathbf {u} ')\right]+{\frac {(\gamma _{v}-1)}{c^{2}(\gamma _{v}+1)}}\left[(\mathbf {v} \cdot \mathbf {v} )(\mathbf {u} '\cdot \mathbf {u} ')-(\mathbf {v} \cdot \mathbf {u} ')^{2}\right]\\&=(\mathbf {v} +\mathbf {u} ')^{2}+2{\frac {1}{c^{2}}}{\frac {\gamma _{v}}{\gamma _{v}+1}}\left[(\mathbf {v} \cdot \mathbf {u} ')^{2}-(\mathbf {v} \cdot \mathbf {v} )(\mathbf {u} '\cdot \mathbf {u} ')\right]+{\frac {(1-\gamma _{v})}{c^{2}(\gamma _{v}+1)}}\left[(\mathbf {v} \cdot \mathbf {u} ')^{2}-(\mathbf {v} \cdot \mathbf {v} )(\mathbf {u} '\cdot \mathbf {u} ')\right]\\&=(\mathbf {v} +\mathbf {u} ')^{2}+{\frac {1}{c^{2}}}{\frac {\gamma _{v}+1}{\gamma _{v}+1}}\left[(\mathbf {v} \cdot \mathbf {u} ')^{2}-(\mathbf {v} \cdot \mathbf {v} )(\mathbf {u} '\cdot \mathbf {u} ')\right]\\&=(\mathbf {v} +\mathbf {u} ')^{2}-{\frac {1}{c^{2}}}|\mathbf {v} \times \mathbf {u} '|^{2}\end{aligned}}} Reverse formula found by using standard procedure of swapping v for − v and u for u ′ .
It is clear that the non-commutativity manifests itself as an additional rotation of the coordinate frame when two boosts are involved, since the norm squared is the same for both orders of boosts.
The gamma factors for the combined velocities are computed as γ u = γ v ⊕ u ′ = [ 1 − 1 c 2 1 ( 1 + v ⋅ u ′ c 2 ) 2 ( ( v + u ′ ) 2 − 1 c 2 ( v 2 u ′ 2 − ( v ⋅ u ′ ) 2 ) ) ] − 1 2 = γ v γ u ′ ( 1 + v ⋅ u ′ c 2 ) , γ u ′ = γ v γ u ( 1 − v ⋅ u c 2 ) {\displaystyle \gamma _{u}=\gamma _{\mathbf {v} \oplus \mathbf {u} '}=\left[1-{\frac {1}{c^{2}}}{\frac {1}{(1+{\frac {\mathbf {v} \cdot \mathbf {u} '}{c^{2}}})^{2}}}\left((\mathbf {v} +\mathbf {u} ')^{2}-{\frac {1}{c^{2}}}(v^{2}u'^{2}-(\mathbf {v} \cdot \mathbf {u} ')^{2})\right)\right]^{-{\frac {1}{2}}}=\gamma _{v}\gamma _{u}'\left(1+{\frac {\mathbf {v} \cdot \mathbf {u} '}{c^{2}}}\right),\quad \quad \gamma _{u}'=\gamma _{v}\gamma _{u}\left(1-{\frac {\mathbf {v} \cdot \mathbf {u} }{c^{2}}}\right)}
γ v ⊕ u ′ = [ c 3 ( 1 + v ⋅ u ′ c 2 ) 2 c 2 ( 1 + v ⋅ u ′ c 2 ) 2 − 1 c 2 ( v + u ′ ) 2 − 1 c 2 ( v 2 u ′ 2 − ( v ⋅ u ′ ) 2 ) ( 1 + v ⋅ u ′ c 2 ) 2 ] − 1 2 = [ c 2 ( 1 + v ⋅ u ′ c 2 ) 2 − ( v + u ′ ) 2 + 1 c 2 ( v 2 u ′ 2 − ( v ⋅ u ′ ) 2 ) c 2 ( 1 + v ⋅ u ′ c 2 ) 2 ] − 1 2 = [ c 2 ( 1 + 2 v ⋅ u ′ c 2 + ( v ⋅ u ′ ) 2 c 4 ) − v 2 − u ′ 2 − 2 ( v ⋅ u ′ ) + 1 c 2 ( v 2 u ′ 2 − ( v ⋅ u ′ ) 2 ) c 2 ( 1 + v ⋅ u ′ c 2 ) 2 ] − 1 2 = [ 1 + 2 v ⋅ u ′ c 2 + ( v ⋅ u ′ ) 2 c 4 − v 2 c 2 − u ′ 2 c 2 − 2 c 2 ( v ⋅ u ′ ) + 1 c 4 ( v 2 u ′ 2 − ( v ⋅ u ′ ) 2 ) ( 1 + v ⋅ u ′ c 2 ) 2 ] − 1 2 = [ 1 + ( v ⋅ u ′ ) 2 c 4 − v 2 c 2 − u ′ 2 c 2 + 1 c 4 ( v 2 u ′ 2 − ( v ⋅ u ′ ) 2 ) ( 1 + v ⋅ u ′ c 2 ) 2 ] − 1 2 = [ ( 1 − v 2 c 2 ) ( 1 − u ′ 2 c 2 ) ( 1 + v ⋅ u ′ c 2 ) 2 ] − 1 2 = [ 1 γ v 2 γ u ′ 2 ( 1 + v ⋅ u ′ c 2 ) 2 ] − 1 2 = γ v γ u ′ ( 1 + v ⋅ u ′ c 2 ) {\displaystyle {\begin{aligned}\gamma _{\mathbf {v} \oplus \mathbf {u} '}&=\left[{\frac {c^{3}(1+{\frac {\mathbf {v} \cdot \mathbf {u} '}{c^{2}}})^{2}}{c^{2}(1+{\frac {\mathbf {v} \cdot \mathbf {u} '}{c^{2}}})^{2}}}-{\frac {1}{c^{2}}}{\frac {(\mathbf {v} +\mathbf {u} ')^{2}-{\frac {1}{c^{2}}}(v^{2}u'^{2}-(\mathbf {v} \cdot \mathbf {u} ')^{2})}{(1+{\frac {\mathbf {v} \cdot \mathbf {u} '}{c^{2}}})^{2}}}\right]^{-{\frac {1}{2}}}\\&=\left[{\frac {c^{2}(1+{\frac {\mathbf {v} \cdot \mathbf {u} '}{c^{2}}})^{2}-(\mathbf {v} +\mathbf {u} ')^{2}+{\frac {1}{c^{2}}}(v^{2}u'^{2}-(\mathbf {v} \cdot \mathbf {u} ')^{2})}{c^{2}(1+{\frac {\mathbf {v} \cdot \mathbf {u} '}{c^{2}}})^{2}}}\right]^{-{\frac {1}{2}}}\\&=\left[{\frac {c^{2}(1+2{\frac {\mathbf {v} \cdot \mathbf {u} '}{c^{2}}}+{\frac {(\mathbf {v} \cdot \mathbf {u} ')^{2}}{c^{4}}})-v^{2}-u'^{2}-2(\mathbf {v} \cdot \mathbf {u} ')+{\frac {1}{c^{2}}}(v^{2}u'^{2}-(\mathbf {v} \cdot \mathbf {u} ')^{2})}{c^{2}(1+{\frac {\mathbf {v} \cdot \mathbf {u} '}{c^{2}}})^{2}}}\right]^{-{\frac {1}{2}}}\\&=\left[{\frac {1+2{\frac {\mathbf {v} \cdot \mathbf {u} '}{c^{2}}}+{\frac {(\mathbf {v} \cdot \mathbf {u} ')^{2}}{c^{4}}}-{\frac {v^{2}}{c^{2}}}-{\frac {u'^{2}}{c^{2}}}-{\frac {2}{c^{2}}}(\mathbf {v} \cdot \mathbf {u} ')+{\frac {1}{c^{4}}}(v^{2}u'^{2}-(\mathbf {v} \cdot \mathbf {u} ')^{2})}{(1+{\frac {\mathbf {v} \cdot \mathbf {u} '}{c^{2}}})^{2}}}\right]^{-{\frac {1}{2}}}\\&=\left[{\frac {1+{\frac {(\mathbf {v} \cdot \mathbf {u} ')^{2}}{c^{4}}}-{\frac {v^{2}}{c^{2}}}-{\frac {u'^{2}}{c^{2}}}+{\frac {1}{c^{4}}}(v^{2}u'^{2}-(\mathbf {v} \cdot \mathbf {u} ')^{2})}{(1+{\frac {\mathbf {v} \cdot \mathbf {u} '}{c^{2}}})^{2}}}\right]^{-{\frac {1}{2}}}\\&=\left[{\frac {\left(1-{\frac {v^{2}}{c^{2}}}\right)\left(1-{\frac {u'^{2}}{c^{2}}}\right)}{\left(1+{\frac {\mathbf {v} \cdot \mathbf {u} '}{c^{2}}}\right)^{2}}}\right]^{-{\frac {1}{2}}}=\left[{\frac {1}{\gamma _{v}^{2}\gamma _{u}'^{2}\left(1+{\frac {\mathbf {v} \cdot \mathbf {u} '}{c^{2}}}\right)^{2}}}\right]^{-{\frac {1}{2}}}\\&=\gamma _{v}\gamma _{u}'\left(1+{\frac {\mathbf {v} \cdot \mathbf {u} '}{c^{2}}}\right)\end{aligned}}}
Reverse formula found by using standard procedure of swapping v for − v and u for u ′ .
Notations and conventions for the velocity addition vary from author to author. Different symbols may be used for the operation, or for the velocities involved, and the operands may be switched for the same expression, or the symbols may be switched for the same velocity. A completely separate symbol may also be used for the transformed velocity, rather than the prime used here. Since the velocity addition is non-commutative, one cannot switch the operands or symbols without changing the result.
Examples of alternative notation include:
Some classical applications of velocity-addition formulas, to the Doppler shift, to the aberration of light, and to the dragging of light in moving water, yielding relativistically valid expressions for these phenomena are detailed below. It is also possible to use the velocity addition formula, assuming conservation of momentum (by appeal to ordinary rotational invariance), the correct form of the 3 -vector part of the momentum four-vector , without resort to electromagnetism, or a priori not known to be valid, relativistic versions of the Lagrangian formalism . This involves experimentalist bouncing off relativistic billiard balls from each other. This is not detailed here, but see for reference Lewis & Tolman (1909) Wikisource version (primary source) and Sard (1970 , Section 3.2).
When light propagates in a medium, its speed is reduced, in the rest frame of the medium, to c m = c / n m , where n m is the index of refraction of the medium m . The speed of light in a medium uniformly moving with speed V in the positive x -direction as measured in the lab frame is given directly by the velocity addition formulas. For the forward direction (standard configuration, drop index m on n ) one gets, [ 13 ] c m = V + c m ′ 1 + V c m ′ c 2 = V + c n 1 + V c n c 2 = c n 1 + n V c 1 + V n c = c n ( 1 + n V c ) 1 1 + V n c = ( c n + V ) ( 1 − V n c + ( V n c ) 2 − ⋯ ) . {\displaystyle {\begin{aligned}c_{m}&={\frac {V+c_{m}'}{1+{\frac {Vc_{m}'}{c^{2}}}}}={\frac {V+{\frac {c}{n}}}{1+{\frac {Vc}{nc^{2}}}}}={\frac {c}{n}}{\frac {1+{\frac {nV}{c}}}{1+{\frac {V}{nc}}}}\\&={\frac {c}{n}}\left(1+{\frac {nV}{c}}\right){\frac {1}{1+{\frac {V}{nc}}}}=\left({\frac {c}{n}}+V\right)\left(1-{\frac {V}{nc}}+\left({\frac {V}{nc}}\right)^{2}-\cdots \right).\end{aligned}}}
Collecting the largest contributions explicitly, c m = c n + V ( 1 − 1 n 2 − V n c + ⋯ ) . {\displaystyle c_{m}={\frac {c}{n}}+V\left(1-{\frac {1}{n^{2}}}-{\frac {V}{nc}}+\cdots \right).} Fizeau found the first three terms. [ 14 ] [ 15 ] The classical result is the first two terms.
Another basic application is to consider the deviation of light, i.e. change of its direction, when transforming to a new reference frame with parallel axes, called aberration of light . In this case, v ′ = v = c , and insertion in the formula for tan θ yields tan θ = 1 − V 2 c 2 c sin θ ′ c cos θ ′ + V = 1 − V 2 c 2 sin θ ′ cos θ ′ + V c . {\displaystyle \tan \theta ={\frac {{\sqrt {1-{\frac {V^{2}}{c^{2}}}}}c\sin \theta '}{c\cos \theta '+V}}={\frac {{\sqrt {1-{\frac {V^{2}}{c^{2}}}}}\sin \theta '}{\cos \theta '+{\frac {V}{c}}}}.}
For this case one may also compute sin θ and cos θ from the standard formulae, [ 16 ] sin θ = 1 − V 2 c 2 sin θ ′ 1 + V c cos θ ′ , {\displaystyle \sin \theta ={\frac {{\sqrt {1-{\frac {V^{2}}{c^{2}}}}}\sin \theta '}{1+{\frac {V}{c}}\cos \theta '}},}
v y v = 1 − V 2 c 2 v y ′ 1 + V c 2 v x ′ v ′ 2 + V 2 + 2 V v ′ cos θ ′ − ( V v ′ sin θ ′ c ) 2 1 + V c 2 v ′ cos θ ′ = c 1 − V 2 c 2 sin θ ′ c 2 + V 2 + 2 V c cos θ ′ − V 2 sin 2 θ ′ = c 1 − V 2 c 2 sin θ ′ c 2 + V 2 + 2 V c cos θ ′ − V 2 ( 1 − cos 2 θ ′ ) = c 1 − V 2 c 2 sin θ ′ c 2 + 2 V c cos θ ′ + V 2 cos 2 θ ′ = 1 − V 2 c 2 sin θ ′ 1 + V c cos θ ′ , {\displaystyle {\begin{aligned}{\frac {v_{y}}{v}}&={\frac {\frac {{\sqrt {1-{\frac {V^{2}}{c^{2}}}}}v_{y}'}{1+{\frac {V}{c^{2}}}v_{x}'}}{\frac {\sqrt {v'^{2}+V^{2}+2Vv'\cos \theta '-({\frac {Vv'\sin \theta '}{c}})^{2}}}{1+{\frac {V}{c^{2}}}v'\cos \theta '}}}\\&={\frac {c{\sqrt {1-{\frac {V^{2}}{c^{2}}}}}\sin \theta '}{\sqrt {c^{2}+V^{2}+2Vc\cos \theta '-V^{2}\sin ^{2}\theta '}}}\\&={\frac {c{\sqrt {1-{\frac {V^{2}}{c^{2}}}}}\sin \theta '}{\sqrt {c^{2}+V^{2}+2Vc\cos \theta '-V^{2}(1-\cos ^{2}\theta ')}}}={\frac {c{\sqrt {1-{\frac {V^{2}}{c^{2}}}}}\sin \theta '}{\sqrt {c^{2}+2Vc\cos \theta '+V^{2}\cos ^{2}\theta '}}}\\&={\frac {{\sqrt {1-{\frac {V^{2}}{c^{2}}}}}\sin \theta '}{1+{\frac {V}{c}}\cos \theta '}},\end{aligned}}}
cos θ = V c + cos θ ′ 1 + V c cos θ ′ , {\displaystyle \cos \theta ={\frac {{\frac {V}{c}}+\cos \theta '}{1+{\frac {V}{c}}\cos \theta '}},}
the trigonometric manipulations essentially being identical in the cos case to the manipulations in the sin case. Consider the difference,
sin θ − sin θ ′ = sin θ ′ ( 1 − V 2 c 2 1 + V c cos θ ′ − 1 ) ≈ sin θ ′ ( 1 − V c cos θ ′ − 1 ) = − V c sin θ ′ cos θ ′ , {\displaystyle {\begin{aligned}\sin \theta -\sin \theta '&=\sin \theta '\left({\frac {\sqrt {1-{\frac {V^{2}}{c^{2}}}}}{1+{\frac {V}{c}}\cos \theta '}}-1\right)\\&\approx \sin \theta '\left(1-{\frac {V}{c}}\cos \theta '-1\right)=-{\frac {V}{c}}\sin \theta '\cos \theta ',\end{aligned}}} correct to order v / c . Employ in order to make small angle approximations a trigonometric formula, sin θ ′ − sin θ = 2 sin 1 2 ( θ ′ − θ ) cos 1 2 ( θ + θ ′ ) ≈ ( θ ′ − θ ) cos θ ′ , {\displaystyle \sin \theta '-\sin \theta =2\sin {\frac {1}{2}}(\theta '-\theta )\cos {\frac {1}{2}}(\theta +\theta ')\approx (\theta '-\theta )\cos \theta ',} where cos 1 / 2 ( θ + θ ′) ≈ cos θ ′, sin 1 / 2 ( θ − θ ′) ≈ 1 / 2 ( θ − θ ′) were used.
Thus the quantity Δ θ ≡ θ ′ − θ = V c sin θ ′ , {\displaystyle \Delta \theta \equiv \theta '-\theta ={\frac {V}{c}}\sin \theta ',} the classical aberration angle , is obtained in the limit V / c → 0 .
Here velocity components will be used as opposed to speed for greater generality, and in order to avoid perhaps seemingly ad hoc introductions of minus signs. Minus signs occurring here will instead serve to illuminate features when speeds less than that of light are considered.
For light waves in vacuum, time dilation together with a simple geometrical observation alone suffices to calculate the Doppler shift in standard configuration (collinear relative velocity of emitter and observer as well of observed light wave).
All velocities in what follows are parallel to the common positive x -direction , so subscripts on velocity components are dropped. In the observers frame, introduce the geometrical observation λ = − s T + V T = ( − s + V ) T {\displaystyle \lambda =-sT+VT=(-s+V)T} as the spatial distance, or wavelength , between two pulses (wave crests), where T is the time elapsed between the emission of two pulses. The time elapsed between the passage of two pulses at the same point in space is the time period τ , and its inverse ν = 1 / τ is the observed (temporal) frequency . The corresponding quantities in the emitters frame are endowed with primes. [ 18 ]
For light waves s = s ′ = − c , {\displaystyle s=s'=-c,} and the observed frequency is [ 2 ] [ 19 ] [ 20 ] ν = − s λ = − s ( V − s ) T = c ( V + c ) γ V T ′ = ν ′ c 1 − V 2 c 2 c + V = ν ′ 1 − β 1 + β . {\displaystyle \nu ={-s \over \lambda }={-s \over (V-s)T}={c \over (V+c)\gamma _{_{V}}T'}=\nu '{\frac {c{\sqrt {1-{V^{2} \over c^{2}}}}}{c+V}}=\nu '{\sqrt {\frac {1-\beta }{1+\beta }}}\,.} where T = γ V T ′ is standard time dilation formula.
Suppose instead that the wave is not composed of light waves with speed c , but instead, for easy visualization, bullets fired from a relativistic machine gun, with velocity s ′ in the frame of the emitter. Then, in general, the geometrical observation is precisely the same . But now, s ′ ≠ s , and s is given by velocity addition, s = s ′ + V 1 + s ′ V c 2 . {\displaystyle s={\frac {s'+V}{1+{s'V \over c^{2}}}}.}
The calculation is then essentially the same, except that here it is easier carried out upside down with τ = 1 / ν instead of ν . One finds
τ = 1 γ V ν ′ ( 1 1 + V s ′ ) , ν = γ V ν ′ ( 1 + V s ′ ) {\displaystyle \tau ={1 \over \gamma _{_{V}}\nu '}\left({\frac {1}{1+{V \over s'}}}\right),\quad \nu =\gamma _{_{V}}\nu '\left(1+{V \over s'}\right)}
L − s = ( − s ′ − V 1 + s ′ V c 2 + V ) T − s ′ − V 1 + s ′ V c 2 = γ V ν ′ − s ′ − V + V ( 1 + s ′ V c 2 ) − s ′ − V = γ V ν ′ ( s ′ ( 1 − V 2 c 2 ) s ′ + V ) = γ V ν ′ ( s ′ γ − 2 s ′ + V ) = 1 γ V ν ′ ( 1 1 + V s ′ ) . {\displaystyle {\begin{aligned}{L \over -s}&={\frac {\left({\frac {-s'-V}{1+{s'V \over c^{2}}}}+V\right)T}{\frac {-s'-V}{1+{s'V \over c^{2}}}}}\\&={\gamma _{_{V}} \over \nu '}{\frac {-s'-V+V(1+{s'V \over c^{2}})}{-s'-V}}\\&={\gamma _{_{V}} \over \nu '}\left({\frac {s'\left(1-{V^{2} \over c^{2}}\right)}{s'+V}}\right)\\&={\gamma _{_{V}} \over \nu '}\left({\frac {s'\gamma ^{-2}}{s'+V}}\right)\\&={1 \over \gamma _{_{V}}\nu '}\left({\frac {1}{1+{V \over s'}}}\right).\\\end{aligned}}}
Observe that in the typical case, the s ′ that enters is negative . The formula has general validity though. [ nb 2 ] When s ′ = − c , the formula reduces to the formula calculated directly for light waves above,
ν = ν ′ γ V ( 1 − β ) = ν ′ 1 − β 1 − β 1 + β = ν ′ 1 − β 1 + β . {\displaystyle \nu =\nu '\gamma _{_{V}}(1-\beta )=\nu '{\frac {1-\beta }{{\sqrt {1-\beta }}{\sqrt {1+\beta }}}}=\nu '{\sqrt {\frac {1-\beta }{1+\beta }}}\,.}
If the emitter is not firing bullets in empty space, but emitting waves in a medium, then the formula still applies , but now, it may be necessary to first calculate s ′ from the velocity of the emitter relative to the medium.
Returning to the case of a light emitter, in the case the observer and emitter are not collinear, the result has little modification, [ 2 ] [ 21 ] [ 22 ] ν = γ V ν ′ ( 1 + V s ′ cos θ ) , {\displaystyle \nu =\gamma _{_{V}}\nu '\left(1+{\frac {V}{s'}}\cos \theta \right),} where θ is the angle between the light emitter and the observer. This reduces to the previous result for collinear motion when θ = 0 , but for transverse motion corresponding to θ = π /2 , the frequency is shifted by the Lorentz factor . This does not happen in the classical optical Doppler effect.
Associated to the relativistic velocity β {\displaystyle {\boldsymbol {\beta }}} of an object is a quantity ζ {\displaystyle {\boldsymbol {\zeta }}} whose norm is called rapidity . These are related through s o ( 3 , 1 ) ⊃ s p a n { K 1 , K 2 , K 3 } ≈ R 3 ∋ ζ = β ^ tanh − 1 β , β ∈ B 3 , {\displaystyle {\mathfrak {so}}(3,1)\supset \mathrm {span} \{K_{1},K_{2},K_{3}\}\approx \mathbb {R} ^{3}\ni {\boldsymbol {\zeta }}={\boldsymbol {\hat {\beta }}}\tanh ^{-1}\beta ,\quad {\boldsymbol {\beta }}\in \mathbb {B} ^{3},} where the vector ζ {\displaystyle {\boldsymbol {\zeta }}} is thought of as being Cartesian coordinates on a 3-dimensional subspace of the Lie algebra s o ( 3 , 1 ) {\displaystyle {\mathfrak {so}}(3,1)} of the Lorentz group spanned by the boost generators K 1 , K 2 , K 3 {\displaystyle K_{1},K_{2},K_{3}} . This space, call it rapidity space , is isomorphic to ℝ 3 as a vector space, and is mapped to the open unit ball, B 3 {\displaystyle \mathbb {B} ^{3}} , velocity space , via the above relation. [ 23 ] The addition law on collinear form coincides with the law of addition of hyperbolic tangents tanh ( ζ v + ζ u ′ ) = tanh ζ v + tanh ζ u ′ 1 + tanh ζ v tanh ζ u ′ {\displaystyle \tanh(\zeta _{v}+\zeta _{u'})={\tanh \zeta _{v}+\tanh \zeta _{u'} \over 1+\tanh \zeta _{v}\tanh \zeta _{u'}}} with v c = tanh ζ v , u ′ c = tanh ζ u ′ , u c = tanh ( ζ v + ζ u ′ ) . {\displaystyle {\frac {v}{c}}=\tanh \zeta _{v}\ ,\quad {\frac {u'}{c}}=\tanh \zeta _{u'}\ ,\quad \,{\frac {u}{c}}=\tanh(\zeta _{v}+\zeta _{u'}).}
The line element in velocity space B 3 {\displaystyle \mathbb {B} ^{3}} follows from the expression for relativistic relative velocity in any frame, [ 24 ] v r = ( v 1 − v 2 ) 2 − ( v 1 × v 2 ) 2 1 − v 1 ⋅ v 2 , {\displaystyle v_{r}={\frac {\sqrt {(\mathbf {v_{1}} -\mathbf {v_{2}} )^{2}-(\mathbf {v_{1}} \times \mathbf {v_{2}} )^{2}}}{1-\mathbf {v_{1}} \cdot \mathbf {v_{2}} }},} where the speed of light is set to unity so that v i {\displaystyle v_{i}} and β i {\displaystyle \beta _{i}} agree. It this expression, v 1 {\displaystyle \mathbf {v} _{1}} and v 2 {\displaystyle \mathbf {v} _{2}} are velocities of two objects in any one given frame. The quantity v r {\displaystyle v_{r}} is the speed of one or the other object relative to the other object as seen in the given frame . The expression is Lorentz invariant, i.e. independent of which frame is the given frame, but the quantity it calculates is not . For instance, if the given frame is the rest frame of object one, then v r = v 2 {\displaystyle v_{r}=v_{2}} .
The line element is found by putting v 2 = v 1 + d v {\displaystyle \mathbf {v} _{2}=\mathbf {v} _{1}+d\mathbf {v} } or equivalently β 2 = β 1 + d β {\displaystyle {\boldsymbol {\beta }}_{2}={\boldsymbol {\beta }}_{1}+d{\boldsymbol {\beta }}} , [ 25 ] d l β 2 = d β 2 − ( β × d β ) 2 ( 1 − β 2 ) 2 = d β 2 ( 1 − β 2 ) 2 + β 2 1 − β 2 ( d θ 2 + sin 2 θ d φ 2 ) , {\displaystyle dl_{\boldsymbol {\beta }}^{2}={\frac {d{\boldsymbol {\beta }}^{2}-({\boldsymbol {\beta }}\times d{\boldsymbol {\beta }})^{2}}{(1-\beta ^{2})^{2}}}={\frac {d\beta ^{2}}{(1-\beta ^{2})^{2}}}+{\frac {\beta ^{2}}{1-\beta ^{2}}}(d\theta ^{2}+\sin ^{2}\theta d\varphi ^{2}),} with θ and φ the usual spherical angle coordinates for β {\displaystyle {\boldsymbol {\beta }}} taken in the z -direction. Now introduce ζ through ζ = | ζ | = tanh − 1 β , {\displaystyle \zeta =|{\boldsymbol {\zeta }}|=\tanh ^{-1}\beta ,} and the line element on rapidity space R 3 {\displaystyle \mathbb {R} ^{3}} becomes d l ζ 2 = d ζ 2 + sinh 2 ζ ( d θ 2 + sin 2 θ d φ 2 ) . {\displaystyle dl_{\boldsymbol {\zeta }}^{2}=d\zeta ^{2}+\sinh ^{2}\zeta (d\theta ^{2}+\sin ^{2}\theta d\varphi ^{2}).}
In scattering experiments the primary objective is to measure the invariant scattering cross section . This enters the formula for scattering of two particle types into a final state f {\displaystyle f} assumed to have two or more particles, [ 26 ] d N f = R f d V d t = σ F d V d t {\displaystyle dN_{f}=R_{f}\,dV\,dt=\sigma F\,dV\,dt} or, in most textbooks, d N f = σ n 1 n 2 v r d V d t {\displaystyle dN_{f}=\sigma n_{1}n_{2}v_{r}\,dV\,dt} where
The objective is to find a correct expression for relativistic relative speed v rel {\displaystyle v_{\text{rel}}} and an invariant expression for the incident flux.
Non-relativistically, one has for relative speed v r = | v 2 − v 1 | {\displaystyle v_{r}=|\mathbf {v} _{2}-\mathbf {v} _{1}|} . If the system in which velocities are measured is the rest frame of particle type 1 {\displaystyle 1} , it is required that v rel = v r = | v 2 | . {\displaystyle v_{\text{rel}}=v_{r}=|\mathbf {v} _{2}|.} Setting the speed of light c = 1 {\displaystyle c=1} , the expression for v rel {\displaystyle v_{\text{rel}}} follows immediately from the formula for the norm (second formula) in the general configuration as [ 27 ] [ 28 ] v rel = ( v 1 − v 2 ) 2 − ( v 1 × v 2 ) 2 1 − v 1 ⋅ v 2 . {\displaystyle v_{\text{rel}}={\frac {\sqrt {(\mathbf {v_{1}} -\mathbf {v_{2}} )^{2}-(\mathbf {v_{1}} \times \mathbf {v_{2}} )^{2}}}{1-\mathbf {v_{1}} \cdot \mathbf {v_{2}} }}.}
The formula reduces in the classical limit to v r = | v 1 − v 2 | {\displaystyle v_{r}=|\mathbf {v} _{1}-\mathbf {v} _{2}|} as it should, and gives the correct result in the rest frames of the particles. The relative velocity is incorrectly given in most, perhaps all books on particle physics and quantum field theory. [ 27 ] This is mostly harmless, since if either one particle type is stationary or the relative motion is collinear, then the right result is obtained from the incorrect formulas. The formula is invariant, but not manifestly so. It can be rewritten in terms of four-velocities as v rel = ( u 1 ⋅ u 2 ) 2 − 1 u 1 ⋅ u 2 . {\displaystyle v_{\text{rel}}={\frac {\sqrt {(u_{1}\cdot u_{2})^{2}-1}}{u_{1}\cdot u_{2}}}.}
The correct expression for the flux, published by Christian Møller [ 29 ] in 1945, is given by [ 30 ] F = n 1 n 2 ( v 1 − v 2 ) 2 − ( v 1 × v 2 ) 2 ≡ n 1 n 2 v ¯ . {\displaystyle F=n_{1}n_{2}{\sqrt {(\mathbf {v} _{1}-\mathbf {v} _{2})^{2}-(\mathbf {v} _{1}\times \mathbf {v} _{2})^{2}}}\equiv n_{1}n_{2}{\bar {v}}.}
One notes that for collinear velocities, F = n 1 n 2 | v 2 − v 1 | = n 1 n 2 v r {\displaystyle F=n_{1}n_{2}|\mathbf {v} _{2}-\mathbf {v} _{1}|=n_{1}n_{2}v_{r}} . In order to get a manifestly Lorentz invariant expression one writes J i = ( n i , n i v i ) {\displaystyle J_{i}=(n_{i},n_{i}\mathbf {v} _{i})} with n i = γ i n i 0 {\displaystyle n_{i}=\gamma _{i}n_{i}^{0}} , where n i 0 {\displaystyle n_{i}^{0}} is the density in the rest frame, for the individual particle fluxes and arrives at [ 31 ] F = ( J 1 ⋅ J 2 ) v rel . {\displaystyle F=(J_{1}\cdot J_{2})v_{\text{rel}}.}
In the literature the quantity v ¯ {\displaystyle {\bar {v}}} as well as v r {\displaystyle v_{r}} are both referred to as the relative velocity. In some cases (statistical physics and dark matter literature), v ¯ {\displaystyle {\bar {v}}} is referred to as the Møller velocity , in which case v r {\displaystyle v_{r}} means relative velocity. The true relative velocity is at any rate v rel {\displaystyle v_{\text{rel}}} . [ 31 ] The discrepancy between v rel {\displaystyle v_{\text{rel}}} and v r {\displaystyle v_{r}} is relevant though in most cases velocities are collinear. At LHC the crossing angle is small, around 300 μrad , but at
the old Intersecting Storage Ring at CERN , it was about 18°. [ 32 ]
For collinear velocities interpreted as rapidity, the sum formula is simple addition. According to Minkowski, the time-like vector ( zeitartiger Vektor ) for a given duration lies on a hyperbola. Since the hyperbola is traced by a hyperbolic angle , and velocity determines a point on the hyperbola, there is a hyperbolic angle called rapidity associated with a particular velocity. For a unit of duration, the unit hyperbola provides a reference, where hyperbolic angle forms a one-parameter group isomorphic to the real number line under addition. The form of the formula for velocity addition can then be accounted in terms of the hyperbolic tangent function tanh which takes hyperbolic angle (rapidity) as an argument. In fact, the hyperbolic tangent of rapidity is the ratio of velocity to the speed of light in vacuum. [ 33 ]
so that
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https://en.wikipedia.org/wiki/Velocity-addition_formula
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