text stringlengths 26 3.6k | page_title stringlengths 1 71 | source stringclasses 1 value | token_count int64 10 512 | id stringlengths 2 8 | url stringlengths 31 117 | topic stringclasses 4 values | section stringlengths 4 49 ⌀ | sublist stringclasses 9 values |
|---|---|---|---|---|---|---|---|---|
Corneal epithelium: an exceedingly thin multicellular epithelial tissue layer (non-keratinized stratified squamous epithelium) of fast-growing and easily regenerated cells, kept moist with tears. Irregularity or edema of the corneal epithelium disrupts the smoothness of the air/tear-film interface, the most significant component of the total refractive power of the eye, thereby reducing visual acuity. Corneal epithelium is continuous with the conjunctival epithelium, and is composed of about 6 layers of cells which are shed constantly on the exposed layer and are regenerated by multiplication in the basal layer.
Bowman's layer (also known as the anterior limiting membrane): when discussed in lieu of a subepithelial basement membrane, Bowman's Layer is a tough layer composed of collagen (mainly type I collagen fibrils), laminin, nidogen, perlecan and other HSPGs that protects the corneal stroma. When discussed as a separate entity from the subepithelial basement membrane, Bowman's Layer can be described as an acellular, condensed region of the apical stroma, composed primarily of randomly organized yet tightly woven collagen fibrils. These fibrils interact with and attach onto each other. This layer is eight to 14 micrometres (μm) thick and is absent or very thin in non-primates.
Corneal stroma (also substantia propria): a thick, transparent middle layer, consisting of regularly arranged collagen fibers along with sparsely distributed interconnected keratocytes, which are the cells for general repair and maintenance. They are parallel and are superimposed like book pages. The corneal stroma consists of approximately 200 layers of mainly type I collagen fibrils. Each layer is 1.5-2.5 μm. Up to 90% of the corneal thickness is composed of stroma. There are 2 theories of how transparency in the cornea comes about:
The lattice arrangements of the collagen fibrils in the stroma. The light scatter by individual fibrils is cancelled by destructive interference from the scattered light from other individual fibrils. | Cornea | Wikipedia | 483 | 311888 | https://en.wikipedia.org/wiki/Cornea | Biology and health sciences | Visual system | Biology |
The spacing of the neighboring collagen fibrils in the stroma must be < 200 nm for there to be transparency. (Goldman and Benedek)
Descemet's membrane (also posterior limiting membrane): a thin acellular layer that serves as the modified basement membrane of the corneal endothelium, from which the cells are derived. This layer is composed mainly of collagen type IV fibrils, less rigid than collagen type I fibrils, and is around 5-20 μm thick, depending on the subject's age. Just anterior to Descemet's membrane, a very thin and strong layer, Dua's layer, 15 microns thick and able to withstand 1.5 to 2 bars of pressure.
Corneal endothelium: a simple squamous or low cuboidal monolayer, approx 5 μm thick, of mitochondria-rich cells. These cells are responsible for regulating fluid and solute transport between the aqueous and corneal stromal compartments. (The term endothelium is a misnomer here. The corneal endothelium is bathed by aqueous humor, not by blood or lymph, and has a very different origin, function, and appearance from vascular endothelia.) Unlike the corneal epithelium, the cells of the endothelium do not regenerate. Instead, they stretch to compensate for dead cells which reduces the overall cell density of the endothelium, which affects fluid regulation. If the endothelium can no longer maintain a proper fluid balance, stromal swelling due to excess fluids and subsequent loss of transparency will occur and this may cause corneal edema and interference with the transparency of the cornea and thus impairing the image formed. Iris pigment cells deposited on the corneal endothelium can sometimes be washed into a distinct vertical pattern by the aqueous currents - this is known as Krukenberg's Spindle. | Cornea | Wikipedia | 423 | 311888 | https://en.wikipedia.org/wiki/Cornea | Biology and health sciences | Visual system | Biology |
Nerve supply
The cornea is one of the most sensitive tissues of the body, as it is densely innervated with sensory nerve fibres via the ophthalmic division of the trigeminal nerve by way of 70–80 long ciliary nerves. Research suggests the density of pain receptors in the cornea is 300–600 times greater than skin and 20–40 times greater than dental pulp, making any injury to the structure excruciatingly painful.
The ciliary nerves run under the endothelium and exit the eye through holes in the sclera apart from the optic nerve (which transmits only optic signals). The nerves enter the cornea via three levels; scleral, episcleral and conjunctival. Most of the bundles give rise by subdivision to a network in the stroma, from which fibres supply the different regions. The three networks are, midstromal, subepithelial/sub-basal, and epithelial. The receptive fields of each nerve ending are very large, and may overlap.
Corneal nerves of the subepithelial layer terminate near the superficial epithelial layer of the cornea in a logarithmic spiral pattern. The density of epithelial nerves decreases with age, especially after the seventh decade.
Function
Refraction
The optical component is concerned with producing a reduced inverted image on the retina. The eye's optical system consists of not only two but four surfaces—two on the cornea, two on the lens. Rays are refracted toward the midline. Distant rays, due to their parallel nature, converge to a point on the retina. The cornea admits light at the greatest angle. The aqueous and vitreous humors both have a refractive index of 1.336-1.339, whereas the cornea has a refractive index of 1.376. Because the change in refractive index between cornea and aqueous humor is relatively small compared to the change at the air–cornea interface, it has a negligible refractive effect, typically -6 dioptres. The cornea is considered to be a positive meniscus lens. Some species of birds and chameleons, and one kinown species of fish, also have corneas which can focus.
Transparency | Cornea | Wikipedia | 483 | 311888 | https://en.wikipedia.org/wiki/Cornea | Biology and health sciences | Visual system | Biology |
Upon death or removal of an eye the cornea absorbs the aqueous humor, thickens, and becomes hazy. Transparency can be restored by putting it in a warm, well-ventilated chamber at 31 °C (88 °F, the normal temperature), allowing the fluid to leave the cornea and become transparent. The cornea takes in fluid from the aqueous humor and the small blood vessels of the limbus, but a pump ejects the fluid immediately upon entry. When energy is deficient the pump may fail, or function too slowly to compensate, leading to swelling. This arises at death, but a dead eye can be placed in a warm chamber with a reservoir of sugar and glycogen that generally keeps the cornea transparent for at least 24 hours.
The endothelium controls this pumping action, and as discussed above, damage thereof is more serious, and is a cause of opaqueness and swelling. When damage to the cornea occurs, such as in a viral infection, the collagen used to repair the process is not regularly arranged, leading to an opaque patch (leukoma).
Clinical significance
The most common corneal disorders are the following:
Corneal abrasion – a medical condition involving the loss of the surface epithelial layer of the eye's cornea as a result of trauma to the surface of the eye.
Corneal dystrophy – a condition in which one or more parts of the cornea lose their normal clarity due to a buildup of cloudy material.
Corneal ulcer – an inflammatory or infective condition of the cornea involving disruption of its epithelial layer with involvement of the corneal stroma.
Corneal neovascularization – excessive ingrowth of blood vessels from the limbal vascular plexus into the cornea, caused by deprivation of oxygen from the air.
Fuchs' dystrophy – cloudy morning vision.
Keratitis – inflammation of the cornea.
Keratoconus – a degenerative disease, the cornea thins and changes shape to be more like a cone.
Corneal foreign body – a foreign object present in the cornea, one of the most common preventable occupational hazards.
Management
Surgical procedures | Cornea | Wikipedia | 458 | 311888 | https://en.wikipedia.org/wiki/Cornea | Biology and health sciences | Visual system | Biology |
Various refractive eye surgery techniques change the shape of the cornea in order to reduce the need for corrective lenses or otherwise improve the refractive state of the eye. In many of the techniques used today, reshaping of the cornea is performed by photoablation using the excimer laser.
There are also synthetic corneas (keratoprostheses) in development. Most are merely plastic inserts, but there are also those composed of biocompatible synthetic materials that encourage tissue ingrowth into the synthetic cornea, thereby promoting biointegration. Other methods, such as magnetic deformable membranes and optically coherent transcranial magnetic stimulation of the human retina are still in very early stages of research.
Other procedures
Orthokeratology is a method using specialized hard or rigid gas-permeable contact lenses to transiently reshape the cornea in order to improve the refractive state of the eye or reduce the need for eyeglasses and contact lenses.
In 2009, researchers at the University of Pittsburgh Medical center demonstrated that stem cell collected from human corneas can restore transparency without provoking a rejection response in mice with corneal damage. For corneal epithelial diseases such as Stevens Johnson Syndrome, persistent corneal ulcer etc., the autologous contralateral (normal) suprabasal limbus derived in vitro expanded corneal limbal stem cells are found to be effective as amniotic membrane based expansion is controversial. For endothelial diseases, such as bullous keratopathy, cadaver corneal endothelial precursor cells have been proven to be efficient. Recently emerging tissue engineering technologies are expected to be capable of making one cadaver-donor's corneal cells be expanded and be usable in more than one patient's eye. | Cornea | Wikipedia | 371 | 311888 | https://en.wikipedia.org/wiki/Cornea | Biology and health sciences | Visual system | Biology |
Corneal retention and permeability in topical drug delivery to the eye
The majority of ocular therapeutic agents are administered to the eye via the topical route. Cornea is one of the main barriers for drug diffusion because of its highly impermeable nature. Its continuous irrigation with a tear fluid also results in poor retention of the therapeutic agents on the ocular surface. Poor permeability of the cornea and quick wash out of therapeutic agents from ocular surface result in very low bioavailability of the drugs administered via topical route (typically less than 5%). Poor retention of formulations on ocular surfaces could potentially be improved with the use of mucoadhesive polymers. Drug permeability through the cornea could be facilitated with addition of penetration enhancers into topical formulations.
Transplantation
If the corneal stroma develops visually significant opacity, irregularity, or edema, a cornea of a deceased donor can be transplanted. Because there are no blood vessels in the cornea, there are also few problems with rejection of the new cornea.
When a cornea is needed for transplant, as from an eye bank, the best procedure is to remove the cornea from the eyeball, preventing the cornea from absorbing the aqueous humor.
There is a global shortage of corneal donations, severely limiting the availability of corneal transplants across most of the world. A 2016 study found that 12.7 million visually impaired people were in need of a corneal transplant, with only 1 cornea available for every 70 needed. Many countries have years-long waitlists for corneal transplant surgery due to the shortage of donated corneas. Only a handful of countries consistently have a large enough supply of donated corneas to meet local demand without a waitlist, including the United States, Italy, and Sri Lanka. | Cornea | Wikipedia | 377 | 311888 | https://en.wikipedia.org/wiki/Cornea | Biology and health sciences | Visual system | Biology |
Fourier optics is the study of classical optics using Fourier transforms (FTs), in which the waveform being considered is regarded as made up of a combination, or superposition, of plane waves. It has some parallels to the Huygens–Fresnel principle, in which the wavefront is regarded as being made up of a combination of spherical wavefronts (also called phasefronts) whose sum is the wavefront being studied. A key difference is that Fourier optics considers the plane waves to be natural modes of the propagation medium, as opposed to Huygens–Fresnel, where the spherical waves originate in the physical medium.
A curved phasefront may be synthesized from an infinite number of these "natural modes" i.e., from plane wave phasefronts oriented in different directions in space. When an expanding spherical wave is far from its sources, it is locally tangent to a planar phase front (a single plane wave out of the infinite spectrum), which is transverse to the radial direction of propagation. In this case, a Fraunhofer diffraction pattern is created, which emanates from a single spherical wave phase center. In the near field, no single well-defined spherical wave phase center exists, so the wavefront isn't locally tangent to a spherical ball. In this case, a Fresnel diffraction pattern would be created, which emanates from an extended source, consisting of a distribution of (physically identifiable) spherical wave sources in space. In the near field, a full spectrum of plane waves is necessary to represent the Fresnel near-field wave, even locally. A "wide" wave moving forward (like an expanding ocean wave coming toward the shore) can be regarded as an infinite number of "plane wave modes", all of which could (when they collide with something such as a rock in the way) scatter independently of one other. These mathematical simplifications and calculations are the realm of Fourier analysis and synthesis – together, they can describe what happens when light passes through various slits, lenses or mirrors that are curved one way or the other, or is fully or partially reflected. | Fourier optics | Wikipedia | 442 | 312008 | https://en.wikipedia.org/wiki/Fourier%20optics | Physical sciences | Optics | Physics |
Fourier optics forms much of the theory behind image processing techniques, as well as applications where information needs to be extracted from optical sources such as in quantum optics. To put it in a slightly complex way, similar to the concept of frequency and time used in traditional Fourier transform theory, Fourier optics makes use of the spatial frequency domain (kx, ky) as the conjugate of the spatial (x, y) domain. Terms and concepts such as transform theory, spectrum, bandwidth, window functions and sampling from one-dimensional signal processing are commonly used.
Fourier optics plays an important role for high-precision optical applications such as photolithography in which a pattern on a reticle to be imaged on wafers for semiconductor chip production is so dense such that light (e.g., DUV or EUV) emanated from the reticle is diffracted and each diffracted light may correspond to a different spatial frequency (kx, ky). Due to generally non-uniform patterns on reticles, a simple diffraction grating analysis may not provide the details of how light is diffracted from each reticle.
Propagation of light in homogeneous, source-free media
Light can be described as a waveform propagating through a free space (vacuum) or a material medium (such as air or glass). Mathematically, a real-valued component of a vector field describing a wave is represented by a scalar wave function u that depends on both space and time:
where
represents a position in a three dimensional space (in the Cartesian coordinate system here), and t represents time.
The wave equation
Fourier optics begins with the homogeneous, scalar wave equation (valid in source-free regions):
where is the speed of light and u(r,t) is a real-valued Cartesian component of an electromagnetic wave propagating through a free space (e.g., for where Ei is the i-axis component of an electric field E in the Cartesian coordinate system).
Sinusoidal steady state | Fourier optics | Wikipedia | 425 | 312008 | https://en.wikipedia.org/wiki/Fourier%20optics | Physical sciences | Optics | Physics |
If light of a fixed frequency in time/wavelength/color (as from a single-mode laser) is assumed, then, based on the engineering time convention, which assumes an time dependence in wave solutions at the angular frequency with where is a time period of the waves, the time-harmonic form of the optical field is given as
where is the imaginary unit, is the operator taking the real part of ,
is the angular frequency (in radians per unit time) of light waves, and
is, in general, a complex quantity, with separate amplitude in non-negative real number and phase .
The Helmholtz equation
Substituting this expression into the scalar wave equation above yields the time-independent form of the wave equation,
where
with the wavelength in vacuum, is the wave number (also called propagation constant), is the spatial part of a complex-valued Cartesian component of an electromagnetic wave. Note that the propagation constant and the angular frequency are linearly related to one another, a typical characteristic of transverse electromagnetic (TEM) waves in homogeneous media.
Since the originally desired real-valued solution of the scalar wave equation can be simply obtained by taking the real part of , solving the following equation, known as the Helmholtz equation, is mostly concerned as treating a complex-valued function is often much easier than treating the corresponding real-valued function.
Solving the Helmholtz equation
Solutions to the Helmholtz equation in the Cartesian coordinate system may readily be found via the principle of separation of variables for partial differential equations. This principle says that in separable orthogonal coordinates, an elementary product solution to this wave equation may be constructed of the following form:
i.e., as the product of a function of x, times a function of y, times a function of z. If this elementary product solution is substituted into the wave equation, using the scalar Laplacian in the Cartesian coordinates system
then the following equation for the 3 individual functions is obtained
which is readily rearranged into the form: | Fourier optics | Wikipedia | 409 | 312008 | https://en.wikipedia.org/wiki/Fourier%20optics | Physical sciences | Optics | Physics |
It may now be argued that each quotient in the equation above must, of necessity, be constant. To justify this, let's say that the first quotient is not a constant, and is a function of x. Since none of the other terms in the equation has any dependence on the variable x, so the first term also must not have any x-dependence; it must be a constant. (If the first term is a function of x, then there is no way to make the left hand side of this equation be zero.) This constant is denoted as -kx2. Reasoning in a similar way for the y and z quotients, three ordinary differential equations are obtained for the fx, fy and fz, along with one separation condition:
Each of these 3 differential equations has the same solution form: sines, cosines or complex exponentials. We'll go with the complex exponential as to be a complex function. As a result, the elementary product solution is
with a generally complex number . This solution is the spatial part of a complex-valued Cartesian component (e.g., , , or as the electric field component along each axis in the Cartesian coordinate system) of a propagating plane wave. (, , or ) is a real number here since waves in a source-free medium has been assumed so each plane wave is not decayed or amplified as it propagates in the medium. The negative sign of (, , or ) in a wave vector (where ) means that the wave propagation direction vector has a positive (, , or )-component, while the positive sign of means a negative (, , or )-component of that vector.
Product solutions to the Helmholtz equation are also readily obtained in cylindrical and spherical coordinates, yielding cylindrical and spherical harmonics (with the remaining separable coordinate systems being used much less frequently).
The complete solution: the superposition integral
A general solution to the homogeneous electromagnetic wave equation at a fixed time frequency in the Cartesian coordinate system may be formed as a weighted superposition of all possible elementary plane wave solutions as
with the constraints of , each as a real number, and where . In this superposition, is the weight factor or the amplitude of the plane wave component with the wave vector where is determined in terms of and by the mentioned constraint.
Next, let
Then: | Fourier optics | Wikipedia | 486 | 312008 | https://en.wikipedia.org/wiki/Fourier%20optics | Physical sciences | Optics | Physics |
The plane wave spectrum representation of a general electromagnetic field (e.g., a spherical wave) in the equation () is the basic foundation of Fourier optics (this point cannot be emphasized strongly enough), because at z = 0, the equation simply becomes a Fourier transform (FT) relationship between the field and its plane wave contents (hence the name, Fourier optics).
Thus:
and
All spatial dependence of each plane wave component is described explicitly by an exponential function. The coefficient of the exponential is a function of only two components of the wave vector for each plane wave (since other remained component can be determined via the above mentioned constraints), for example and , just as in ordinary Fourier analysis and Fourier transforms. | Fourier optics | Wikipedia | 144 | 312008 | https://en.wikipedia.org/wiki/Fourier%20optics | Physical sciences | Optics | Physics |
Connection between Fourier optics and imaging resolution
Let's consider an imaging system where the z-axis is the optical axis of the system and the object plane (to be imaged on the image plane of the system) is the plane at . On the object plane, the spatial part of a complex-valued Cartesian component of a wave is, as shown above, with the constraints of , each as a real number, and where . The imaging is the reconstruction of a wave on the object plane (having information about a pattern on the object plane to be imaged) on the image plane via the proper wave propagation from the object to the image planes, (E.g., think about the imaging of an image in an aerial space.) and the wave on the object plane, that fully follows the pattern to be imaged, is in principle, described by the unconstrained inverse Fourier transform where takes an infinite range of real numbers. It means that, for a given light frequency, only a part of the full feature of the pattern can be imaged because of the above-mentioned constraints on ; (1) a fine feature which representation in the inverse Fourier transform requires spatial frequencies , where are transverse wave numbers satisfying , can not be fully imaged since waves with such do not exist for the given light of (This phenomenon is known as the diffraction limit.), and (2) spatial frequencies with but close to so higher wave outgoing angles with respect to the optical axis, requires a high NA (Numerical Aperture) imaging system that is expensive and difficult to build. For (1), even if complex-valued longitudinal wavenumbers are allowed (by an unknown interaction between light and the object plane pattern that is usually a solid material), give rise to light decay along the axis (Light amplification along the axis does not physically make sense if there is no amplification material between the object and image planes, and this is a usual case.) so waves with such may not reach the image plane that is usually sufficiently far way from the object plane. | Fourier optics | Wikipedia | 421 | 312008 | https://en.wikipedia.org/wiki/Fourier%20optics | Physical sciences | Optics | Physics |
In connection with photolithography of electronic components, these (1) and (2) are the reasons why light of a higher frequency (smaller wavelength, thus larger magnitude of ) or a higher NA imaging system is required to image finer features of integrated circuits on a photoresist on a wafer. As a result, machines realizing such an optical lithography have become more and more complex and expensive, significantly increasing the cost of the electronic component production.
The paraxial approximation
Paraxial wave propagation (optic axis assumed as z axis)
A solution to the Helmholtz equation as the spatial part of a complex-valued Cartesian component of a single frequency wave is assumed to take the form:
where is the wave vector, and
and
is the wave number. Next, use the paraxial approximation, that is a small-angle approximation such that
so, up to the second order approximation of trigonometric functions (that is, taking only up to the second term in the Taylor series expansion of each trigonometric function),
where is the angle (in radian) between the wave vector k and the z-axis as the optical axis of an optical system under discussion.
As a result,
and
The paraxial wave equation
Substituting this expression into the Helmholtz equation, the paraxial wave equation is derived:
where
is the transverse Laplace operator in the Cartesian coordinates system. In the derivation of the paraxial wave equation, the following approximations are used.
is small () so a term with is ignored.
Terms with and are much smaller than a term with (or ) so these two terms are ignored.
so a term with is ignored. It is the slowly varying envelope approximation, means that the amplitude or envelope of a wave is slowly varying compared with the major period of the wave .
The far field approximation
The equation () above may be evaluated asymptotically in the far field (using the stationary phase method) to show that the field at a distant point is indeed due solely to the plane wave component with the wave vector which propagates parallel to the vector , and whose plane is tangent to the phasefront at . The mathematical details of this process may be found in Scott [1998] or Scott [1990]. The result of performing a stationary phase integration on the expression above is the following expression,
which clearly indicates that the field at is directly proportional to the spectral component in the direction of , where,
and | Fourier optics | Wikipedia | 500 | 312008 | https://en.wikipedia.org/wiki/Fourier%20optics | Physical sciences | Optics | Physics |
Stated another way, the radiation pattern of any planar field distribution is the FT (Fourier Transform) of that source distribution (see Huygens–Fresnel principle, wherein the same equation is developed using a Green's function approach). Note that this is NOT a plane wave. The radial dependence is a spherical wave - both in magnitude and phase - whose local amplitude is the FT of the source plane distribution at that far field angle. A plane wave spectrum does not necessarily mean that the field as the superposition of the plane wave components in that spectrum behaves something like a plane wave at far distances.
Spatial versus angular bandwidth
The equation () above is critical to making the connection between spatial bandwidth (on the one hand) and angular bandwidth (on the other), in the far field. Note that the term "far field" usually means we're talking about a converging or diverging spherical wave with a pretty well defined phase center. The connection between spatial and angular bandwidth in the far field is essential in understanding the low pass filtering property of thin lenses. See the section 6.1.3 for the condition defining the far field region.
Once the concept of angular bandwidth is understood, the optical scientist can "jump back and forth" between the spatial and spectral domains to quickly gain insights which would ordinarily not be so readily available just through spatial domain or ray optics considerations alone. For example, any source bandwidth which lies past the edge angle to the first lens (This edge angle sets the bandwidth of the optical system.) will not be captured by the system to be processed.
As a side note, electromagnetics scientists have devised an alternative means to calculate an electric field in a far zone which does not involve stationary phase integration. They have devised a concept known as "fictitious magnetic currents" usually denoted by M, and defined as | Fourier optics | Wikipedia | 374 | 312008 | https://en.wikipedia.org/wiki/Fourier%20optics | Physical sciences | Optics | Physics |
In this equation, it is assumed that the unit vector in the z-direction points into the half-space where the far field calculations will be made. These equivalent magnetic currents are obtained using equivalence principles which, in the case of an infinite planar interface, allow any electric currents J to be "imaged away" while the fictitious magnetic currents are obtained from twice the aperture electric field (see Scott [1998]). Then the radiated electric field is calculated from the magnetic currents using an equation similar to the equation for the magnetic field radiated by an electric current. In this way, a vector equation is obtained for the radiated electric field in terms of the aperture electric field, and the derivation requires no use of stationary phase ideas.
The plane wave spectrum: the foundation of Fourier optics
The plane wave spectrum concept is the basic foundation of Fourier Optics. The plane wave spectrum is a continuous spectrum of uniform plane waves, and there is one plane wave component in the spectrum for every tangent point on the far-field phase front. The amplitude of that plane wave component would be the amplitude of the optical field at that tangent point. Again, this is true only in the far field, roughly defined as the range beyond where is the maximum linear extent of the optical sources and is the wavelength (Scott [1998]). The plane wave spectrum is often regarded as being discrete for certain types of periodic gratings, though in reality, the spectra from gratings are continuous as well, since no physical device can have the infinite extent required to produce a true line spectrum. | Fourier optics | Wikipedia | 313 | 312008 | https://en.wikipedia.org/wiki/Fourier%20optics | Physical sciences | Optics | Physics |
Likely to electrical signals, bandwidth in optics is a measure of how finely detailed an image is; the finer the detail, the greater the bandwidth required to represent it. A DC (Direct Current) electrical signal is constant and has no oscillations; a plane wave propagating parallel to the optic () axis has constant value in any x-y plane, and therefore is analogous to the (constant) DC component of an electrical signal. Bandwidth in electrical signals relates to the difference between the highest and lowest frequencies present in the spectrum of a signal, practically with a criterion to cut off high and low frequency edges of the spectrum to represent bandwidth in a number. For optical systems, bandwidth also relates to spatial frequency content (spatial bandwidth), but it also has a secondary meaning. It also measures how far from the optic axis the corresponding plane waves are tilted, and so this type of bandwidth is often referred to also as angular bandwidth. It takes more frequency bandwidth to produce a short pulse in an electrical circuit, and more angular (or, spatial frequency) bandwidth to produce a sharp spot in an optical system (see discussion related to Point spread function).
The plane wave spectrum arises naturally as the eigenfunction or "natural mode" solution to the homogeneous electromagnetic wave equation in rectangular coordinates (see also Electromagnetic radiation, which derives the wave equation from Maxwell's equations in source-free media, or Scott [1998]). In the frequency domain, with an assumed time convention of , the homogeneous electromagnetic wave equation becomes what is known as the Helmholtz equation and takes the form
where and is the wavenumber of the medium.
Eigenfunction (natural mode) solutions: background and overview | Fourier optics | Wikipedia | 344 | 312008 | https://en.wikipedia.org/wiki/Fourier%20optics | Physical sciences | Optics | Physics |
In the case of differential equations, as in the case of matrix equations, whenever the right-hand side of an equation is zero (For example, a forcing function, forcing vector, or the source of a force is zero.), the equation may still admit a non-trivial solution, known in applied mathematics as an eigenfunction solution, in physics as a "natural mode" solution, and in electrical circuit theory as the "zero-input response." This is a concept that spans a wide range of physical disciplines. Common physical examples of resonant natural modes would include the resonant vibrational modes of stringed instruments (1D), percussion instruments (2D) or the former Tacoma Narrows Bridge (3D). Examples of propagating natural modes would include waveguide modes, optical fiber modes, solitons and Bloch waves. an Infinite homogeneous media admits the rectangular, circular and spherical harmonic solutions to the Helmholtz equation, depending on the coordinate system under consideration. The propagating plane waves that we'll study in this article are perhaps the simplest type of propagating waves found in any type of media.
There is a striking similarity between the Helmholtz equation () above, which may be written
and the usual equation form for the eigenvalues / eigenvectors of a square matrix A,
particularly since both the scalar Laplacian and the matrix A are linear operators on their respective functions / vector spaces. (The minus sign in this matrix equation is, for all intents and purposes, immaterial. However, the plus sign in the Helmholtz equation is significant.) It is perhaps worthwhile to note that the eigenfunction solutions / eigenvector solutions to the Helmholtz equation / the matrix equation, often yield an orthogonal set of the eigenfunctions / the eigenvectors which span (i.e., form a basis set for) the function space / vector space under consideration. The interested reader may investigate other functional linear operators (so for different equations than the Helmholtz equation) which give rise to different kinds of orthogonal eigenfunctions such as Legendre polynomials, Chebyshev polynomials and Hermite polynomials. | Fourier optics | Wikipedia | 457 | 312008 | https://en.wikipedia.org/wiki/Fourier%20optics | Physical sciences | Optics | Physics |
In the matrix equation case in which A is a square matrix, eigenvalues may be found by setting the determinant of the matrix equal to zero, i.e. finding where the matrix has no inverse. (Such a square matrix is said to be singular.) Finite matrices have only a finite number of eigenvalues/eigenvectors, whereas linear operators can have a countably infinite number of eigenvalues/eigenfunctions (in confined regions) or uncountably infinite (continuous) spectra of solutions, as in unbounded regions.
In certain physics applications such as in the computation of bands in a periodic volume, it is often a case that the elements of a matrix will be very complicated functions of frequency and wavenumber, and the matrix will be non-singular (I.e., it has the inverse matrix.) for most combinations of frequency and wavenumber, but will also be singular (I.e., it does not have the inverse matrix.) for certain specific combinations. By finding which combinations of frequency and wavenumber drive the determinant of the matrix to zero, the propagation characteristics of the medium may be determined. Relations of this type, between frequency and wavenumber, are known as dispersion relations and some physical systems may admit many different kinds of dispersion relations. An example from electromagnetics is an ordinary waveguide, which may admit numerous dispersion relations, each associated with a unique propagation mode of the waveguide. Each propagation mode of the waveguide is known as an eigenfunction solution (or eigenmode solution) to Maxwell's equations in the waveguide. Free space also admits eigenmode (natural mode) solutions (known more commonly as plane waves), but with the distinction that for any given frequency, free space admits a continuous modal spectrum, whereas waveguides have a discrete mode spectrum. In this case, the dispersion relation is linear, as in section 1.3.
K-space
For a given such as for a homogeneous vacuum space, the separation condition,
which is identical to the equation for the Euclidean metric in a three-dimensional configuration space, suggests the notion of a k-vector in a three-dimensional "k-space", defined (for propagating plane waves) in rectangular coordinates as:
and in the spherical coordinate system as
Use will be made of these spherical coordinate system relations in the next section. | Fourier optics | Wikipedia | 508 | 312008 | https://en.wikipedia.org/wiki/Fourier%20optics | Physical sciences | Optics | Physics |
The notion of k-space is central to many disciplines in engineering and physics, especially in the study of periodic volumes, such as in crystallography and the band theory of semiconductor materials.
The two-dimensional Fourier transform
A spectrum analysis equation (calculating the spectrum of a function ):
A synthesis equation (reconstructing the function from its spectrum):
The normalizing factor of is present whenever angular frequency (radians) is used, but not when ordinary frequency (cycles) is used.
Optical systems: general overview and analogy with electrical signal processing systems
In a high level overview, an optical system consists of three parts; an input plane, and output plane, and a set of components between these planes that transform an image f formed in the input plane into a different image g formed in the output plane. The optical system output image g is related to the input image f by convolving the input image with the optical impulse response function of the optical system, h (known as the point-spread function, for focused optical systems). The impulse response function uniquely defines the input-output behavior of the optical system. By convention, the optical axis of the system is taken as the z-axis. As a result, the two images and the impulse response function are all functions of the transverse coordinates, x and y.
The impulse response of an optical imaging system is the output plane field which is produced when an ideal mathematical optical field point source of light, that is an impulse input to the system, is placed in the input plane (usually on-axis, i.e., on the optical axis). In practice, it is not necessary to have an ideal point source in order to determine an exact impulse response. This is because any source bandwidth which lies outside the bandwidth of the optical system under consideration won't matter anyway (since it cannot even be captured by the optical system), so therefore it's not necessary in determining the impulse response. The source only needs to have at least as much (angular) bandwidth as the optical system. | Fourier optics | Wikipedia | 415 | 312008 | https://en.wikipedia.org/wiki/Fourier%20optics | Physical sciences | Optics | Physics |
Optical systems typically fall into one of two different categories. The first is ordinary focused optical imaging systems (e.g., cameras), wherein the input plane is called the object plane and the output plane is called the image plane. An optical field in the image plane (the output plane of the imaging system) is desired to be a high-quality reproduction of an optical field in the object plane (the input plane of the imaging system). The impulse response function of an optical imaging system is desired to approximate a 2D delta function, at the location (or a linearly scaled location) in the output plane corresponding to the location of the impulse (an ideal point source) in the input plane. The actual impulse response function of an imaging system typically resembles an Airy function, whose radius is on the order of the wavelength of the light used. The impulse response function in this case is typically referred to as a point spread function, since the mathematical point of light in the object plane has been spread out into an Airy function in the image plane.
The second type is optical image processing systems, in which a significant feature in the input plane optical field is to be located and isolated. In this case, the impulse response of such a system is desired to be a close replica (picture) of that feature which is being searched for in the input plane field, so that a convolution of the impulse response (an image of the desired feature) against the input plane field will produce a bright spot at the feature location in the output plane. It is this latter type of optical image processing system that is the subject of this section. The section 6.2 presents one hardware implementation of the optical image processing operations described in this section.
Input plane
The input plane is defined as the locus of all points such that z = 0. The input image f is therefore
Output plane
The output plane is defined as the locus of all points such that z = d. The output image g is therefore
The 2D convolution of input function against the impulse response function
i.e., | Fourier optics | Wikipedia | 419 | 312008 | https://en.wikipedia.org/wiki/Fourier%20optics | Physical sciences | Optics | Physics |
The alert reader will note that the integral above tacitly assumes that the impulse response is NOT a function of the position (x',y') of the impulse of light in the input plane (if this were not the case, this type of convolution would not be possible). This property is known as shift invariance (Scott [1998]). No optical system is perfectly shift invariant: as the ideal, mathematical point of light is scanned away from the optic axis, aberrations will eventually degrade the impulse response (known as a coma in focused imaging systems). However, high quality optical systems are often "shift invariant enough" over certain regions of the input plane that we may regard the impulse response as being a function of only the difference between input and output plane coordinates, and thereby use the equation above with impunity.
Also, this equation assumes unit magnification. If magnification is present, then eqn. () becomes
which basically translates the impulse response function, hM(), from x′ to x = Mx′. In eqn. (), hM will be a magnified version of the impulse response function h of a similar, unmagnified system, so that hM(x,y) = h(x/M,y/M).
Derivation of the convolution equation
The extension to two dimensions is trivial, except for the difference that causality exists in the time domain, but not in the spatial domain. Causality means that the impulse response h(t − t′) of an electrical system, due to an impulse applied at time t', must of necessity be zero for all times t such that t − t′ < 0.
Obtaining the convolution representation of the system response requires representing the input signal as a weighted superposition over a train of impulse functions by using the sifting property of Dirac delta functions. | Fourier optics | Wikipedia | 394 | 312008 | https://en.wikipedia.org/wiki/Fourier%20optics | Physical sciences | Optics | Physics |
It is then presumed that the system under consideration is linear, that is to say that the output of the system due to two different inputs (possibly at two different times) is the sum of the individual outputs of the system to the two inputs, when introduced individually. Thus the optical system may contain no nonlinear materials nor active devices (except possibly, extremely linear active devices). The output of the system, for a single delta function input is defined as the impulse response of the system, h(t − t′). And, by our linearity assumption (i.e., that the output of system to a pulse train input is the sum of the outputs due to each individual pulse), we can now say that the general input function f(t) produces the output:
where h(t − t′) is the (impulse) response of the linear system to the delta function input δ(t − t′), applied at time t. This is where the convolution equation above comes from. The convolution equation is useful because it is often much easier to find the response of a system to a delta function input - and then perform the convolution above to find the response to an arbitrary input - than it is to try to find the response to the arbitrary input directly. Also, the impulse response (in either time or frequency domains) usually yields insight to relevant figures of merit of the system. In the case of most lenses, the point spread function (PSF) is a pretty common figure of merit for evaluation purposes.
The same logic is used in connection with the Huygens–Fresnel principle, or Stratton-Chu formulation, wherein the "impulse response" is referred to as the Green's function of the system. So the spatial domain operation of a linear optical system is analogous in this way to the Huygens–Fresnel principle.
System transfer function
If the last equation above is Fourier transformed, it becomes:
where
is the spectrum of the output signal
is the system transfer function
is the spectrum of the input signal
In like fashion, eqn. () may be Fourier transformed to yield:
The system transfer function, . In optical imaging this function is better known as the optical transfer function (Goodman).
Once again it may be noted from the discussion on the Abbe sine condition, that this equation assumes unit magnification. | Fourier optics | Wikipedia | 490 | 312008 | https://en.wikipedia.org/wiki/Fourier%20optics | Physical sciences | Optics | Physics |
This equation takes on its real meaning when the Fourier transform, is associated with the coefficient of the plane wave whose transverse wavenumbers are . Thus, the input-plane plane wave spectrum is transformed into the output-plane plane wave spectrum through the multiplicative action of the system transfer function. It is at this stage of understanding that the previous background on the plane wave spectrum becomes invaluable to the conceptualization of Fourier optical systems.
Applications of Fourier optics principles
Fourier optics is used in the field of optical information processing, the staple of which is the classical 4F processor.
The Fourier transform properties of a lens provide numerous applications in optical signal processing such as spatial filtering, optical correlation and computer generated holograms.
Fourier optical theory is used in interferometry, optical tweezers, atom traps, and quantum computing. Concepts of Fourier optics are used to reconstruct the phase of light intensity in the spatial frequency plane (see adaptive-additive algorithm).
Fourier transforming property of lenses
If a transmissive object is placed at one focal length in front of a lens, then its Fourier transform will be formed at one focal length behind the lens. Consider the figure to the right (click to enlarge)
In this figure, a plane wave incident from the left is assumed. The transmittance function in the front focal plane (i.e., Plane 1) spatially modulates the incident plane wave in magnitude and phase, like on the left-hand side of eqn. () (specified to z = 0), and in so doing, produces a spectrum of plane waves corresponding to the FT of the transmittance function, like on the right-hand side of eqn. () (for z > 0). The various plane wave components propagate at different tilt angles with respect to the optic axis of the lens (i.e., the horizontal axis). The finer the features in the transparency, the broader the angular bandwidth of the plane wave spectrum. We'll consider one such plane wave component, propagating at angle θ with respect to the optic axis. It is assumed that θ is small (paraxial approximation), so that
and
and
In the figure, the plane wave phase, moving horizontally from the front focal plane to the lens plane, is
and the spherical wave phase from the lens to the spot in the back focal plane is: | Fourier optics | Wikipedia | 487 | 312008 | https://en.wikipedia.org/wiki/Fourier%20optics | Physical sciences | Optics | Physics |
and the sum of the two path lengths is f (1 + θ2/2 + 1 − θ2/2) = 2f; i.e., it is a constant value, independent of tilt angle, θ, for paraxial plane waves. Each paraxial plane wave component of the field in the front focal plane appears as a point spread function spot in the back focal plane, with an intensity and phase equal to the intensity and phase of the original plane wave component in the front focal plane. In other words, the field in the back focal plane is the Fourier transform of the field in the front focal plane.
All FT components are computed simultaneously - in parallel - at the speed of light. As an example, light travels at a speed of roughly per nanosecond, so if a lens has a focal length, an entire 2D FT can be computed in about 2 ns (2 × 10−9 seconds). If the focal length is 1 in, then the time is under 200 ps. No electronic computer can compete with these kinds of numbers or perhaps ever hope to, although supercomputers may actually prove faster than optics, as improbable as that may seem. However, their speed is obtained by combining numerous computers which, individually, are still slower than optics. The disadvantage of the optical FT is that, as the derivation shows, the FT relationship only holds for paraxial plane waves, so this FT "computer" is inherently bandlimited. On the other hand, since the wavelength of visible light is so minute in relation to even the smallest visible feature dimensions in the image i.e.,
(for all kx, ky within the spatial bandwidth of the image, so that kz is nearly equal to k), the paraxial approximation is not terribly limiting in practice. And, of course, this is an analog - not a digital - computer, so precision is limited. Also, phase can be challenging to extract; often it is inferred interferometrically.
Optical processing is especially useful in real time applications where rapid processing of massive amounts of 2D data is required, particularly in relation to pattern recognition.
Object truncation and Gibbs phenomenon | Fourier optics | Wikipedia | 448 | 312008 | https://en.wikipedia.org/wiki/Fourier%20optics | Physical sciences | Optics | Physics |
The spatially modulated electric field, shown on the left-hand side of eqn. (), typically only occupies a finite (usually rectangular) aperture in the x,y plane. The rectangular aperture function acts like a 2D square-top filter, where the field is assumed to be zero outside this 2D rectangle. The spatial domain integrals for calculating the FT coefficients on the right-hand side of eqn. () are truncated at the boundary of this aperture. This step truncation can introduce inaccuracies in both theoretical calculations and measured values of the plane wave coefficients on the RHS of eqn. ().
Whenever a function is discontinuously truncated in one FT domain, broadening and rippling are introduced in the other FT domain. A perfect example from optics is in connection with the point spread function, which for on-axis plane wave illumination of a quadratic lens (with circular aperture), is an Airy function, J1(x)/x. Literally, the point source has been "spread out" (with ripples added), to form the Airy point spread function (as the result of truncation of the plane wave spectrum by the finite aperture of the lens). This source of error is known as Gibbs phenomenon and it may be mitigated by simply ensuring that all significant content lies near the center of the transparency, or through the use of window functions which smoothly taper the field to zero at the frame boundaries. By the convolution theorem, the FT of an arbitrary transparency function - multiplied (or truncated) by an aperture function - is equal to the FT of the non-truncated transparency function convolved against the FT of the aperture function, which in this case becomes a type of "Greens function" or "impulse response function" in the spectral domain. Therefore, the image of a circular lens is equal to the object plane function convolved against the Airy function (the FT of a circular aperture function is J1(x)/x and the FT of a rectangular aperture function is a product of sinc functions, sinx/x).
Fourier analysis and functional decomposition | Fourier optics | Wikipedia | 445 | 312008 | https://en.wikipedia.org/wiki/Fourier%20optics | Physical sciences | Optics | Physics |
Even though the input transparency only occupies a finite portion of the x-y plane (Plane 1), the uniform plane waves comprising the plane wave spectrum occupy the entire x-y plane, which is why (for this purpose) only the longitudinal plane wave phase (in the z-direction, from Plane 1 to Plane 2) must be considered, and not the phase transverse to the z-direction. It is of course, very tempting to think that if a plane wave emanating from the finite aperture of the transparency is tilted too far from horizontal, it will somehow "miss" the lens altogether but again, since the uniform plane wave extends infinitely far in all directions in the transverse (x-y) plane, the planar wave components cannot miss the lens.
This issue brings up perhaps the predominant difficulty with Fourier analysis, namely that the input-plane function, defined over a finite support (i.e., over its own finite aperture), is being approximated with other functions (sinusoids) which have infinite support (i.e., they are defined over the entire infinite x-y plane). This is unbelievably inefficient computationally, and is the principal reason why wavelets were conceived, that is to represent a function (defined on a finite interval or area) in terms of oscillatory functions which are also defined over finite intervals or areas. Thus, instead of getting the frequency content of the entire image all at once (along with the frequency content of the entire rest of the x-y plane, over which the image has zero value), the result is instead the frequency content of different parts of the image, which is usually much simpler. Unfortunately, wavelets in the x-y plane don't correspond to any known type of propagating wave function, in the same way that Fourier's sinusoids (in the x-y plane) correspond to plane wave functions in three dimensions. However, the FTs of most wavelets are well known and could possibly be shown to be equivalent to some useful type of propagating field. | Fourier optics | Wikipedia | 431 | 312008 | https://en.wikipedia.org/wiki/Fourier%20optics | Physical sciences | Optics | Physics |
On the other hand, sinc functions and Airy functions - which are not only the point spread functions of rectangular and circular apertures, respectively, but are also cardinal functions commonly used for functional decomposition in interpolation/sampling theory [Scott 1990] - do''' correspond to converging or diverging spherical waves, and therefore could potentially be implemented as a whole new functional decomposition of the object plane function, thereby leading to another point of view similar in nature to Fourier optics. This would basically be the same as conventional ray optics, but with diffraction effects included. In this case, each point spread function would be a type of "smooth pixel," in much the same way that a soliton on a fiber is a "smooth pulse."
Perhaps a lens figure-of-merit in this "point spread function" viewpoint would be to ask how well a lens transforms an Airy function in the object plane into an Airy function in the image plane, as a function of radial distance from the optic axis, or as a function of the size of the object plane Airy function. This is somewhat like the point spread function, except now we're really looking at it as a kind of input-to-output plane transfer function (like MTF), and not so much in absolute terms, relative to a perfect point. Similarly, Gaussian wavelets, which would correspond to the waist of a propagating Gaussian beam, could also potentially be used in still another functional decomposition of the object plane field.
Far-field range and the 2D2 / λ criterion | Fourier optics | Wikipedia | 326 | 312008 | https://en.wikipedia.org/wiki/Fourier%20optics | Physical sciences | Optics | Physics |
In the figure above, illustrating the Fourier transforming property of lenses, the lens is in the near field of the object plane transparency, therefore the object plane field at the lens may be regarded as a superposition of plane waves, each one of which propagates at some angle with respect to the z-axis. In this regard, the far-field criterion is loosely defined as: Range = 2D2/λ where D is the maximum linear extent of the optical sources and λ is the wavelength (Scott [1998]). The D of the transparency is on the order of cm (10−2 m) and the wavelength of light is on the order of 10−6 m, therefore D/λ for the whole transparency is on the order of 104. This times D is on the order of 102 m, or hundreds of meters. On the other hand, the far field distance from a PSF spot is on the order of λ. This is because D for the spot is on the order of λ, so that D/λ is on the order of unity; this times D (i.e., λ) is on the order of λ (10−6 m).
Since the lens is in the far field of any PSF spot, the field incident on the lens from the spot may be regarded as being a spherical wave, as in eqn. (), not as a plane wave spectrum, as in eqn. (). On the other hand, the lens is in the near field of the entire input plane transparency, therefore eqn. () - the full plane wave spectrum - accurately represents the field incident on the lens from that larger, extended source.
Lens as a low-pass filter | Fourier optics | Wikipedia | 352 | 312008 | https://en.wikipedia.org/wiki/Fourier%20optics | Physical sciences | Optics | Physics |
A lens is basically a low-pass plane wave filter (see Low-pass filter). Consider a "small" light source located on-axis in the object plane of the lens. It is assumed that the source is small enough that, by the far-field criterion, the lens is in the far field of the "small" source. Then, the field radiated by the small source is a spherical wave which is modulated by the FT of the source distribution, as in eqn. (), Then, the lens passes - from the object plane over onto the image plane - only that portion of the radiated spherical wave which lies inside the edge angle of the lens. In this far-field case, truncation of the radiated spherical wave is equivalent to truncation of the plane wave spectrum of the small source. So, the plane wave components in this far-field spherical wave, which lie beyond the edge angle of the lens, are not captured by the lens and are not transferred over to the image plane. Note: this logic is valid only for small sources, such that the lens is in the far field region of the source, according to the 2D2/λ criterion mentioned previously. If an object plane transparency is imagined as a summation over small sources (as in the Whittaker–Shannon interpolation formula, Scott [1990]), each of which has its spectrum truncated in this fashion, then every point of the entire object plane transparency suffers the same effects of this low pass filtering.
Loss of the high (spatial) frequency content causes blurring and loss of sharpness (see discussion related to point spread function). Bandwidth truncation causes a (fictitious, mathematical, ideal) point source in the object plane to be blurred (or, spread out) in the image plane, giving rise to the term, "point spread function." Whenever bandwidth is expanded or contracted, image size is typically contracted or expanded accordingly, in such a way that the space-bandwidth product remains constant, by Heisenberg's principle (Scott [1998] and Abbe sine condition).
Coherence and Fourier transforming | Fourier optics | Wikipedia | 437 | 312008 | https://en.wikipedia.org/wiki/Fourier%20optics | Physical sciences | Optics | Physics |
While working in the frequency domain, with an assumed ejωt (engineering) time dependence, coherent (laser) light is implicitly assumed, which has a delta function dependence in the frequency domain. Light at different (delta function) frequencies will "spray" the plane wave spectrum out at different angles, and as a result these plane wave components will be focused at different places in the output plane. The Fourier transforming property of lenses works best with coherent light, unless there is some special reason to combine light of different frequencies, to achieve some special purpose.
Hardware implementation of the system transfer function: The 4F correlator
The theory on optical transfer functions presented in the section 5 is somewhat abstract. However, there is one very well known device which implements the system transfer function H in hardware using only 2 identical lenses and a transparency plate - the 4F correlator. Although one important application of this device would certainly be to implement the mathematical operations of cross-correlation and convolution, this device - 4 focal lengths long - actually serves a wide variety of image processing operations that go well beyond what its name implies. A diagram of a typical 4F correlator is shown in the figure below (click to enlarge). This device may be readily understood by combining the plane wave spectrum representation of the electric field (section 1.5) with the Fourier transforming property of quadratic lenses (section 6.1) to yield the optical image processing operations described in the section 5. | Fourier optics | Wikipedia | 305 | 312008 | https://en.wikipedia.org/wiki/Fourier%20optics | Physical sciences | Optics | Physics |
The 4F correlator is based on the convolution theorem from Fourier transform theory, which states that convolution in the spatial (x,y) domain is equivalent to direct multiplication in the spatial frequency (kx, ky) domain (aka: spectral domain). Once again, a plane wave is assumed incident from the left and a transparency containing one 2D function, f(x,y), is placed in the input plane of the correlator, located one focal length in front of the first lens. The transparency spatially modulates the incident plane wave in magnitude and phase, like on the left-hand side of eqn. (), and in so doing, produces a spectrum of plane waves corresponding to the FT of the transmittance function, like on the right-hand side of eqn. (). That spectrum is then formed as an "image" one focal length behind the first lens, as shown. A transmission mask containing the FT of the second function, g(x,y), is placed in this same plane, one focal length behind the first lens, causing the transmission through the mask to be equal to the product, F(kx,ky) × G(kx,ky). This product now lies in the "input plane" of the second lens (one focal length in front), so that the FT of this product (i.e., the convolution of f(x,y) and g(x,y)), is formed in the back focal plane of the second lens.
If an ideal, mathematical point source of light is placed on-axis in the input plane of the first lens, then there will be a uniform, collimated field produced in the output plane of the first lens. When this uniform, collimated field is multiplied by the FT plane mask, and then Fourier transformed by the second lens, the output plane field (which in this case is the impulse response of the correlator) is just our correlating function, g(x,y). In practical applications, g(x,y) will be some type of feature which must be identified and located within the input plane field (see Scott [1998]). In military applications, this feature may be a tank, ship or airplane which must be quickly identified within some more complex scene. | Fourier optics | Wikipedia | 492 | 312008 | https://en.wikipedia.org/wiki/Fourier%20optics | Physical sciences | Optics | Physics |
The 4F correlator is an excellent device for illustrating the "systems" aspects of optical instruments, alluded to in the section 5 above. The FT plane mask function, G(kx,ky) is the system transfer function of the correlator, which we'd in general denote as H(kx,ky), and it is the FT of the impulse response function of the correlator, h(x,y) which is just our correlating function g(x,y). And, as mentioned above, the impulse response of the correlator is just a picture of the feature we're trying to find in the input image. In the 4F correlator, the system transfer function H(kx,ky) is directly multiplied against the spectrum F(kx,ky) of the input function, to produce the spectrum of the output function. This is how electrical signal processing systems operate on 1D temporal signals.
Image restoration
Image blurring by a point spread function is studied extensively in optical information processing, one way to alleviate the blurring is to adopt Wiener Filter. For example, assume that is the intensity distribution from an incoherent object, is the intensity distribution of its image which is blurred by a space-invariant point-spread function and a noise introduced in the detection process:
The goal of image restoration is to find a linear restoration filter that minimize the mean-squared error between the true distribution and the estimation . That is, to minimize
The solution of this optimization problem is Wiener filter:
where , , are the power spectral densities of the point-spread function, the object and the noise.
Ragnarsson proposed a method to realize Wiener restoration filters optically by holographic technique like setup shown in the figure. The derivation of the function of the setup is described as follows.
Assume there is a transparency as the recording plane and an impulse emitted from a point source S. The wave of impulse is collimated by lens L1, forming a distribution equal to the impulse response . Then the distribution is then split into two parts: | Fourier optics | Wikipedia | 433 | 312008 | https://en.wikipedia.org/wiki/Fourier%20optics | Physical sciences | Optics | Physics |
The upper portion is first focused (i.e., Fourier transformed) by a lens L2 to a spot in the front focal plan of lens L3, forming a virtual point source generating a spherical wave. The wave is then collimated by lens L3 and produces a tilted plane wave with the form at the recording plane.
The lower portion is directly collimated by lens L3'', yielding an amplitude distribution .
Therefore, the total intensity distribution is
Assume has an amplitude distribution and a phase distribution such that
then we can rewrite intensity as follows:
Note that for the point at the origin of the film plane (), the recorded wave from the lower portion should be much stronger than that from the upper portion because the wave passing through the lower path is focused, which leads to the relationship .
In Ragnarsson' s work, this method is based on the following postulates:
Assume there is a transparency, with its amplitude transmittance proportional to , that has recorded the known impulse response of the blurred system.
The maximum phase shift introduced by the filter is much smaller than radians so that .
The phase shift of the transparency after bleaching is linearly proportional to the silver density present before bleaching.
The density is linearly proportional to the logarithm of exposure
The average exposure is much stronger than varying exposure
By these postulates, we have the following relationship:
Finally, we get a amplitude transmittance with the form of a Wiener filter:
Afterword: Plane wave spectrum within the broader context of functional decomposition | Fourier optics | Wikipedia | 313 | 312008 | https://en.wikipedia.org/wiki/Fourier%20optics | Physical sciences | Optics | Physics |
Electrical fields can be represented mathematically in many different ways. In the Huygens–Fresnel or Stratton-Chu viewpoints, the electric field is represented as a superposition of point sources, each one of which gives rise to a Green's function field. The total field is then the weighted sum of all of the individual Green's function fields. That seems to be the most natural way of viewing the electric field for most people - no doubt because most of us have, at one time or another, drawn out the circles with protractor and paper, much the same way Thomas Young did in his classic paper on the double-slit experiment. However, it is by no means the only way to represent the electric field, which may also be represented as a spectrum of sinusoidally varying plane waves. In addition, Frits Zernike proposed still another functional decomposition based on his Zernike polynomials, defined on the unit disc. The third-order (and lower) Zernike polynomials correspond to the normal lens aberrations. And still another functional decomposition could be made in terms of Sinc functions and Airy functions, as in the Whittaker–Shannon interpolation formula and the Nyquist–Shannon sampling theorem. All of these functional decompositions have utility in different circumstances. The optical scientist having access to these various representational forms has available a richer insight to the nature of these marvelous fields and their properties. These different ways of looking at the field are not conflicting or contradictory, rather, by exploring their connections, one can often gain deeper insight into the nature of wave fields.
Functional decomposition and eigenfunctions
The twin subjects of eigenfunction expansions and functional decomposition, both briefly alluded to here, are not completely independent. The eigenfunction expansions to certain linear operators defined over a given domain, will often yield a countably infinite set of orthogonal functions which will span that domain. Depending on the operator and the dimensionality (and shape, and boundary conditions) of its domain, many different types of functional decompositions are, in principle, possible. | Fourier optics | Wikipedia | 433 | 312008 | https://en.wikipedia.org/wiki/Fourier%20optics | Physical sciences | Optics | Physics |
VMware LLC is an American cloud computing and virtualization technology company headquartered in Palo Alto, California. VMware was the first commercially successful company to virtualize the x86 architecture.
VMware's desktop software runs on Microsoft Windows, Linux, and macOS. VMware ESXi, its enterprise software hypervisor, is an operating system that runs on server hardware.
On November 22, 2023, Broadcom Inc. acquired VMware in a cash-and-stock transaction valued at US$69 billion, with the End-User Computing (EUC) division of VMware then sold to KKR and rebranded to Omnissa.
History
Early history
In 1998, VMware was founded by Diane Greene, Mendel Rosenblum, Scott Devine, Ellen Wang, and Edouard Bugnion. Greene and Rosenblum were graduate students at the University of California, Berkeley. Edouard Bugnion remained the chief architect and CTO of VMware until 2005 and went on to found Nuova Systems (now part of Cisco). VMware operated in stealth mode for the first year, with roughly 20 employees by the end of 1998. The company was launched officially early in the second year, in February 1999, at the DEMO conference organized by Chris Shipley. The first product, VMware Workstation, was delivered in May 1999, and the company entered the server market in 2001 with VMware GSX Server (hosted) and VMware ESX Server (host-less).
In 2003, VMware launched VMware Virtual Center, vMotion, and Virtual Symmetric Multi-Processing (SMP) technology. 64-bit support was introduced in 2004.
Acquisition by EMC
On January 9, 2004, under the terms of the definitive agreement announced on December 15, 2003, EMC (now Dell EMC) acquired the company with US$625 million in cash. On August 14, 2007, EMC sold 15% of VMware to the public via an initial public offering. Shares were priced at per share and closed the day at . | VMware | Wikipedia | 430 | 312018 | https://en.wikipedia.org/wiki/VMware | Technology | Virtualization | null |
On July 8, 2008, after disappointing financial performance, the board of directors fired VMware co-founder, president and CEO Diane Greene, who was replaced by Paul Maritz, a 14-year Microsoft veteran who was heading EMC's cloud computing business unit. Greene had been CEO since the company's founding, ten years earlier. On September 10, 2008, Mendel Rosenblum, the company's co-founder, chief scientist, and the husband of Diane Greene, resigned.
On September 16, 2008, VMware announced a collaboration with Cisco Systems. One result was the Cisco Nexus 1000V, a distributed virtual software switch, an integrated option in the VMware infrastructure.
In April 2011, EMC transferred control of the Mozy backup service to VMware.
On April 12, 2011, VMware released an open-source platform-as-a-service system called Cloud Foundry, as well as a hosted version of the service. This supported application deployment for Java, Ruby on Rails, Sinatra, Node.js, and Scala, as well as database support for MySQL, MongoDB, Redis, PostgreSQL, and RabbitMQ.
In August 2012, Pat Gelsinger was appointed as the new CEO of VMware, coming over from EMC. Paul Maritz went over to EMC as Head of Strategy before moving on to lead the Pivotal spin-off.
In March 2013, VMware announced the corporate spin-off of Pivotal Software, with General Electric investing in the company. Most of VMware's application- and developer-oriented products, including Spring, tc Server, Cloud Foundry, RabbitMQ, GemFire, and SQLFire were transferred to this organization.
In May 2013, VMware launched its own IaaS service, vCloud Hybrid Service, at its new Palo Alto headquarters (vCloud Hybrid Service was rebranded vCloud Air and later sold to cloud provider OVH), announcing an early access program in a Las Vegas data center. The service is designed to function as an extension of its customer's existing vSphere installations, with full compatibility with existing virtual machines virtualized with VMware software and tightly integrated networking. The service is based on vCloud Director 5.1/vSphere 5.1. | VMware | Wikipedia | 481 | 312018 | https://en.wikipedia.org/wiki/VMware | Technology | Virtualization | null |
In September 2013, at VMworld San Francisco, VMware announced the general availability of vCloud Hybrid Service and expansion to Sterling, Virginia, Santa Clara, California, Dallas, Texas, and a service beta in the UK. It announced the acquisition of Desktone in October 2013.
Acquisition by Dell
In January 2016, in anticipation of Dell's acquisition of EMC, VMware announced a restructuring to reduce about 800 positions, and some executives resigned. The entire development team behind VMware Workstation and Fusion was disbanded and all US developers were immediately fired. On April 24, 2016, maintenance release 12.1.1 was released. On September 8, 2016, VMware announced the release of Workstation 12.5 and Fusion 8.5 as a free upgrade supporting Windows 10 and Windows Server 2016.
In April 2016, VMware president and COO Carl Eschenbach left VMware to join Sequoia Capital, and Martin Casado, VMware's general manager for its Networking and Security business, left to join Andreessen Horowitz. Analysts commented that the cultures at Dell and EMC, and at EMC and VMware, are different, and said that they had heard that impending corporate cultural collisions and potentially radical product overlap pruning, would cause many EMC and VMware personnel to leave; VMware CEO Pat Gelsinger, following rumors, categorically denied that he would leave.
In August 2016 VMware introduced the VMware Cloud Provider website.
Mozy was transferred to Dell in 2016 after the merger of Dell and EMC.
In April 2017, according to Glassdoor, VMware was ranked 3rd on the list of highest paying companies in the United States.
In Q2 2017, VMware sold vCloud Air to French cloud service provider OVH.
On January 13, 2021, VMware announced that CEO Pat Gelsinger would be leaving to step in at Intel. Intel is where Gelsinger spent 30 years of his career and was Intel's first chief technology officer. CFO Zane Rowe became interim CEO while the board searched for a replacement.
Spinoff from Dell
On April 15, 2021, it was reported that Dell would spin off its remaining stake in VMware to shareholders and that the two companies would continue to operate without major changes for at least five years. The spinoff was completed on November 1, 2021. | VMware | Wikipedia | 498 | 312018 | https://en.wikipedia.org/wiki/VMware | Technology | Virtualization | null |
On May 12, 2021, VMware announced that Raghu Raghuram would take over as CEO.
In May 2022, VMware announced that the company had partnered with Formula One motor racing team, McLaren Racing.
Log4Shell vulnerability
Beginning in January 2022, hackers infiltrated servers using the Log4Shell vulnerability at organizations who failed to implement available patches released by VMware according to PCMag. ZDNET reported in March 2022 that hackers utilized Log4Shell on some customers' VMware servers to install backdoors and for cryptocurrency mining. In May 2022, Bleeping Computer reported that the Lazarus Group cybercrime group, which is possibly linked to North Korea, was actively using Log4Shell "to inject backdoors that fetch information-stealing payloads on VMware Horizon servers", including VMware Horizon.
Acquisition by Broadcom
On May 26, 2022, Broadcom announced its intention to acquire VMware for approximately $61 billion in cash and stock in addition to assuming $8 billion of VMware's net debt, and that Broadcom Software Group would rebrand and operate as VMware.
In November 2022, the UK's Competition and Markets Authority regulator announced it would investigate whether the acquisition would "result in a substantial lessening of competition within any market or markets in the United Kingdom for goods or services".
The transaction closed on November 22, 2023, after a prolonged delay in getting approval from the Chinese regulator on an additional condition that VMware's server software should maintain compatibility with third-party hardware and not require the use of Broadcom's hardware products. On completion, Broadcom reorganized the company into four divisions: VMware Cloud Foundation, Tanzu, Software-Defined Edge, and Application Networking and Security, and subsequently laid off over 2,800 employees. Broadcom also relocated its headquarters from North San Jose to VMware's headquarters campus in Palo Alto.
On December 13, 2023, VMware ended availability for perpetually licensed products such as vSphere and Cloud Foundation, moving exclusively to subscription-based offerings. The company stated that this had been planned as an eventuality prior to the Broadcom acquisition. | VMware | Wikipedia | 467 | 312018 | https://en.wikipedia.org/wiki/VMware | Technology | Virtualization | null |
In February 2024 private equity firm KKR and Broadcom agreed for KKR to acquire Broadcom's End-User Computing (EUC) Division, formerly a division of VMware, for about $4 billion. This includes the VDI product Horizon and the device management suite Workspace ONE, formerly AirWatch.
On May 14, 2024, it was announced that VMware Workstation Pro and VMware Fusion Pro would be made free for personal use, with commercial use still requiring payment. In November 2024, VMware announced that commercial use would be free too.
Acquisitions
Litigation
In March 2015, the Software Freedom Conservancy announced it was funding litigation by Christoph Hellwig in Hamburg, Germany against VMware for alleged violation of his copyrights in its ESXi product.
Hellwig's core claim is that ESXi is a derivative work of the GPLv2-licensed Linux kernel 2.4, and therefore VMware is not in compliance
with GPLv2 because it does not publish the source code to ESXi. VMware publicly stated that ESXi is not a derivative of the Linux kernel, denying Hellwig's
core claim. VMware said it offered a way to use Linux device drivers with ESXi, and that code does use some Linux GPLv2-licensed code and so it had published the source, meeting GPLv2 requirements.
The lawsuit was dismissed by the court in July 2016 and Hellwig announced he would file an appeal. The appeal was decided February 2019 and again dismissed by German court, on the basis of not meeting "procedural requirements for the burden of proof of the plaintiff."
In May 2023, VMware was ordered to pay $84.5 million for patent infringement on two patents belonging to Densify, a Canadian software company.
Current products
VMware's most notable products are its hypervisors. VMware became well known for its first type 2 hypervisor known as VMware Workstation. This product has since evolved into two additional hypervisor product lines: VMware's type 1 hypervisors running directly on hardware (ESX/ESXi) and their discontinued hosted type 2 hypervisors (GSX). | VMware | Wikipedia | 462 | 312018 | https://en.wikipedia.org/wiki/VMware | Technology | Virtualization | null |
VMware software provides a completely virtualized set of hardware to the guest operating system. VMware software virtualizes the hardware for a video adapter, a network adapter, and hard disk adapters. The host provides pass-through drivers for guest USB, serial, and parallel devices. In this way, VMware virtual machines become highly portable between computers, because every host looks nearly identical to the guest. In practice, a system administrator can pause operations on a virtual machine guest, move or copy that guest to another physical computer, and there resume execution exactly at the point of suspension. Alternatively, for enterprise servers, a feature called vMotion allows the migration of operational guest virtual machines between similar but separate hardware hosts sharing the same storage (or, with vMotion Storage, separate storage can be used, too). Each of these transitions is completely transparent to any users on the virtual machine at the time it is being migrated.
VMware's products predate the virtualization extensions to the x86 instruction set, and do not require virtualization-enabled processors. On newer processors, the hypervisor is now designed to take advantage of the extensions. However, unlike many other hypervisors, VMware still supports older processors. In such cases, it uses the CPU to run code directly whenever possible (as, for example, when running user-mode and virtual 8086 mode code on x86). When direct execution cannot operate, such as with kernel-level and real-mode code, VMware products use binary translation (BT) to re-write the code dynamically. The translated code gets stored in spare memory, typically at the end of the address space, which segmentation mechanisms can protect and make invisible. For these reasons, VMware operates dramatically faster than emulators, running at more than 80% of the speed that the virtual guest operating system would run directly on the same hardware. In one study VMware claims a slowdown over native ranging from 0–6 percent for the VMware ESX Server. | VMware | Wikipedia | 421 | 312018 | https://en.wikipedia.org/wiki/VMware | Technology | Virtualization | null |
Desktop software
VMware Workstation, introduced in 1999, was the first product launched by VMware. This software suite allows users to run multiple instances of x86 or x86-64-compatible operating systems on a single physical personal computer. Version 17.0 was released on November 17, 2022. Originally a commercial app, WMware Workstation has become freeware in December 2024.
VMware Fusion (discontinued on 30 April 2024), provides similar functionality for users of the Intel Mac platform, along with full compatibility with virtual machines created by other VMware products.
VMware Workstation Player (discontinued) was freeware for non-commercial use, without requiring a license, and available for commercial use with permission. It is similar to VMware Workstation, with some features not available, including support for UEFI Secure Boot, snapshots, encrypted virtual machines, and some advanced features.
Server software
VMware ESXi, an enterprise software product, can deliver greater performance than the freeware VMware Server, due to lower system computational overhead. VMware ESXi, as a "bare-metal" product, runs directly on the server hardware, allowing virtual servers to also use hardware more or less directly. In addition, VMware ESXi integrates into VMware vCenter, which offers extra services.
Cloud management software
VMware Suite – a cloud management platform purpose-built for a hybrid cloud. VMware vRealize Hyperic was acquired from SpringSource and subsequently discontinued in 2020.
VMware Go is a web-based service to guide users of any expertise level through the installation and configuration of VMware vSphere Hypervisor.
VMware Cloud Foundation – Cloud Foundation provides an easy way to deploy and operate a private cloud on an integrated SDDC system.
VMware Horizon View is a virtual desktop infrastructure (VDI) product.
vSphere+ and vSAN+ – activates add-on hybrid cloud services for business-critical applications running on-premises, including IT disaster recovery and ransomware protection
Application management
VMware Workspace Portal was a self-service app store for workspace management.
Provisioning
PlateSpin (does Provisioning)
Storage and availability
VMware's storage and availability products are composed of two primary offerings: | VMware | Wikipedia | 480 | 312018 | https://en.wikipedia.org/wiki/VMware | Technology | Virtualization | null |
VMware vSAN (previously called VMware Virtual SAN) is software-defined storage that is embedded in VMware's ESXi hypervisor. The vSphere and vSAN software runs on industry-standard x86 servers to form a hyper-converged infrastructure (or HCI). However, network operators need to have servers from HCL (Hardware Compatibility List) to put one into production. The first release, version 5.5, was released in March 2014. The 6th generation, version 6.6, was released in April 2017. New features available in VMware vSAN 6.6 include native data at rest encryption, local protection for stretched clusters, analytics, and optimized solid-state drive performance. The VMWare 6.7 version was released in April 2018.
VMware Site Recovery Manager (SRM) automates the failover and failback of virtual machines to and from a secondary site using policy-based management.
Networking and security products
VMware NSX is VMware's network virtualization product marketed using the term software-defined data center (SDDC). The technology included some acquired from the 2012 purchase of Nicira. Software Defined Networking (SDN) allows the same policies that govern Identity and Access Management (IAM) to dictate levels of access to applications and data through a totally converged infrastructure not possible with legacy network and system access methods.
Other products
Workspace ONE allows mobile users to access apps and data.
The VIX (Virtual Infrastructure eXtension) API allows automated or scripted management of a computer virtualized using either VMware's vSphere, Workstation, Player, or Fusion products. VIX provides bindings for the programming languages C, Perl, Visual Basic, VBScript and C#.
Herald is a communications protocol from VMware for more reliable Bluetooth communication and range finding for mobile devices. Herald code is available under an open-source license and was implemented in the Australian Government's COVIDSafe app for contact tracing on 19 December 2020. | VMware | Wikipedia | 425 | 312018 | https://en.wikipedia.org/wiki/VMware | Technology | Virtualization | null |
Cowrie or cowry () is the common name for a group of small to large sea snails in the family Cypraeidae.
The term porcelain derives from the old Italian term for the cowrie shell () due to their similar appearance.
Cowrie shells have held cultural, economic, and ornamental significance in various cultures. The cowrie was the shell most widely used worldwide as shell money. It is most abundant in the Indian Ocean, and was collected in the Maldive Islands, in Sri Lanka, along the Indian Malabar coast, in Borneo and on other East Indian islands, in Maluku in the Pacific, and in various parts of the African coast from Ras Hafun to Mozambique. Cowrie shell money was important in the trade networks of Africa, South Asia, and East Asia.
In the United States and Mexico, cowrie species inhabit the waters off Central California to Baja California (the chestnut cowrie is the only cowrie species native to the eastern Pacific Ocean off the coast of the United States; further south, off the coast of Mexico, Central America and Peru, Little Deer Cowrie habitat can be found; and further into the Pacific from Central America, the Pacific habitat range of Money Cowrie can be reached) as well as the waters south of the Southeastern United States.
Some species in the family Ovulidae are also often referred to as cowries. In the British Isles the local Trivia species (family Triviidae, species Trivia monacha and Trivia arctica) are sometimes called cowries. The Ovulidae and the Triviidae are other families within Cypraeoidea, the superfamily of cowries and their close relatives.
Etymology
The word cowrie comes from Hindi (), which is itself derived from Sanskrit ().
Shell description
The shells of cowries are usually smooth and shiny and more or less egg-shaped. The round side of the shell is called the Dorsal Face, whereas the flat under side is called the Ventral Face, which shows a long, narrow, slit-like opening (aperture), which is often toothed at the edges. The narrower end of the egg-shaped cowrie shell is the anterior end, and the broader end of the shell is called the posterior. The spire of the shell is not visible in the adult shell of most species, but is visible in juveniles, which have a different shape from the adults. | Cowrie | Wikipedia | 486 | 312029 | https://en.wikipedia.org/wiki/Cowrie | Biology and health sciences | Gastropods | Animals |
Nearly all cowries have a porcelain-like shine, with some exceptions such as Hawaii's granulated cowrie, Nucleolaria granulata. Many have colorful patterns. Lengths range from for some species up to for the Atlantic deer cowrie, Macrocypraea cervus.
Human use
Monetary use
Cowrie shells, especially Monetaria moneta, were used for centuries as currency by native Africans. In his book Marriage and Morals, Bertrand Russell attributed the use of cowrie shells as currency in ancient Egypt to the similarity between shape of the shell and that of female genitalia. Additionally, the money cowrie was almost impossible to counterfeit until the late 19th Century. After the 1500s, however, the shell's use as currency became even more common. Western nations, chiefly through the slave trade, introduced huge numbers of Maldivian cowries in Africa. The Ghanaian cedi was named after cowrie shells. Starting over three thousand years ago, cowrie shells, or copies of the shells, were used as Chinese currency. They were also used as means of exchange in India.
The Classical Chinese character for money (貝) originated as a stylized drawing of a Maldivian cowrie shell. Words and characters concerning money, property or wealth usually have this as a radical. Before the Spring and Autumn period the cowrie was used as a type of trade token awarding access to a feudal lord's resources to a worthy vassal.
Ritual use
The Ojibwe aboriginal people in North America use cowrie shells which are called sacred miigis shells or whiteshells in Midewiwin ceremonies, and the Whiteshell Provincial Park in Manitoba, Canada is named after this type of shell. There is some debate about how the Ojibway traded for or found these shells, so far inland and so far north, very distant from the natural habitat. Oral stories and birch bark scrolls seem to indicate that the shells were found in the ground, or washed up on the shores of lakes or rivers. Finding the cowrie shells so far inland could indicate the previous use of them by an earlier tribe or group in the area, who may have obtained them through an extensive trade network in the ancient past.
In Eastern India, particularly in West Bengal, it is given as a token price for the ferry ride of the departed soul to cross the river "Vaitarani". Cowries are used during cremation. Cowries are also used in the worship of Goddess Laxmi. | Cowrie | Wikipedia | 505 | 312029 | https://en.wikipedia.org/wiki/Cowrie | Biology and health sciences | Gastropods | Animals |
In Brazil, as a result of the Atlantic slave trade from Africa, cowrie shells (called búzios) are also used to consult the Orixás divinities and hear their replies.
Cowrie shells were among the devices used for divination by the Kaniyar Panicker astrologers of Kerala, India.
In certain parts of Africa, cowries were prized charms, and they were said to be associated with fecundity, sexual pleasure and good luck. It is also used in the treatment of certain diseases such as rashes and ringworm when it is burnt into ashes.
In Pre-dynastic Egypt and Neolithic Southern Levant, cowrie shells were placed in the graves of young girls. The modified Levantine cowries were discovered ritually arranged around the skull in female burials. During the Bronze Age, cowries became more common as funerary goods, also associated with burials of women and children.
Jewelry
Cowrie shells are also worn as jewelry or otherwise used as ornaments or charms. In Mende culture, cowrie shells are viewed as symbols of womanhood, fertility, birth and wealth. Its underside is supposed, by one modern ethnographic author, to represent a vulva or an eye.
On the Fiji Islands, a shell of the golden cowrie or bulikula, Cypraea aurantium, was drilled at the ends and worn on a string around the neck by chieftains as a badge of rank. The women of Tuvalu use cowrie and other shells in traditional handicrafts.
Games and gambling
Cowrie shells are sometimes used in a way similar to dice, e.g., in board games like Pachisi, Ashta Chamma or in divination (cf. Ifá and the annual customs of Dahomey of Benin). A number of shells (6 or 7 in Pachisi) are thrown, with those landing aperture upwards indicating the actual number rolled.
In Nepal cowries are used for a gambling game, where 16 pieces of cowries are tossed by four different bettors (and sub-bettors under them). This game is usually played at homes and in public during the Hindu festival of Tihar or Deepawali. In the same festival these shells are also worshiped as a symbol of Goddess Lakshmi and wealth. | Cowrie | Wikipedia | 472 | 312029 | https://en.wikipedia.org/wiki/Cowrie | Biology and health sciences | Gastropods | Animals |
Other
Large cowrie shells such as that of a Cypraea tigris have been used in Europe in the recent past as a darning egg over which sock heels were stretched. The cowrie's smooth surface allows the needle to be positioned under the cloth more easily.
In the 1940s and 1950s, small cowry shells were used as a teaching aid in infant schools e.g counting, adding, subtracting. | Cowrie | Wikipedia | 86 | 312029 | https://en.wikipedia.org/wiki/Cowrie | Biology and health sciences | Gastropods | Animals |
Bryophytes () are a group of land plants (embryophytes), sometimes treated as a taxonomic division, that contains three groups of non-vascular land plants: the liverworts, hornworts, and mosses. In the strict sense, the division Bryophyta consists of the mosses only. Bryophytes are characteristically limited in size and prefer moist habitats although some species can survive in drier environments. The bryophytes consist of about 20,000 plant species. Bryophytes produce enclosed reproductive structures (gametangia and sporangia), but they do not produce flowers or seeds. They reproduce sexually by spores and asexually by fragmentation or the production of gemmae.
Though bryophytes were considered a paraphyletic group in recent years, almost all of the most recent phylogenetic evidence supports the monophyly of this group, as originally classified by Wilhelm Schimper in 1879.
The term bryophyte comes .
Features
The defining features of bryophytes are:
Their life cycles are dominated by a multicellular haploid gametophyte stage
Their sporophytes are diploid and unbranched
They do not have a true vascular tissue containing lignin (although some have specialized tissues for the transport of water)
Habitat
Bryophytes exist in a wide variety of habitats. They can be found growing in a range of temperatures (cold arctics and in hot deserts), elevations (sea-level to alpine), and moisture (dry deserts to wet rain forests). Bryophytes can grow where vascularized plants cannot because they do not depend on roots for uptake of nutrients from soil. Bryophytes can survive on rocks and bare soil.
Life cycle
Like all land plants (embryophytes), bryophytes have life cycles with alternation of generations. In each cycle, a haploid gametophyte, each of whose cells contains a fixed number of unpaired chromosomes, alternates with a diploid sporophyte, whose cells contain two sets of paired chromosomes. Gametophytes produce haploid sperm and eggs which fuse to form diploid zygotes that grow into sporophytes. Sporophytes produce haploid spores by meiosis, that grow into gametophytes. | Bryophyte | Wikipedia | 480 | 312249 | https://en.wikipedia.org/wiki/Bryophyte | Biology and health sciences | Bryophytes | null |
Bryophytes are gametophyte dominant, meaning that the more prominent, longer-lived plant is the haploid gametophyte. The diploid sporophytes appear only occasionally and remain attached to and nutritionally dependent on the gametophyte. In bryophytes, the sporophytes are always unbranched and produce a single sporangium (spore producing capsule), but each gametophyte can give rise to several sporophytes at once.
Liverworts, mosses and hornworts spend most of their lives as gametophytes. Gametangia (gamete-producing organs), archegonia and antheridia, are produced on the gametophytes, sometimes at the tips of shoots, in the axils of leaves or hidden under thalli. Some bryophytes, such as the liverwort Marchantia, create elaborate structures to bear the gametangia that are called gametangiophores. Sperm are flagellated and must swim from the antheridia that produce them to archegonia which may be on a different plant. Arthropods can assist in transfer of sperm.
Fertilized eggs become zygotes, which develop into sporophyte embryos inside the archegonia. Mature sporophytes remain attached to the gametophyte. They consist of a stalk called a seta and a single sporangium or capsule. Inside the sporangium, haploid spores are produced by meiosis. These are dispersed, most commonly by wind, and if they land in a suitable environment can develop into a new gametophyte. Thus bryophytes disperse by a combination of swimming sperm and spores, in a manner similar to lycophytes, ferns and other cryptogams. | Bryophyte | Wikipedia | 381 | 312249 | https://en.wikipedia.org/wiki/Bryophyte | Biology and health sciences | Bryophytes | null |
The sporophyte develops differently in the three groups. Both mosses and hornworts have a meristem zone where cell division occurs. In hornworts, the meristem starts at the base where the foot ends, and the division of cells pushes the sporophyte body upwards. In mosses, the meristem is located between the capsule and the top of the stalk (seta), and produces cells downward, elongating the stalk and elevating the capsule. In liverworts the meristem is absent and the elongation of the sporophyte is caused almost exclusively by cell expansion.
Sexuality
The arrangement of antheridia and archegonia on an individual bryophyte plant is usually constant within a species, although in some species it may depend on environmental conditions. The main division is between species in which the antheridia and archegonia occur on the same plant and those in which they occur on different plants. The term monoicous may be used where antheridia and archegonia occur on the same gametophyte and the term dioicous where they occur on different gametophytes.
In seed plants, "monoecious" is used where flowers with anthers (microsporangia) and flowers with ovules (megasporangia) occur on the same sporophyte and "dioecious" where they occur on different sporophytes. These terms occasionally may be used instead of "monoicous" and "dioicous" to describe bryophyte gametophytes. "Monoecious" and "monoicous" are both derived from the Greek for "one house", "dioecious" and "dioicous" from the Greek for two houses. The use of the "-oicy" terminology refers to the gametophyte sexuality of bryophytes as distinct from the sporophyte sexuality of seed plants. | Bryophyte | Wikipedia | 406 | 312249 | https://en.wikipedia.org/wiki/Bryophyte | Biology and health sciences | Bryophytes | null |
Monoicous plants are necessarily hermaphroditic, meaning that the same plant produces gametes of both sexes. The exact arrangement of the antheridia and archegonia in monoicous plants varies. They may be borne on different shoots (autoicous), on the same shoot but not together in a common structure (paroicous or paroecious), or together in a common "inflorescence" (synoicous or synoecious). Dioicous plants are unisexual, meaning that an individual plant has only one sex. All four patterns (autoicous, paroicous, synoicous and dioicous) occur in species of the moss genus Bryum.
Classification and phylogeny
Traditionally, all living land plants without vascular tissues were classified in a single taxonomic group, often a division (or phylum). The term "Bryophyta" was first suggested by Braun in 1864. As early as 1879, the term Bryophyta was used by German bryologist Wilhelm Schimper to describe a group containing all three bryophyte clades (though at the time, hornworts were considered part of the liverworts). G.M. Smith placed this group between Algae and Pteridophyta. Although a 2005 study supported this traditional monophyletic view, by 2010 a broad consensus had emerged among systematists that bryophytes as a whole are not a natural group (i.e., are paraphyletic). However, a 2014 study concluded that these previous phylogenies, which were based on nucleic acid sequences, were subject to composition biases, and that, furthermore, phylogenies based on amino acid sequences suggested that the bryophytes are monophyletic after all. Since then, partially thanks to a proliferation of genomic and transcriptomic datasets, almost all phylogenetics studies based on nuclear and chloroplastic sequences have concluded that the bryophytes form a monophyletic group. Nevertheless, phylogenies based on mitochondrial sequences fail to support the monophyletic view. | Bryophyte | Wikipedia | 445 | 312249 | https://en.wikipedia.org/wiki/Bryophyte | Biology and health sciences | Bryophytes | null |
The three bryophyte clades are the Marchantiophyta (liverworts), Bryophyta (mosses) and Anthocerotophyta (hornworts). However, it has been proposed that these clades are de-ranked to the classes Marchantiopsida, Bryopsida, and Anthocerotopsida, respectively. There is now strong evidence that the liverworts and mosses belong to a monophyletic clade, called Setaphyta.
Monophyletic view
The favored model, based on amino acids phylogenies, indicates bryophytes as a monophyletic group:
Consistent with this view, compared to other living land plants, all three lineages lack vascular tissue containing lignin and branched sporophytes bearing multiple sporangia. The prominence of the gametophyte in the life cycle is also a shared feature of the three bryophyte lineages (extant vascular plants are all sporophyte dominant). However, if this phylogeny is correct, then the complex sporophyte of living vascular plants might have evolved independently of the simpler unbranched sporophyte present in bryophytes. Furthermore, this view implies that stomata evolved only once in plant evolution, before being subsequently lost in the liverworts.
Paraphyletic view
In this alternative view, the Setaphyta grouping is retained, but hornworts instead are sister to vascular plants. (Another paraphyletic view involves hornworts branching out first.)
Traditional morphology
Traditionally, when basing classifications on morphological characters, bryophytes have been distinguished by their lack of vascular structure. However, this distinction is problematic, firstly because some of the earliest-diverging (but now extinct) non-bryophytes, such as the horneophytes, did not have true vascular tissue, and secondly because many mosses have well-developed water-conducting vessels. A more useful distinction may lie in the structure of their sporophytes. In bryophytes, the sporophyte is a simple unbranched structure with a single spore-forming organ (sporangium), whereas in all other land plants, the polysporangiophytes, the sporophyte is branched and carries many sporangia. The contrast is shown in the cladogram below: | Bryophyte | Wikipedia | 501 | 312249 | https://en.wikipedia.org/wiki/Bryophyte | Biology and health sciences | Bryophytes | null |
Evolution
There have probably been several different terrestrialization events, in which originally aquatic organisms colonized the land, just within the lineage of the Viridiplantae. Between 510 and 630 million years ago, however, land plants emerged within the green algae. Molecular phylogenetic studies conclude that bryophytes are the earliest diverging lineages of the extant land plants. They provide insights into the migration of plants from aquatic environments to land. A number of physical features link bryophytes to both land plants and aquatic plants.
Similarities to algae and vascular plants
Green algae, bryophytes and vascular plants all have chlorophyll a and b, and the chloroplast structures are similar. Like green algae and land plants, bryophytes also produce starch stored in the plastids and contain cellulose in their walls. Distinct adaptations observed in bryophytes have allowed plants to colonize Earth's terrestrial environments. To prevent desiccation of plant tissues in a terrestrial environment, a waxy cuticle covering the soft tissue of the plant may be present, providing protection. In hornworts and mosses, stomata provide gas exchange between the atmosphere and an internal intercellular space system. The development of gametangia provided further protection specifically for gametes, the zygote and the developing sporophyte. The bryophytes and vascular plants (embryophytes) also have embryonic development which is not seen in green algae. While bryophytes have no truly vascularized tissue, they do have organs that are specialized for transport of water and other specific functions, analogous for example to the functions of leaves and stems in vascular land plants.
Bryophytes depend on water for reproduction and survival. In common with ferns and lycophytes, a thin layer of water is required on the surface of the plant to enable the movement of the flagellated sperm between gametophytes and the fertilization of an egg.
Comparative morphology
Summary of the morphological characteristics of the gametophytes of the three groups of bryophytes:
Summary of the morphological characteristics of the sporophytes of the three groups of bryophytes: | Bryophyte | Wikipedia | 451 | 312249 | https://en.wikipedia.org/wiki/Bryophyte | Biology and health sciences | Bryophytes | null |
Uses
Environmental
Characteristics of bryophytes make them useful to the environment. Depending on the specific plant texture, bryophytes have been shown to help improve the water retention and air space within soil. Bryophytes are used in pollution studies to indicate soil pollution (such as the presence of heavy metals), air pollution, and UV-B radiation. Gardens in Japan are designed with moss to create peaceful sanctuaries. Some bryophytes have been found to produce natural pesticides. The liverwort, Plagiochila, produces a chemical that is poisonous to mice. Other bryophytes produce chemicals that are antifeedants which protect them from being eaten by slugs. When Phythium sphagnum is sprinkled on the soil of germinating seeds, it inhibits growth of "damping off fungus" which would otherwise kill young seedlings.
Commercial
Peat is a fuel produced from dried bryophytes, typically Sphagnum. Bryophytes' antibiotic properties and ability to retain water make them a useful packaging material for vegetables, flowers, and bulbs. Also, because of its antibiotic properties, Sphagnum was used as a surgical dressing in World War I. | Bryophyte | Wikipedia | 254 | 312249 | https://en.wikipedia.org/wiki/Bryophyte | Biology and health sciences | Bryophytes | null |
In physics, a partition function describes the statistical properties of a system in thermodynamic equilibrium. Partition functions are functions of the thermodynamic state variables, such as the temperature and volume. Most of the aggregate thermodynamic variables of the system, such as the total energy, free energy, entropy, and pressure, can be expressed in terms of the partition function or its derivatives. The partition function is dimensionless.
Each partition function is constructed to represent a particular statistical ensemble (which, in turn, corresponds to a particular free energy). The most common statistical ensembles have named partition functions. The canonical partition function applies to a canonical ensemble, in which the system is allowed to exchange heat with the environment at fixed temperature, volume, and number of particles. The grand canonical partition function applies to a grand canonical ensemble, in which the system can exchange both heat and particles with the environment, at fixed temperature, volume, and chemical potential. Other types of partition functions can be defined for different circumstances; see partition function (mathematics) for generalizations. The partition function has many physical meanings, as discussed in Meaning and significance.
Canonical partition function
Definition
Initially, let us assume that a thermodynamically large system is in thermal contact with the environment, with a temperature T, and both the volume of the system and the number of constituent particles are fixed. A collection of this kind of system comprises an ensemble called a canonical ensemble. The appropriate mathematical expression for the canonical partition function depends on the degrees of freedom of the system, whether the context is classical mechanics or quantum mechanics, and whether the spectrum of states is discrete or continuous.
Classical discrete system
For a canonical ensemble that is classical and discrete, the canonical partition function is defined as
where
is the index for the microstates of the system;
is Euler's number;
is the thermodynamic beta, defined as where is the Boltzmann constant;
is the total energy of the system in the respective microstate.
The exponential factor is otherwise known as the Boltzmann factor.
Classical continuous system
In classical mechanics, the position and momentum variables of a particle can vary continuously, so the set of microstates is actually uncountable. In classical statistical mechanics, it is rather inaccurate to express the partition function as a sum of discrete terms. In this case we must describe the partition function using an integral rather than a sum. For a canonical ensemble that is classical and continuous, the canonical partition function is defined as | Partition function (statistical mechanics) | Wikipedia | 508 | 312252 | https://en.wikipedia.org/wiki/Partition%20function%20%28statistical%20mechanics%29 | Physical sciences | Statistical mechanics | Physics |
where
is the Planck constant;
is the thermodynamic beta, defined as ;
is the Hamiltonian of the system;
is the canonical position;
is the canonical momentum.
To make it into a dimensionless quantity, we must divide it by h, which is some quantity with units of action (usually taken to be the Planck constant).
Classical continuous system (multiple identical particles)
For a gas of identical classical non-interacting particles in three dimensions, the partition function is
where
is the Planck constant;
is the thermodynamic beta, defined as ;
is the index for the particles of the system;
is the Hamiltonian of a respective particle;
is the canonical position of the respective particle;
is the canonical momentum of the respective particle;
is shorthand notation to indicate that and are vectors in three-dimensional space.
is the classical continuous partition function of a single particle as given in the previous section.
The reason for the factorial factor N! is discussed below. The extra constant factor introduced in the denominator was introduced because, unlike the discrete form, the continuous form shown above is not dimensionless. As stated in the previous section, to make it into a dimensionless quantity, we must divide it by h3N (where h is usually taken to be the Planck constant).
Quantum mechanical discrete system
For a canonical ensemble that is quantum mechanical and discrete, the canonical partition function is defined as the trace of the Boltzmann factor:
where:
is the trace of a matrix;
is the thermodynamic beta, defined as ;
is the Hamiltonian operator.
The dimension of is the number of energy eigenstates of the system.
Quantum mechanical continuous system
For a canonical ensemble that is quantum mechanical and continuous, the canonical partition function is defined as
where:
is the Planck constant;
is the thermodynamic beta, defined as ;
is the Hamiltonian operator;
is the canonical position;
is the canonical momentum.
In systems with multiple quantum states s sharing the same energy Es, it is said that the energy levels of the system are degenerate. In the case of degenerate energy levels, we can write the partition function in terms of the contribution from energy levels (indexed by j) as follows:
where gj is the degeneracy factor, or number of quantum states s that have the same energy level defined by Ej = Es. | Partition function (statistical mechanics) | Wikipedia | 484 | 312252 | https://en.wikipedia.org/wiki/Partition%20function%20%28statistical%20mechanics%29 | Physical sciences | Statistical mechanics | Physics |
The above treatment applies to quantum statistical mechanics, where a physical system inside a finite-sized box will typically have a discrete set of energy eigenstates, which we can use as the states s above. In quantum mechanics, the partition function can be more formally written as a trace over the state space (which is independent of the choice of basis):
where is the quantum Hamiltonian operator. The exponential of an operator can be defined using the exponential power series.
The classical form of Z is recovered when the trace is expressed in terms of coherent states and when quantum-mechanical uncertainties in the position and momentum of a particle are regarded as negligible. Formally, using bra–ket notation, one inserts under the trace for each degree of freedom the identity:
where is a normalised Gaussian wavepacket centered at position x and momentum p. Thus
A coherent state is an approximate eigenstate of both operators and , hence also of the Hamiltonian , with errors of the size of the uncertainties. If and can be regarded as zero, the action of reduces to multiplication by the classical Hamiltonian, and reduces to the classical configuration integral.
Connection to probability theory
For simplicity, we will use the discrete form of the partition function in this section. Our results will apply equally well to the continuous form.
Consider a system S embedded into a heat bath B. Let the total energy of both systems be E. Let pi denote the probability that the system S is in a particular microstate, i, with energy Ei. According to the fundamental postulate of statistical mechanics (which states that all attainable microstates of a system are equally probable), the probability pi will be inversely proportional to the number of microstates of the total closed system (S, B) in which S is in microstate i with energy Ei. Equivalently, pi will be proportional to the number of microstates of the heat bath B with energy :
Assuming that the heat bath's internal energy is much larger than the energy of S (), we can Taylor-expand to first order in Ei and use the thermodynamic relation , where here , are the entropy and temperature of the bath respectively:
Thus
Since the total probability to find the system in some microstate (the sum of all pi) must be equal to 1, we know that the constant of proportionality must be the normalization constant, and so, we can define the partition function to be this constant: | Partition function (statistical mechanics) | Wikipedia | 511 | 312252 | https://en.wikipedia.org/wiki/Partition%20function%20%28statistical%20mechanics%29 | Physical sciences | Statistical mechanics | Physics |
Calculating the thermodynamic total energy
In order to demonstrate the usefulness of the partition function, let us calculate the thermodynamic value of the total energy. This is simply the expected value, or ensemble average for the energy, which is the sum of the microstate energies weighted by their probabilities:
or, equivalently,
Incidentally, one should note that if the microstate energies depend on a parameter λ in the manner
then the expected value of A is
This provides us with a method for calculating the expected values of many microscopic quantities. We add the quantity artificially to the microstate energies (or, in the language of quantum mechanics, to the Hamiltonian), calculate the new partition function and expected value, and then set λ to zero in the final expression. This is analogous to the source field method used in the path integral formulation of quantum field theory.
Relation to thermodynamic variables
In this section, we will state the relationships between the partition function and the various thermodynamic parameters of the system. These results can be derived using the method of the previous section and the various thermodynamic relations.
As we have already seen, the thermodynamic energy is
The variance in the energy (or "energy fluctuation") is
The heat capacity is
In general, consider the extensive variable X and intensive variable Y where X and Y form a pair of conjugate variables. In ensembles where Y is fixed (and X is allowed to fluctuate), then the average value of X will be:
The sign will depend on the specific definitions of the variables X and Y. An example would be X = volume and Y = pressure. Additionally, the variance in X will be
In the special case of entropy, entropy is given by
where A is the Helmholtz free energy defined as , where is the total energy and S is the entropy, so that
Furthermore, the heat capacity can be expressed as
Partition functions of subsystems
Suppose a system is subdivided into N sub-systems with negligible interaction energy, that is, we can assume the particles are essentially non-interacting. If the partition functions of the sub-systems are ζ1, ζ2, ..., ζN, then the partition function of the entire system is the product of the individual partition functions:
If the sub-systems have the same physical properties, then their partition functions are equal, ζ1 = ζ2 = ... = ζ, in which case | Partition function (statistical mechanics) | Wikipedia | 512 | 312252 | https://en.wikipedia.org/wiki/Partition%20function%20%28statistical%20mechanics%29 | Physical sciences | Statistical mechanics | Physics |
However, there is a well-known exception to this rule. If the sub-systems are actually identical particles, in the quantum mechanical sense that they are impossible to distinguish even in principle, the total partition function must be divided by a N! (N factorial):
This is to ensure that we do not "over-count" the number of microstates. While this may seem like a strange requirement, it is actually necessary to preserve the existence of a thermodynamic limit for such systems. This is known as the Gibbs paradox.
Meaning and significance
It may not be obvious why the partition function, as we have defined it above, is an important quantity. First, consider what goes into it. The partition function is a function of the temperature T and the microstate energies E1, E2, E3, etc. The microstate energies are determined by other thermodynamic variables, such as the number of particles and the volume, as well as microscopic quantities like the mass of the constituent particles. This dependence on microscopic variables is the central point of statistical mechanics. With a model of the microscopic constituents of a system, one can calculate the microstate energies, and thus the partition function, which will then allow us to calculate all the other thermodynamic properties of the system.
The partition function can be related to thermodynamic properties because it has a very important statistical meaning. The probability Ps that the system occupies microstate s is
Thus, as shown above, the partition function plays the role of a normalizing constant (note that it does not depend on s), ensuring that the probabilities sum up to one: | Partition function (statistical mechanics) | Wikipedia | 343 | 312252 | https://en.wikipedia.org/wiki/Partition%20function%20%28statistical%20mechanics%29 | Physical sciences | Statistical mechanics | Physics |
This is the reason for calling Z the "partition function": it encodes how the probabilities are partitioned among the different microstates, based on their individual energies. Other partition functions for different ensembles divide up the probabilities based on other macrostate variables. As an example: the partition function for the isothermal-isobaric ensemble, the generalized Boltzmann distribution, divides up probabilities based on particle number, pressure, and temperature. The energy is replaced by the characteristic potential of that ensemble, the Gibbs Free Energy. The letter Z stands for the German word Zustandssumme, "sum over states". The usefulness of the partition function stems from the fact that the macroscopic thermodynamic quantities of a system can be related to its microscopic details through the derivatives of its partition function. Finding the partition function is also equivalent to performing a Laplace transform of the density of states function from the energy domain to the β domain, and the inverse Laplace transform of the partition function reclaims the state density function of energies.
Grand canonical partition function
We can define a grand canonical partition function for a grand canonical ensemble, which describes the statistics of a constant-volume system that can exchange both heat and particles with a reservoir. The reservoir has a constant temperature T, and a chemical potential μ.
The grand canonical partition function, denoted by , is the following sum over microstates
Here, each microstate is labelled by , and has total particle number and total energy . This partition function is closely related to the grand potential, , by the relation
This can be contrasted to the canonical partition function above, which is related instead to the Helmholtz free energy.
It is important to note that the number of microstates in the grand canonical ensemble may be much larger than in the canonical ensemble, since here we consider not only variations in energy but also in particle number. Again, the utility of the grand canonical partition function is that it is related to the probability that the system is in state :
An important application of the grand canonical ensemble is in deriving exactly the statistics of a non-interacting many-body quantum gas (Fermi–Dirac statistics for fermions, Bose–Einstein statistics for bosons), however it is much more generally applicable than that. The grand canonical ensemble may also be used to describe classical systems, or even interacting quantum gases. | Partition function (statistical mechanics) | Wikipedia | 488 | 312252 | https://en.wikipedia.org/wiki/Partition%20function%20%28statistical%20mechanics%29 | Physical sciences | Statistical mechanics | Physics |
The grand partition function is sometimes written (equivalently) in terms of alternate variables as
where is known as the absolute activity (or fugacity) and is the canonical partition function. | Partition function (statistical mechanics) | Wikipedia | 37 | 312252 | https://en.wikipedia.org/wiki/Partition%20function%20%28statistical%20mechanics%29 | Physical sciences | Statistical mechanics | Physics |
Chinampa ( ) is a technique used in Mesoamerican agriculture which relies on small, rectangular areas of fertile arable land to grow crops on the shallow lake beds in the Valley of Mexico. The word chinampa has Nahuatl origins, chinampa meaning “in the fence of reeds”. They are built up on wetlands of a lake or freshwater swamp for agricultural purposes, and their proportions ensure optimal moisture retention. This method was also used and occupied most of Lake Xochimilco. The United Nations designated it a Globally Important Agricultural Heritage System in 2018.
Although different technologies existed during the Post-classic and Colonial periods in the basin, chinampas have raised many questions on agricultural production and political development. After the Aztec Triple Alliance formed, the conquest of southern basin city-states, such as Xochimilco, was one of the first strategies of imperial expansion. Before this time, farmers maintained small-scale chinampas adjacent to their households and communities in the freshwater lakes of Xochimilco and Chalco. The Aztecs did not invent chinampas but rather were the first to develop it to a large scale cultivation. Sometimes referred to as "floating gardens," chinampas are artificial islands that were created by interweaving reeds with stakes beneath the lake's surface, creating underwater fences. A buildup of soil and aquatic vegetation would be piled into these "fences" until the top layer of soil was visible on the water's surface. | Chinampa | Wikipedia | 306 | 312259 | https://en.wikipedia.org/wiki/Chinampa | Technology | Buildings and infrastructure | null |
When creating chinampas, in addition to building up masses of land, a drainage system was developed. This drainage system was multi-purposed. A ditch was created to allow for the flow of water and sediments (likely including night soil). Over time, the ditch would slowly accumulate piles of mud. This mud would then be dug up and placed on top of the chinampas, clearing the blockage. The soil from the bottom of the lake was also rich in nutrients, thus acting as an efficient and effective way of fertilizing the chinampas. Replenishing the topsoil with lost nutrients provided for bountiful harvests. Embarcadero-Jiménez and colleagues tested the correlation between environmental parameters and bacterial diversity in the soil. It is speculated that a diverse array of bacteria can affect the nutrients in the soil. The results found that bacterial diversity was more abundant in cultivated soils than non-cultivated soils. Also, "the structure of the bacterial communities showed that the chinampas are a transition system between sediment and soil and revealed an interesting association of the S-cycle and iron-oxidizing bacteria with the rhizosphere of plants grown in the chinampa soil". | Chinampa | Wikipedia | 248 | 312259 | https://en.wikipedia.org/wiki/Chinampa | Technology | Buildings and infrastructure | null |
Evidence from Nahuatl wills from late seventeenth-century Pueblo Culhuacán suggests chinampas were measured in matl (one matl = 1.67 meters), often listed in groups of seven. One scholar has calculated the size of chinampas using Codex Vergara as a source, finding that they usually measured roughly . In Tenochtitlan, the chinampas ranged from to They were created by staking out the shallow lake bed and then fencing in the rectangle with wattle. The fenced-off area was then layered with mud, lake sediment, and decaying vegetation, eventually bringing it above the level of the lake. Often trees such as āhuexōtl (Salix bonplandiana) (a willow) and āhuēhuētl (Taxodium mucronatum) (a cypress) were planted at the corners to secure the chinampa. In some places, the long raised beds had ditches in between them, giving plants continuous access to water and making crops grown there independent of rainfall. Chinampas were separated by channels wide enough for a canoe to pass. These raised, well-watered beds had very high crop yields with up to 7 harvests a year. Chinampas were commonly used in pre-colonial Mexico and Central America. There is evidence that the Nahua settlement of Culhuacan, on the south side of the Ixtapalapa peninsula that divided Lake Texcoco from Lake Xochimilco, constructed the first chinampas in C.E. 1100.
History | Chinampa | Wikipedia | 318 | 312259 | https://en.wikipedia.org/wiki/Chinampa | Technology | Buildings and infrastructure | null |
The earliest fields that have been securely dated are from the Middle Postclassic period, 1150 – 1350 CE. Chinampas were used primarily in Lakes Xochimilco and Chalco near the springs that lined the south shore of those lakes. The Aztecs not only conducted military campaigns to obtain control over these regions but, according to some researchers, undertook significant state-led efforts to increase their extent. There is some strong evidence to suggest state-led operations for the “expansion” of the chinampas. This is sometimes referred to as the hydraulic hypothesis, which is directly related to a hydraulic empire, which is an empire that maintains power and control through the regulation and distribution of water. There is evidence to support the idea of state involvement, primarily the amount of manpower and materials it would take to build, turn, and maintain the chinampas. However, arguments about state control of the chinampas rely upon the assumption that dikes were necessary to control the water levels and to keep the saline water of Lake Texcoco away from the freshwater of the chinampa zone. This is plausible, but there is evidence that the chinampas were functional before the construction of a dike that protected them from the saline water. It is suggested that the dike was meant to drastically improve the size of the chinampa operation.
Chinampa farms also ringed Tenochtitlán, the Aztec capital, which was considerably enlarged over time. Smaller-scale farms have also been identified near the island-city of Xaltocan and on the east side of Lake Texcoco. With the destruction of the dams and sluice gates during the Spanish conquest of the Aztec Empire, many chinampas fields were abandoned. However, many lakeshore towns retained their chinampas through the end of the colonial era since cultivation was highly labor-intensive and less attractive for Spaniards to acquire.
The Aztecs built Tenochtitlan on an island around 1325. Issues arose when the cities' constant expansion eventually caused them to run out of room to build. As the empire grew, more sources of food were required. At times this meant conquering more land; at other times it meant expanding the chinampa system. With this expansion, chinampas' multiple crops per year became a large factor in the production and supply of food. Empirical records suggest that farmers had a relatively light tribute to pay compared to others because the annual tribute may have been only a fraction of the amount necessary for local needs. | Chinampa | Wikipedia | 509 | 312259 | https://en.wikipedia.org/wiki/Chinampa | Technology | Buildings and infrastructure | null |
The extent to which Tenochtitlan depended on chinampas for its fresh food supply has been the topic of a number of scholarly studies.
Among the crops grown on chinampas were maize, beans, squash, amaranth, tomatoes, chili peppers, and flowers. Maize was planted with digging stick huictli with a wooden blade on one end.
The word chinampa comes from the Nahuatl word chināmitl, meaning "square made of canes" and the Nahuatl locative, "pan." In documentation by Spaniards, they used the word camellones, "ridges between the rows." However, Franciscan Fray Juan de Torquemada described them with the Nahua term, chinampa, saying "without much trouble [the Indians] plant and harvest their maize and greens, for all over there are ridges called chinampas; these were strips built above water and surrounded by ditches, which obviates watering."
Chinampas are depicted in pictorial Aztec codices, including Codex Vergara, Codex Santa María Asunción, the so-called Uppsala Map, and the Maguey Plan (from Azcapotzalco). In alphabetic Nahuatl documentation, The Testaments of Culhuacan from the late sixteenth century have numerous references to chinampas as property that individuals bequeathed to their heirs in written wills.
There are still remnants of the chinampa system in Xochimilco, the southern portion of greater Mexico City. Chinampas have been promoted as a model for modern sustainable agriculture, although some sources have disputed the applicability of this model. One anthropologist, for instance, reports that attempts by Mexico to develop chinampas among the Chontal Maya people in the 1970s failed until the technicians modified their goals in order to suit the Chontales' interests.
Construction: | Chinampa | Wikipedia | 379 | 312259 | https://en.wikipedia.org/wiki/Chinampa | Technology | Buildings and infrastructure | null |
According to Antonio Vera, through the UH Hilo website, within the framework of chinampas, there was two versions; inland and irrigated chinampas. Inland’s are created on banks, irrigated is built on water. Through steps, the structure of chinampas is to locate shallow land by the bank and surround said area with stakes of a common wetland tree [ahuejote]. The urbanization of Mexico lost this tradition and new challenges are created within the urbanization of Mexico. (https://hilo.hawaii.edu/nihopeku/2018/02/02/chinampa-an-ancient-agricultural-system/)
Modern chinampas
As of 1998, chinampas are still present in San Gregorio, a small town east of Xochimilco, in addition to San Luis, Tlahuac, and Mixquic. Although many of these gardens were constructed and thoroughly tended to from the Postclassic Period through the Spanish conquest, many of these plots of land still exist and are in active use.
Many of these chinampas have been allowed by present-day farmers to become overgrown. Some choose to use canoes to farm, but many are becoming increasingly dependent on wheelbarrows and bicycles for transportation. Other fields, such as some located in San Gregorio and San Luis areas, have been deliberately filled up. As the canals dry up, several of the fields are naturally joined. Although not used for their original purpose, they are commonly used for cattle feed.
Other fields, both dried and surrounded by canals, produce foods such as lettuce, cilantro, spinach, chard, squash, parsley, coriander, cauliflower, celery, mint, chives, rosemary, corn, and radishes. The young leaves of quelites and quintoniles, which are often mistaken for weeds, are grown and harvested as ingredients of sauces. Flowers also continue to be grown on these plots. Some chinampa fields are also used as tourist sites. | Chinampa | Wikipedia | 428 | 312259 | https://en.wikipedia.org/wiki/Chinampa | Technology | Buildings and infrastructure | null |
Challenges
Although many locals and farmers are happy to return to their agricultural roots, they are faced with several challenges. During the Spanish conquest, many lakes were drained, limiting their agricultural capacity, such as the lake at Xochimilco. In addition, in 1985 an earthquake struck, further damaging several canals. Other challenges include limited water supply, the use of pesticides, climate change, urban sprawl, and water pollution caused by untreated sewage and toxic waste. | Chinampa | Wikipedia | 95 | 312259 | https://en.wikipedia.org/wiki/Chinampa | Technology | Buildings and infrastructure | null |
In complex analysis, Liouville's theorem, named after Joseph Liouville (although the theorem was first proven by Cauchy in 1844), states that every bounded entire function must be constant. That is, every holomorphic function for which there exists a positive number such that for all is constant. Equivalently, non-constant holomorphic functions on have unbounded images.
The theorem is considerably improved by Picard's little theorem, which says that every entire function whose image omits two or more complex numbers must be constant.
Statement
Liouville's theorem: Every holomorphic function for which there exists a positive number such that for all is constant.
More succinctly, Liouville's theorem states that every bounded entire function must be constant.
Proof
This important theorem has several proofs.
A standard analytical proof uses the fact that holomorphic functions are analytic.
Another proof uses the mean value property of harmonic functions.
The proof can be adapted to the case where the harmonic function is merely bounded above or below. See Harmonic function#Liouville's theorem.
Corollaries
Fundamental theorem of algebra
There is a short proof of the fundamental theorem of algebra using Liouville's theorem.
No entire function dominates another entire function
A consequence of the theorem is that "genuinely different" entire functions cannot dominate each other, i.e. if and are entire, and everywhere, then for some complex number . Consider that for the theorem is trivial so we assume . Consider the function . It is enough to prove that can be extended to an entire function, in which case the result follows by Liouville's theorem. The holomorphy of is clear except at points in . But since is bounded and all the zeroes of are isolated, any singularities must be removable. Thus can be extended to an entire bounded function which by Liouville's theorem implies it is constant.
If f is less than or equal to a scalar times its input, then it is linear
Suppose that is entire and , for . We can apply Cauchy's integral formula; we have that
where is the value of the remaining integral. This shows that is bounded and entire, so it must be constant, by Liouville's theorem. Integrating then shows that is affine and then, by referring back to the original inequality, we have that the constant term is zero. | Liouville's theorem (complex analysis) | Wikipedia | 498 | 312293 | https://en.wikipedia.org/wiki/Liouville%27s%20theorem%20%28complex%20analysis%29 | Mathematics | Complex analysis | null |
Non-constant elliptic functions cannot be defined on the complex plane
The theorem can also be used to deduce that the domain of a non-constant elliptic function cannot be . Suppose it was. Then, if and are two periods of such that is not real, consider the parallelogram whose vertices are 0, , , and . Then the image of is equal to . Since is continuous and is compact, is also compact and, therefore, it is bounded. So, is constant.
The fact that the domain of a non-constant elliptic function cannot be is what Liouville actually proved, in 1847, using the theory of elliptic functions. In fact, it was Cauchy who proved Liouville's theorem.
Entire functions have dense images
If is a non-constant entire function, then its image is dense in . This might seem to be a much stronger result than Liouville's theorem, but it is actually an easy corollary. If the image of is not dense, then there is a complex number and a real number
such that the open disk centered at with radius has no element of the image of . Define
Then is a bounded entire function, since for all ,
So, is constant, and therefore is constant.
On compact Riemann surfaces
Any holomorphic function on a compact Riemann surface is necessarily constant.
Let be holomorphic on a compact Riemann surface . By compactness, there is a point where attains its maximum. Then we can find a chart from a neighborhood of to the unit disk such that is holomorphic on the unit disk and has a maximum at , so it is constant, by the maximum modulus principle.
Remarks
Let be the one-point compactification of the complex plane . In place of holomorphic functions defined on regions in , one can consider regions in . Viewed this way, the only possible singularity for entire functions, defined on , is the point . If an entire function is bounded in a neighborhood of , then is a removable singularity of , i.e. cannot blow up or behave erratically at . In light of the power series expansion, it is not surprising that Liouville's theorem holds. | Liouville's theorem (complex analysis) | Wikipedia | 450 | 312293 | https://en.wikipedia.org/wiki/Liouville%27s%20theorem%20%28complex%20analysis%29 | Mathematics | Complex analysis | null |
Similarly, if an entire function has a pole of order at —that is, it grows in magnitude comparably to in some neighborhood of
—then is a polynomial. This extended version of Liouville's theorem can be more precisely stated: if for sufficiently large, then is a polynomial of degree at most . This can be proved as follows. Again take the Taylor series representation of ,
The argument used during the proof using Cauchy estimates shows that for all ,
So, if , then
Therefore, .
Liouville's theorem does not extend to the generalizations of complex numbers known as double numbers and dual numbers. | Liouville's theorem (complex analysis) | Wikipedia | 127 | 312293 | https://en.wikipedia.org/wiki/Liouville%27s%20theorem%20%28complex%20analysis%29 | Mathematics | Complex analysis | null |
An avalanche photodiode (APD) is a highly sensitive type of photodiode, which in general are semiconductor diodes that convert light into electricity via interband excitation coupled with impact ionization. APDs use materials and a structure optimised for operating with high reverse bias, approaching the reverse breakdown voltage, such that charge carriers generated by the photoelectric effect are multiplied by an avalanche breakdown; thus they can be used to detect relatively small amounts of light.
From a functional standpoint, they can be regarded as the semiconductor analog of photomultiplier tubes; unlike solar cells, they are not optimised for generating electricity from light but rather for detection of incoming photons. Typical applications for APDs are laser rangefinders, long-range fiber-optic telecommunication, positron emission tomography, and particle physics.
History
The avalanche photodiode was invented by Japanese engineer Jun-ichi Nishizawa in 1952. However, study of avalanche breakdown, micro-plasma defects in silicon and germanium and the investigation of optical detection using p-n junctions predate this patent.
Principle of operation
Photodiodes generally operate by impact ionization, whereby a photon provides the energy to separate charge carriers in the semiconductor material into a positive and negative pair, which can thus cause a charge flow through the diode. By applying a high reverse bias voltage, any photoelectric effect in the diode can be multiplied by the avalanche effect. Thus, the APD can be thought of as applying a high gain effect to the induced photocurrent.
In general, the higher the reverse voltage, the higher the gain. A standard silicon APD typically can sustain 100–200 V of reverse bias before breakdown, leading to a gain factor of around 100. However, by employing alternative doping and bevelling (structural) techniques compared to traditional APDs, a it is possible to create designs where greater voltage can be applied (> 1500 V) before breakdown is reached, and hence a greater operating gain (> 1000) is achieved.
Among the various expressions for the APD multiplication factor (M), an instructive expression is given by the formula
where L is the space-charge boundary for electrons, and is the multiplication coefficient for electrons (and holes). This coefficient has a strong dependence on the applied electric field strength, temperature, and doping profile. Since APD gain varies strongly with the applied reverse bias and temperature, it is necessary to closely monitor the reverse voltage to keep a stable gain. | Avalanche photodiode | Wikipedia | 512 | 312299 | https://en.wikipedia.org/wiki/Avalanche%20photodiode | Technology | Components | null |
Geiger mode counting
If very high gain is needed (105 to 106), detectors related to APDs called SPADs (single-photon avalanche diodes) can be used and operated with a reverse voltage above a typical APD's breakdown voltage. In this case, the photodetector needs to have its signal current limited and quickly diminished. Active and passive current-quenching techniques have been used for this purpose. SPADs that operate in this high-gain regime are sometimes referred to being in Geiger mode. This mode is particularly useful for single-photon detection, provided that the dark count event rate and afterpulsing probability are sufficiently low.
Materials
In principle, any semiconductor material can be used as a multiplication region:
Silicon will detect in the visible and near infrared, with low multiplication noise (excess noise).
Germanium (Ge) will detect infrared out to a wavelength of 1.7 μm, but has high multiplication noise.
InGaAs will detect out to longer than 1.6 μm and has less multiplication noise than Ge. It is normally used as the absorption region of a heterostructure diode, most typically involving InP as a substrate and as a multiplication layer. This material system is compatible with an absorption window of roughly 0.9–1.7 μm. InGaAs exhibits a high absorption coefficient at the wavelengths appropriate to high-speed telecommunications using optical fibers, so only a few micrometres of InGaAs are required for nearly 100% light absorption. The excess noise factor is low enough to permit a gain-bandwidth product in excess of 100 GHz for a simple InP/InGaAs system, and up to 400 GHz for InGaAs on silicon. Therefore, high-speed operation is possible: commercial devices are available to speeds of at least 10 Gbit/s.
Gallium-nitride–based diodes have been used for operation with ultraviolet light.
HgCdTe-based diodes operate in the infrared, typically at wavelengths up to about 14 μm, but require cooling to reduce dark currents. Very low excess noise can be achieved in this material system.
Structure
APDs are often not constructed as simple p-n junctions but have more complex designs such as p+-i-p-n+. | Avalanche photodiode | Wikipedia | 467 | 312299 | https://en.wikipedia.org/wiki/Avalanche%20photodiode | Technology | Components | null |
Performance limits
APD applicability and usefulness depends on many parameters. Two of the larger factors are: quantum efficiency, which indicates how well incident optical photons are absorbed and then used to generate primary charge carriers; and total leakage current, which is the sum of the dark current, photocurrent and noise. Electronic dark-noise components are series and parallel noise. Series noise, which is the effect of shot noise, is basically proportional to the APD capacitance, while the parallel noise is associated with the fluctuations of the APD bulk and surface dark currents.
Gain noise, excess noise factor
Another noise source is the excess noise factor, ENF. It is a multiplicative correction applied to the noise that describes the increase in the statistical noise, specifically Poisson noise, due to the multiplication process. The ENF is defined for any device, such as photomultiplier tubes, silicon solid-state photomultipliers, and APDs, that multiplies a signal, and is sometimes referred to as "gain noise". At a gain M, it is denoted by ENF(M) and can often be expressed as
where is the ratio of the hole impact ionization rate to that of electrons. For an electron multiplication device it is given by the hole impact ionization rate divided by the electron impact ionization rate. It is desirable to have a large asymmetry between these rates to minimize ENF(M), since ENF(M) is one of the main factors that limit, among other things, the best possible energy resolution obtainable.
Conversion noise, Fano factor | Avalanche photodiode | Wikipedia | 333 | 312299 | https://en.wikipedia.org/wiki/Avalanche%20photodiode | Technology | Components | null |
The noise term for an APD may also contain a Fano factor, which is a multiplicative correction applied to the Poisson noise associated with the conversion of the energy deposited by a charged particle to the electron-hole pairs, which is the signal before multiplication. The correction factor describes the decrease in the noise, relative to Poisson statistics, due to the uniformity of conversion process and the absence of, or weak coupling to, bath states in the conversion process. In other words, an "ideal" semiconductor would convert the energy of the charged particle into an exact and reproducible number of electron hole pairs to conserve energy; in reality, however, the energy deposited by the charged particle is divided into the generation of electron hole pairs, the generation of sound, the generation of heat, and the generation of damage or displacement. The existence of these other channels introduces a stochastic process, where the amount of energy deposited into any single process varies from event to event, even if the amount of energy deposited is the same.
Further influences
The underlying physics associated with the excess noise factor (gain noise) and the Fano factor (conversion noise) is very different. However, the application of these factors as multiplicative corrections to the expected Poisson noise is similar. In addition to excess noise, there are limits to device performance associated with the capacitance, transit times and avalanche multiplication time. The capacitance increases with increasing device area and decreasing thickness. The transit times (both electrons and holes) increase with increasing thickness, implying a tradeoff between capacitance and transit time for performance. The avalanche multiplication time times the gain is given to first order by the gain-bandwidth product, which is a function of the device structure and most especially . | Avalanche photodiode | Wikipedia | 358 | 312299 | https://en.wikipedia.org/wiki/Avalanche%20photodiode | Technology | Components | null |
The Tarantula Nebula (also known as 30 Doradus) is a large H II region in the Large Magellanic Cloud (LMC), forming its south-east corner (from Earth's perspective).
Discovery
The Tarantula Nebula was observed by Nicolas-Louis de Lacaille during an expedition to the Cape of Good Hope between 1751 and 1753. He cataloged it as the second of the "Nebulae of the First Class", "Nebulosities not accompanied by any star visible in the telescope of two feet". It was described as a diffuse nebula 20' across.
Johann Bode included the Tarantula in his 1801 Uranographia star atlas and listed it in the accompanying Allgemeine Beschreibung und Nachweisung der Gestirne catalog as number 30 in the constellation "Xiphias or Dorado". Instead of being given a stellar magnitude, it was noted to be nebulous.
The name Tarantula Nebula arose in the mid-20th century from its appearance in deep photographic exposures.
30 Doradus has often been treated as the designation of a star, or of the central star cluster NGC 2070, but is now generally treated as referring to the whole nebula area of the Tarantula Nebula.
Properties
The Tarantula Nebula has an apparent magnitude of 8. Considering its distance of about 49 kpc (160,000 light-years), this is an extremely luminous non-stellar object. Its luminosity is so great that if it were as close to Earth as the Orion Nebula, the Tarantula Nebula would cast visible shadows. In fact, it is the most active starburst region known in the Local Group of galaxies.
It is also one of the largest H II regions in the Local Group with an estimated diameter around 200 to 570 pc (650 to 1860 light years), and also because of its very large size, it is sometimes described as the largest. However, other H II regions such as NGC 604, which is in the Triangulum Galaxy, could be larger. The nebula resides on the leading edge of the LMC where ram pressure stripping, and the compression of the interstellar medium likely resulting from this, is at a maximum.
NGC 2070 | Tarantula Nebula | Wikipedia | 463 | 312335 | https://en.wikipedia.org/wiki/Tarantula%20Nebula | Physical sciences | Notable nebulae | Astronomy |
30 Doradus has at its centre the star cluster NGC 2070 which includes the compact concentration of stars known as R136 that produces most of the energy that makes the nebula visible. The estimated mass of the cluster is 450,000 solar masses, suggesting it will likely become a globular cluster in the future. In addition to NGC 2070, the Tarantula Nebula contains several other star clusters including the much older Hodge 301. The most massive stars of Hodge 301 have already exploded in supernovae.
Supernova 1987A
The closest supernova observed since the invention of the telescope, Supernova 1987A, occurred in the outskirts of the Tarantula Nebula. There is a prominent supernova remnant enclosing the open cluster NGC 2060. Still, the remnants of many other supernovae are difficult to detect in the complex nebulosity.
Black hole VFTS 243
An x-ray quiet black hole was discovered in the Tarantula Nebula, the first outside of the Milky Way Galaxy that does not radiate strongly. The black hole has a mass of at least 9 solar masses and is in a circular orbit with its 25 solar mass blue giant companion VFTS 243. | Tarantula Nebula | Wikipedia | 243 | 312335 | https://en.wikipedia.org/wiki/Tarantula%20Nebula | Physical sciences | Notable nebulae | Astronomy |
In economics, unit of account is one of the functions of money. A unit of account is a standard numerical monetary unit of measurement of the market value of goods, services, and other transactions. Also known as a "measure" or "standard" of relative worth and deferred payment, a unit of account is a necessary prerequisite for the formulation of commercial agreements that involve debt.
Money acts as a standard measure and a common denomination of trade. It is thus a basis for quoting and bargaining of prices. It is necessary for developing efficient accounting systems.
Economics
Unit of account in economics allows a somewhat meaningful interpretation of prices, costs, and profits, so that an entity can monitor its own performance. It allows shareholders to make sense of its past performance and have an idea of its future profitability. The use of money, as a relatively stable unit of measure, can tend to drive market economies toward efficiency.
Historically, prices were often given in a dominant currency used as a unit of account, but transactions actually settled by using a variety of coins that were available, and often goods, all converted into their value in the unit of account. Many international transactions continue to be settled in this way, using a national value (most often expressed in the US dollar or euro) but with the actual settlement in something else.
In historical cost accounting, currencies are assumed to be perfectly stable in real value during non-hyperinflationary conditions under in terms of which the stable measuring unit assumption is applied. The Daily Consumer Price Index (Daily CPI) – or a monetized daily indexed unit of account – can be used to index monetary values on a daily basis when it is required to maintain the purchasing power or real value of monetary values constant during inflation and deflation.
Problems
Money is rarely perfectly stable in real value which is the fundamental problem with traditional historical cost accounting which is based on the stable measuring unit assumption. The unit of account in economics suffers from the pitfall of not being stable in real value over time because money is generally not perfectly stable in real value during inflation and deflation. Inflation destroys the assumption that the real value of the unit of account is stable which is the basis of classic accountancy. In such circumstances, historical values registered in accountancy books become heterogeneous amounts measured in different units. The use of such data under traditional accounting methods without previous correction can lead to confusing — (or even meaningless) — results. | Unit of account | Wikipedia | 494 | 312461 | https://en.wikipedia.org/wiki/Unit%20of%20account | Physical sciences | Measurement systems | Basics and measurement |
History
Historic examples of units of measure include the livre tournois, used in France from 1302 to 1794 whether or not livre coins were minted. In the 14th century Naples used the grossi gigliati, and Bohemia used the Prague groschen. (2021)
At any one time there might be two or three units of account in one region based on the local base, silver and sometimes gold coins, and each often expressed in L.S.D units in ratio 240:12:1. The Florentine gold florin, the French franc and the electoral rheingulden all became pounds (240 denari) of account. Units of account would often survive over 100 years despite the original coins changing composition and availability (e.g. the Castilian maravedi).
In 1921, Henry Ford proposed the use of energy as the basis for currency instead of Gold Standard. Thomas Edison similarly put forward commodities as a basis. At the onset of the Great Depression, John P. Norton restated the "Electric Dollars" standard alongside gold.
A modern unit of account is the European Currency Unit, used in the European Union from 1979 to 1998; its replacement in 1999, the Euro, was also just a unit of account until the introduction of notes and coins in 2002.
Unit of account is the main way of calculating a carrier or ship owner's liability in relation to carriage of goods contracts in which the Hague-Visby Rules apply.
In economics, a standard unit of account is used for statistical purposes to describe economic activity. Indexes such as GDP and the CPI are so broad in their scope that compiling them would be impossible without a standard unit of account. After being compiled, these figures are often used to guide governmental policy; especially monetary and fiscal policy.
In calculating the opportunity cost of a policy, a standard unit of account allows for the creation of a composite good. A composite good is a theoretical abstraction that represents an aggregation of all other opportunities that are not realized by the first good. It allows an economic decision's benefits to be weighed against the costs of all other possible goods in that society, without having to refer to any directly. Often, this is most easily accomplished with money.
Finance
The use of a unit of account in financial accounting, according to the American business model, allows investors to invest capital into those companies that provide the highest rate of return. The use of a unit of account in managerial accounting enables firms to choose between activities that yield the highest profit. | Unit of account | Wikipedia | 509 | 312461 | https://en.wikipedia.org/wiki/Unit%20of%20account | Physical sciences | Measurement systems | Basics and measurement |
Accounting
The unit of account in financial accounting refers to the words used to describe the specific assets and liabilities that are reported in financial statements rather than the units used to measure them. That is, unit of account refers to the object of recognition or display whereas unit of measure refers to the tool for measuring it.
Unit of measure and unit of account are sometimes treated as synonyms in financial accounting and economics. Unit of measure in financial accounting refers to the monetary unit to be used; that is, whether it should be nominal units of money as opposed to units that are adjusted for changes in purchasing power over time. | Unit of account | Wikipedia | 122 | 312461 | https://en.wikipedia.org/wiki/Unit%20of%20account | Physical sciences | Measurement systems | Basics and measurement |
Dots per inch (DPI, or dpi) is a measure of spatial printing, video or image scanner dot density, in particular the number of individual dots that can be placed in a line within the span of . Similarly, dots per centimetre (d/cm or dpcm) refers to the number of individual dots that can be placed within a line of .
DPI measurement in printing
DPI is used to describe the resolution number of dots per inch in a digital print and the printing resolution of a hard copy print dot gain, which is the increase in the size of the halftone dots during printing. This is caused by the spreading of ink on the surface of the media.
Up to a point, printers with higher DPI produce clearer and more detailed output. A printer does not necessarily have a single DPI measurement; it is dependent on print mode, which is usually influenced by driver settings. The range of DPI supported by a printer is most dependent on the print head technology it uses. A dot matrix printer, for example, applies ink via tiny rods striking an ink ribbon, and has a relatively low resolution, typically in the range of . An inkjet printer sprays ink through tiny nozzles, and is typically capable of 300–720 DPI. A laser printer applies toner through a controlled electrostatic charge, and may be in the range of 600 to 2,400 DPI.
The DPI measurement of a printer often needs to be considerably higher than the pixels per inch (PPI) measurement of a video display in order to produce similar-quality output. This is due to the limited range of colours for each dot typically available on a printer. At each dot position, the simplest type of color printer can either print no dot, or print a dot consisting of a fixed volume of ink in each of four color channels (typically CMYK with cyan, magenta, yellow and black ink) or 24 = 16 colours on laser, wax and most inkjet printers, of which only 14 or 15 (or as few as 8 or 9) may be actually discernible depending on the strength of the black component, the strategy used for overlaying and combining it with the other colours, and whether it is in "color" mode. | Dots per inch | Wikipedia | 461 | 312833 | https://en.wikipedia.org/wiki/Dots%20per%20inch | Physical sciences | Visual | Basics and measurement |
Higher-end inkjet printers can offer 5, 6 or 7 ink colours giving 32, 64 or 128 possible tones per dot location (and again, it can be that not all combinations will produce a unique result). Contrast this to a standard sRGB monitor where each pixel produces 256 intensities of light in each of three channels (RGB).
While some color printers can produce variable drop volumes at each dot position, and may use additional ink-color channels, the number of colours is still typically less than on a monitor. Most printers must therefore produce additional colours through a halftone or dithering process, and rely on their base resolution being high enough to "fool" the human observer's eye into perceiving a patch of a single smooth colour.
The exception to this rule is dye-sublimation printers, which can apply a much more variable amount of dye—close to or exceeding the number of the 256 levels per channel available on a typical monitor—to each "pixel" on the page without dithering, but with other limitations:
lower spatial resolution (typically 200 to 300 dpi), which can make text and lines look somewhat rough
lower output speed (a single page requiring three or four complete passes, one for each dye colour, each of which may take more than fifteen seconds—generally quicker, however, than most inkjet printers' "photo" modes)
a wasteful (and, for confidential documents, insecure) dye-film roll cartridge system
occasional color registration errors (mainly along the long axis of the page), which necessitate recalibrating the printer to account for slippage and drift in the paper feed system.
These disadvantages mean that, despite their marked superiority in producing good photographic and non-linear diagrammatic output, dye-sublimation printers remain niche products, and thus other devices using higher resolution, lower color depth, and dither patterns remain the norm. | Dots per inch | Wikipedia | 392 | 312833 | https://en.wikipedia.org/wiki/Dots%20per%20inch | Physical sciences | Visual | Basics and measurement |
This dithered printing process could require a region of four to six dots (measured across each side) to accurately reproduce the color in a single pixel. An image that is 100 pixels wide may need to be 400 to 600 dots in width in the printed output; if a 100 × 100-pixel image is to be printed in a one-inch square, the printer must be capable of 400 to 600 dots per inch to reproduce the image. As such, 600 dpi (sometimes 720) is now the typical output resolution of entry-level laser printers and some utility inkjet printers, with 1,200–1,440 and 2,400–2,880 being common "high" resolutions. This contrasts with the 300–360 (or 240) dpi of early models, and the approximate 200 dpi of dot-matrix printers and fax machines, which gave faxed and computer-printed documents—especially those that made heavy use of graphics or coloured block text—a characteristic "digitized" appearance, because of their coarse, obvious dither patterns, inaccurate colours, loss of clarity in photographs, and jagged ("aliased") edges on some text and line art.
DPI or PPI in digital image files | Dots per inch | Wikipedia | 249 | 312833 | https://en.wikipedia.org/wiki/Dots%20per%20inch | Physical sciences | Visual | Basics and measurement |
In printing, DPI (dots per inch) refers to the output resolution of a printer or imagesetter, and PPI (pixels per inch) refers to the input resolution of a photograph or image.
DPI refers to the physical dot density of an image when it is reproduced as a real physical entity, for example printed onto paper. A digitally stored image has no inherent physical dimensions, measured in inches or centimetres. Some digital file formats record a DPI value, or more commonly a PPI (pixels per inch) value, which is to be used when printing the image. This number lets the printer or software know the intended size of the image, or in the case of scanned images, the size of the original scanned object. For example, a bitmap image may measure 1,000 × 1,000 pixels, a resolution of 1 megapixel. If it is labelled as 250 PPI, that is an instruction to the printer to print it at a size of 4 × 4 inches. Changing the PPI to 100 in an image editing program would tell the printer to print it at a size of 10 × 10 inches. However, changing the PPI value would not change the size of the image in pixels which would still be 1,000 × 1,000. An image may also be resampled to change the number of pixels and therefore the size or resolution of the image, but this is quite different from simply setting a new PPI for the file. | Dots per inch | Wikipedia | 301 | 312833 | https://en.wikipedia.org/wiki/Dots%20per%20inch | Physical sciences | Visual | Basics and measurement |
For vector images, since the file is resolution independent, there is no need to resample the image before resizing it as it prints equally well at all sizes. However, there is still a target printing size. Some image formats, such as Photoshop format, can contain both bitmap and vector data in the same file. Adjusting the PPI in a Photoshop file will change the intended printing size of the bitmap portion of the data and also change the intended printing size of the vector data to match. This way the vector and bitmap data maintain a consistent size relationship when the target printing size is changed. Text stored as outline fonts in bitmap image formats is handled in the same way. Other formats, such as PDF, are primarily vector formats that can contain images, potentially at a mixture of resolutions. In these formats the target PPI of the bitmaps is adjusted to match when the target print size of the file is changed. This is the converse of how it works in a primarily bitmap format like Photoshop, but has exactly the same result of maintaining the relationship between the vector and bitmap portions of the data. | Dots per inch | Wikipedia | 238 | 312833 | https://en.wikipedia.org/wiki/Dots%20per%20inch | Physical sciences | Visual | Basics and measurement |
Computer monitor DPI standards
Since the 1980s, Macs have set the default display "DPI" to 72 PPI, while the Microsoft Windows operating system has used a default of 96 PPI. These default specifications arose out of the problems rendering standard fonts in the early display systems of the 1980s, including the IBM-based CGA, EGA, VGA and 8514 displays as well as the Macintosh displays featured in the 128K computer and its successors. The choice of 72 PPI by Macintosh for their displays arose from existing convention: the official 72 points per inch mirrored the 72 pixels per inch that appeared on their display screens. (Points are a physical unit of measure in typography, dating from the days of printing presses, where 1 point by the modern definition is of the international inch (25.4 mm), which therefore makes 1 point approximately 0.0139 in or 352.8 μm). Thus, the 72 pixels per inch seen on the display had exactly the same physical dimensions as the 72 points per inch later seen on a printout, with 1 pt in printed text equal to 1 px on the display screen. As it is, the Macintosh 128K featured a screen measuring 512 pixels in width by 342 pixels in height, and this corresponded to the width of standard office paper (512 px ÷ 72 px/in ≈ 7.1 in, with a 0.7 in margin down each side when assuming in × 11 in North American paper size; in the rest of the world, it is 210 mm × 297 mm – called A4. B5 is 176 mm × 250 mm).
A consequence of Apple's decision was that the widely used 10-point fonts from the typewriter era had to be allotted 10 display pixels in em height, and 5 display pixels in x-height. This is technically described as 10 pixels per em (PPEm). This made 10-point fonts be rendered crudely and made them difficult to read on the display screen, particularly the lowercase characters. Furthermore, there was the consideration that computer screens are typically viewed (at a desk) at a distance 30% greater than printed materials, causing a mismatch between the perceived sizes seen on the computer screen and those on the printouts. | Dots per inch | Wikipedia | 465 | 312833 | https://en.wikipedia.org/wiki/Dots%20per%20inch | Physical sciences | Visual | Basics and measurement |
Microsoft tried to solve both problems with a hack that has had long-term consequences for the understanding of what DPI and PPI mean. Microsoft began writing its software to treat the screen as though it provided a PPI characteristic that is of what the screen actually displayed. Because most screens at the time provided around 72 PPI, Microsoft essentially wrote its software to assume that every screen provides 96 PPI (because 72 × = 96). The short-term gain of this trickery was twofold:
It would seem to the software that one-third more pixels were available for rendering an image, thereby allowing for bitmap fonts to be created with greater detail.
On every screen that actually provided 72 PPI, each graphical element (such as a character of text) would be rendered at a size one third larger than it "should" be, thereby allowing a person to sit a comfortable distance from the screen. However, larger graphical elements meant less screen space was available for programs to draw. Indeed, the default 720-pixel wide mode of a Hercules mono graphics adaptor (the one-time gold standard for high resolution PC graphics) – or a "tweaked" VGA adaptor – provided an apparent -inch page width at this resolution. However, the more common and colour-capable display adaptors of the time all provided a 640-pixel wide image in their high resolution modes, enough for a bare inches at 100% zoom, with barely any greater visible page height – a maximum of 5 inches, versus . Consequently, the default margins in Microsoft Word were set, and still remain at 1 full inch on all sides of the page, keeping the "text width" for standard size printer paper within visible limits; despite most computer monitors now being both larger and finer-pitched, and printer paper transports having become more sophisticated, the Mac-standard half-inch borders remain listed in Word 2010's page layout presets as the "narrow" option (versus the 1-inch default). | Dots per inch | Wikipedia | 407 | 312833 | https://en.wikipedia.org/wiki/Dots%20per%20inch | Physical sciences | Visual | Basics and measurement |
Without using supplemental, software-provided zoom levels, the 1:1 relationship between display and print size was (deliberately) lost; the availability of different-sized, user-adjustable monitors and display adaptors with varying output resolutions exacerbated this, as it was not possible to rely on a properly-adjusted "standard" monitor and adaptor having a known PPI. For example, a 12-inch Hercules monitor and adaptor with a thick bezel and a little underscan may offer 90 "physical" PPI, with the displayed image appearing nearly identical to hardcopy (assuming the H-scan density was properly adjusted to give square pixels) but a thin-bezel 14-inch VGA monitor adjusted to give a borderless display may be closer to 60, with the same bitmap image thus appearing 50% larger; yet, someone with an 8514 ("XGA") adaptor and the same monitor could achieve 100 DPI using its 1024-pixel wide mode and adjusting the image to be underscanned. A user who wanted to directly compare on-screen elements against those on an existing printed page by holding it up against the monitor would therefore first need to determine the correct zoom level to use, largely by trial and error, and often not be able to obtain an exact match in programs that only allowed integer per cent settings, or even fixed pre-programmed zoom levels. For the examples above, they may need to use respectively 94% (precisely, 93.75) – or , 63% (62.5) – or ; and 104% (104.167) – or , with the more commonly accessible 110% actually being a less precise match. | Dots per inch | Wikipedia | 344 | 312833 | https://en.wikipedia.org/wiki/Dots%20per%20inch | Physical sciences | Visual | Basics and measurement |
Thus, for example, a 10-point font on a Macintosh (at 72 PPI) was represented with 10 pixels (i.e., 10 PPEm), whereas a 10-point font on a Windows platform (at 96 PPI) at the same zoom level is represented with 13 pixels (i.e., Microsoft rounded to 13 pixels, or 13 PPEm) – and, on a typical consumer grade monitor, would have physically appeared around to inch high instead of . Likewise, a 12-point font was represented with 12 pixels on a Macintosh, and 16 pixels (or a physical display height of maybe inch) on a Windows platform at the same zoom, and so on. The negative consequence of this standard is that with 96 PPI displays, there is no longer a one-to-one relationship between the font size in pixels and the printout size in points. This difference is accentuated on more recent displays that feature higher pixel densities. This has been less of a problem with the advent of vector graphics and fonts being used in place of bitmap graphics and fonts. Moreover, many Windows software programs have been written since the 1980s which assume that the screen provides 96 PPI. Accordingly, these programs do not display properly at common alternative resolutions such as 72 PPI or 120 PPI. The solution has been to introduce two concepts:
logical PPI: The PPI that software claims a screen provides. This can be thought of as the PPI provided by a virtual screen created by the operating system.
physical PPI: The PPI that a physical screen actually provides.
Software programs render images to the virtual screen and then the operating system renders the virtual screen onto the physical screen. With a logical PPI of 96 PPI, older programs can still run properly regardless of the actual physical PPI of the display screen, although they may exhibit some visual distortion thanks to the effective 133.3% pixel zoom level (requiring either that every third pixel be doubled in width/height, or heavy-handed smoothing be employed).
How Microsoft Windows handles DPI scaling | Dots per inch | Wikipedia | 425 | 312833 | https://en.wikipedia.org/wiki/Dots%20per%20inch | Physical sciences | Visual | Basics and measurement |
Displays with high pixel densities were not common up to the Windows XP era. High DPI displays became mainstream around the time Windows 8 was released. Display scaling by entering a custom DPI irrespective of the display resolution has been a feature of Microsoft Windows since Windows 95. Windows XP introduced the GDI+ library which allows resolution-independent text scaling. In Microsoft Windows, the DPI higher than 96 DPI is called High DPI.
Windows Vista introduced support for programs to declare themselves to the OS that they are high-DPI aware via a manifest file or using an API. For programs that do not declare themselves as DPI-aware, Windows Vista supports a compatibility feature called DPI virtualization so system metrics and UI elements are presented to applications as if they are running at 96 DPI and the Desktop Window Manager then scales the resulting application window to match the DPI setting. Windows Vista retains the Windows XP style scaling option which when enabled turns off DPI virtualization for all applications globally. DPI virtualization is a compatibility option as application developers are all expected to update their apps to support high DPI without relying on DPI virtualization.
Windows Vista also introduces Windows Presentation Foundation. WPF .NET applications are vector-based, not pixel-based and are designed to be resolution-independent. Developers using the old GDI API and Windows Forms on .NET Framework runtime need to update their apps to be DPI aware and flag their applications as DPI-aware.
Windows 7 adds the ability to change the DPI by doing only a log off, not a full reboot and makes it a per-user setting. Additionally, Windows 7 reads the pixel density related information from the EDID and automatically sets the system DPI value to match the monitor's physical pixel density, unless the effective resolution is less than 1024 × 768. Also, Windows 7 adds DirectWrite that optimised for monitors that larger than 1080p. | Dots per inch | Wikipedia | 399 | 312833 | https://en.wikipedia.org/wiki/Dots%20per%20inch | Physical sciences | Visual | Basics and measurement |
In Windows 8, only the DPI scaling percentage is shown in the DPI changing dialog and the display of the raw DPI value has been removed. In Windows 8.1, the global setting to disable DPI virtualization (only use XP-style scaling) is removed and a per-app setting added for the user to disable DPI virtualization from the Compatibility tab. When the DPI scaling setting is set to be higher than 120 PPI (125%), DPI virtualization is enabled for all applications unless the application opts out of it by specifying a DPI aware flag (manifest) as "true" inside the EXE. Windows 8.1 retains a per-application option to disable DPI virtualization of an app. Windows 8.1 also adds the ability for different displays to use independent DPI scaling factors, although it calculates this automatically for each display and turns on DPI virtualization for all monitors at any scaling level.
Windows 10 adds manual control over DPI scaling for individual monitors.
Proposed metrication
There are some ongoing efforts to abandon the DPI Image resolution unit in favour of a metric unit, giving the inter-dot spacing in dots per centimetre (px/cm or dpcm), as used in CSS3 media queries or micrometres (μm) between dots. A resolution of 72 DPI, for example, equals a resolution of about 28 dpcm or an inter-dot spacing of about 353 μm. | Dots per inch | Wikipedia | 314 | 312833 | https://en.wikipedia.org/wiki/Dots%20per%20inch | Physical sciences | Visual | Basics and measurement |
In physics, action is a scalar quantity that describes how the balance of kinetic versus potential energy of a physical system changes with trajectory. Action is significant because it is an input to the principle of stationary action, an approach to classical mechanics that is simpler for multiple objects. Action and the variational principle are used in Feynman's formulation of quantum mechanics and in general relativity. For systems with small values of action similar to the Planck constant, quantum effects are significant.
In the simple case of a single particle moving with a constant velocity (thereby undergoing uniform linear motion), the action is the momentum of the particle times the distance it moves, added up along its path; equivalently, action is the difference between the particle's kinetic energy and its potential energy, times the duration for which it has that amount of energy.
More formally, action is a mathematical functional which takes the trajectory (also called path or history) of the system as its argument and has a real number as its result. Generally, the action takes different values for different paths. Action has dimensions of energy × time or momentum × length, and its SI unit is joule-second (like the Planck constant h).
Introduction
Introductory physics often begins with Newton's laws of motion, relating force and motion; action is part of a completely equivalent alternative approach with practical and educational advantages. However, the concept took many decades to supplant Newtonian approaches and remains a challenge to introduce to students.
Simple example
For a trajectory of a ball moving in the air on Earth the action is defined between two points in time, and as the kinetic energy (KE) minus the potential energy (PE), integrated over time.
The action balances kinetic against potential energy.
The kinetic energy of a ball of mass is where is the velocity of the ball; the potential energy is where is the gravitational constant. Then the action between and is
The action value depends upon the trajectory taken by the ball through and . This makes the action an input to the powerful stationary-action principle for classical and for quantum mechanics. Newton's equations of motion for the ball can be derived from the action using the stationary-action principle, but the advantages of action-based mechanics only begin to appear in cases where Newton's laws are difficult to apply. Replace the ball with an electron: classical mechanics fails but stationary action continues to work. The energy difference in the simple action definition, kinetic minus potential energy, is generalized and called the Lagrangian for more complex cases. | Action (physics) | Wikipedia | 509 | 312881 | https://en.wikipedia.org/wiki/Action%20%28physics%29 | Physical sciences | Classical mechanics | Physics |
Planck's quantum of action
The Planck constant, written as or when including a factor of , is called the quantum of action. Like action, this constant has unit of energy times time. It figures in all significant quantum equations, like the uncertainty principle and the de Broglie wavelength. Whenever the value of the action approaches the Planck constant, quantum effects are significant.
History
Pierre Louis Maupertuis and Leonhard Euler working in the 1740s developed early versions of the action principle. Joseph Louis Lagrange clarified the mathematics when he invented the calculus of variations. William Rowan Hamilton made the next big breakthrough, formulating Hamilton's principle in 1853. Hamilton's principle became the cornerstone for classical work with different forms of action until Richard Feynman and Julian Schwinger developed quantum action principles.
Definitions
Expressed in mathematical language, using the calculus of variations, the evolution of a physical system (i.e., how the system actually progresses from one state to another) corresponds to a stationary point (usually, a minimum) of the action.
Action has the dimensions of [energy] × [time], and its SI unit is joule-second, which is identical to the unit of angular momentum.
Several different definitions of "the action" are in common use in physics. The action is usually an integral over time. However, when the action pertains to fields, it may be integrated over spatial variables as well. In some cases, the action is integrated along the path followed by the physical system.
The action is typically represented as an integral over time, taken along the path of the system between the initial time and the final time of the development of the system:
where the integrand L is called the Lagrangian. For the action integral to be well-defined, the trajectory has to be bounded in time and space.
Action (functional)
Most commonly, the term is used for a functional which takes a function of time and (for fields) space as input and returns a scalar. In classical mechanics, the input function is the evolution q(t) of the system between two times t1 and t2, where q represents the generalized coordinates. The action is defined as the integral of the Lagrangian L for an input evolution between the two times: | Action (physics) | Wikipedia | 467 | 312881 | https://en.wikipedia.org/wiki/Action%20%28physics%29 | Physical sciences | Classical mechanics | Physics |
where the endpoints of the evolution are fixed and defined as and . According to Hamilton's principle, the true evolution qtrue(t) is an evolution for which the action is stationary (a minimum, maximum, or a saddle point). This principle results in the equations of motion in Lagrangian mechanics.
Abbreviated action (functional)
In addition to the action functional, there is another functional called the abbreviated action. In the abbreviated action, the input function is the path followed by the physical system without regard to its parameterization by time. For example, the path of a planetary orbit is an ellipse, and the path of a particle in a uniform gravitational field is a parabola; in both cases, the path does not depend on how fast the particle traverses the path.
The abbreviated action (sometime written as ) is defined as the integral of the generalized momenta,
for a system Lagrangian along a path in the generalized coordinates :
where and are the starting and ending coordinates.
According to Maupertuis's principle, the true path of the system is a path for which the abbreviated action is stationary.
Hamilton's characteristic function
When the total energy E is conserved, the Hamilton–Jacobi equation can be solved with the additive separation of variables:
where the time-independent function W(q1, q2, ..., qN) is called Hamilton's characteristic function. The physical significance of this function is understood by taking its total time derivative
This can be integrated to give
which is just the abbreviated action.
Action of a generalized coordinate
A variable Jk in the action-angle coordinates, called the "action" of the generalized coordinate qk, is defined by integrating a single generalized momentum around a closed path in phase space, corresponding to rotating or oscillating motion:
The corresponding canonical variable conjugate to Jk is its "angle" wk, for reasons described more fully under action-angle coordinates. The integration is only over a single variable qk and, therefore, unlike the integrated dot product in the abbreviated action integral above. The Jk variable equals the change in Sk(qk) as qk is varied around the closed path. For several physical systems of interest, Jk is either a constant or varies very slowly; hence, the variable Jk is often used in perturbation calculations and in determining adiabatic invariants. For example, they are used in the calculation of planetary and satellite orbits.
Single relativistic particle | Action (physics) | Wikipedia | 511 | 312881 | https://en.wikipedia.org/wiki/Action%20%28physics%29 | Physical sciences | Classical mechanics | Physics |
When relativistic effects are significant, the action of a point particle of mass m travelling a world line C parametrized by the proper time is
If instead, the particle is parametrized by the coordinate time t of the particle and the coordinate time ranges from t1 to t2, then the action becomes
where the Lagrangian is
Action principles and related ideas
Physical laws are frequently expressed as differential equations, which describe how physical quantities such as position and momentum change continuously with time, space or a generalization thereof. Given the initial and boundary conditions for the situation, the "solution" to these empirical equations is one or more functions that describe the behavior of the system and are called equations of motion.
Action is a part of an alternative approach to finding such equations of motion. Classical mechanics postulates that the path actually followed by a physical system is that for which the action is minimized, or more generally, is stationary. In other words, the action satisfies a variational principle: the principle of stationary action (see also below). The action is defined by an integral, and the classical equations of motion of a system can be derived by minimizing the value of that integral.
The action principle provides deep insights into physics, and is an important concept in modern theoretical physics. Various action principles and related concepts are summarized below.
Maupertuis's principle
In classical mechanics, Maupertuis's principle (named after Pierre Louis Maupertuis) states that the path followed by a physical system is the one of least length (with a suitable interpretation of path and length). Maupertuis's principle uses the abbreviated action between two generalized points on a path.
Hamilton's principal function
Hamilton's principle states that the differential equations of motion for any physical system can be re-formulated as an equivalent integral equation. Thus, there are two distinct approaches for formulating dynamical models.
Hamilton's principle applies not only to the classical mechanics of a single particle, but also to classical fields such as the electromagnetic and gravitational fields. Hamilton's principle has also been extended to quantum mechanics and quantum field theory—in particular the path integral formulation of quantum mechanics makes use of the concept—where a physical system explores all possible paths, with the phase of the probability amplitude for each path being determined by the action for the path; the final probability amplitude adds all paths using their complex amplitude and phase.
Hamilton–Jacobi equation | Action (physics) | Wikipedia | 498 | 312881 | https://en.wikipedia.org/wiki/Action%20%28physics%29 | Physical sciences | Classical mechanics | Physics |
Hamilton's principal function is obtained from the action functional by fixing the initial time and the initial endpoint while allowing the upper time limit and the second endpoint to vary. The Hamilton's principal function satisfies the Hamilton–Jacobi equation, a formulation of classical mechanics. Due to a similarity with the Schrödinger equation, the Hamilton–Jacobi equation provides, arguably, the most direct link with quantum mechanics.
Euler–Lagrange equations
In Lagrangian mechanics, the requirement that the action integral be stationary under small perturbations is equivalent to a set of differential equations (called the Euler–Lagrange equations) that may be obtained using the calculus of variations.
Classical fields
The action principle can be extended to obtain the equations of motion for fields, such as the electromagnetic field or gravitational field.
Maxwell's equations can be derived as conditions of stationary action.
The Einstein equation utilizes the Einstein–Hilbert action as constrained by a variational principle. The trajectory (path in spacetime) of a body in a gravitational field can be found using the action principle. For a free falling body, this trajectory is a geodesic.
Conservation laws
Implications of symmetries in a physical situation can be found with the action principle, together with the Euler–Lagrange equations, which are derived from the action principle. An example is Noether's theorem, which states that to every continuous symmetry in a physical situation there corresponds a conservation law (and conversely). This deep connection requires that the action principle be assumed.
Path integral formulation of quantum field theory
In quantum mechanics, the system does not follow a single path whose action is stationary, but the behavior of the system depends on all permitted paths and the value of their action. The action corresponding to the various paths is used to calculate the path integral, which gives the probability amplitudes of the various outcomes.
Although equivalent in classical mechanics with Newton's laws, the action principle is better suited for generalizations and plays an important role in modern physics. Indeed, this principle is one of the great generalizations in physical science. It is best understood within quantum mechanics, particularly in Richard Feynman's path integral formulation, where it arises out of destructive interference of quantum amplitudes.
Modern extensions | Action (physics) | Wikipedia | 463 | 312881 | https://en.wikipedia.org/wiki/Action%20%28physics%29 | Physical sciences | Classical mechanics | Physics |
The action principle can be generalized still further. For example, the action need not be an integral, because nonlocal actions are possible. The configuration space need not even be a functional space, given certain features such as noncommutative geometry. However, a physical basis for these mathematical extensions remains to be established experimentally. | Action (physics) | Wikipedia | 66 | 312881 | https://en.wikipedia.org/wiki/Action%20%28physics%29 | Physical sciences | Classical mechanics | Physics |
A tenrec () is a mammal belonging to any species within the afrotherian family Tenrecidae, which is endemic to Madagascar. Tenrecs are a very diverse group; as a result of adaptive radiation and exhibit convergent evolution, some resemble hedgehogs, shrews, opossums, rats, and mice. They occupy aquatic, arboreal, terrestrial, and fossorial environments. Some of these species, including the greater hedgehog tenrec, can be found in the Madagascar dry deciduous forests. However, the speciation rate in this group has been higher in humid forests.
All tenrecs are believed to descend from a common ancestor that lived 29–37 million years ago after rafting over from Africa. The split from their closest relatives, African otter shrews, is estimated to have occurred about 47–53 million years ago.
Etymology
The word tenrec is borrowed, via French, from the Malagasy word (variant of ), which refers to the tailless tenrec (Tenrec ecaudatus); it has been speculated that the Malagasy word is related to .
Evolution
Tenrecs are believed to have evolved from a single species that colonized Madagascar between 42 and 25 million years ago. The question of how this family reached Madagascar is still unresolved, but the leading hypothesis suggests a small number of individuals may have found themselves on floating vegetation and crossed the Mozambique Channel, which separates Madagascar from southeastern Africa. The Tenrecidae family is one of only four extant terrestrial mammal lineages to have colonized and diversified on Madagascar.
Once established on Madagascar, tenrecs diversified to occupy various niches on the island. Many evolved resemblances to familiar but unrelated mammals that are not found on Madagascar. For instance, the two species of hedgehog tenrec possess coats of hardened spines and the ability to roll into a ball when threatened, characteristics similar to those of true hedgehogs. This example, along with others, demonstrates convergent evolution; it has provided evolutionary biologists with opportunities to study adaptation over evolutionary timescales.
Characteristics | Tenrec | Wikipedia | 428 | 312933 | https://en.wikipedia.org/wiki/Tenrec | Biology and health sciences | Other afrotheres | Animals |
Tenrecs are small mammals of variable body form. The smallest species are the size of shrews, with a body length of around , and weighing just , while the largest, the common or tailless tenrec, is in length, and can weigh over . Although they may resemble shrews, hedgehogs, or opossums, they are not closely related to any of these groups, their closest relatives being the otter shrews, and after that other African insectivorous mammals including golden moles and elephant shrews. The common ancestry of these animals, which are classified together in the clade Afrotheria, was not recognized until the late 1990s. Continuing work on the molecular and morphological diversity of afrotherian mammals has provided ever increasing support for their common ancestry.
Tenrecs are among the few terrestrial mammals that echolocate.
Unusual among placental mammals, the rectum and urogenital tracts of tenrecs share a common opening, or cloaca which is a feature more commonly seen in birds, reptiles, and amphibians. They have a low body temperature, sufficiently low that they do not require a scrotum to cool their sperm as do most other mammals.
All species appear to be at least somewhat omnivorous, with invertebrates forming the largest part of their diets. One species, Microgale mergulus, is semiaquatic (similar to the lifestyle of their closest relatives, the otter shrews). All of the species, semiaquatic or not, appear to have evolved from a single, common ancestor with the otter shrews comprising the next, most-closely related mammalian species. While the fossil record of tenrecs is scarce, at least some specimens from the early Miocene of Kenya show close affinities to living species from Madagascar, such as Geogale aurita.
Most species are nocturnal and have poor eyesight. Their other senses are well developed, however and they have especially sensitive whiskers. As with many of their other features, the dental formula of tenrecs varies greatly between species; they can have from 32 to 42 teeth in total. Unusual for mammals, the permanent dentition in tenrecs tends not to completely erupt until well after adult body size has been reached. This is one of several anatomical features shared by elephants, hyraxes, sengis, and golden moles (but apparently not aardvarks), consistent with their descent from a common ancestor. | Tenrec | Wikipedia | 507 | 312933 | https://en.wikipedia.org/wiki/Tenrec | Biology and health sciences | Other afrotheres | Animals |
Tenrecs have a gestation period of 50 to 64 days, and give birth to a number of relatively undeveloped young. While the otter shrews have just two young per litter, the tailless tenrec can have as many as 32, and females possess up to 29 teats, more than any other mammal. Some tenrec species are social, living in multigenerational family groups with over a dozen individuals.
Interaction with humans
In the island nation of Mauritius, and also on the Comoran island of Mayotte, some of the inhabitants eat tenrec meat, although it is difficult to obtain (as it is not sold in shops or markets) and difficult to prepare correctly.
The lesser hedgehog tenrec (Echinops telfairi) is one of 16 mammalian species that will have its genome sequenced as part of the Mammalian Genome Project. It is increasingly popular in the pet trade, and in the future may serve as an important model organism in biomedicine, as it is only distantly related to the mice, rats, guinea pigs, and rhesus macaques which comprise the most common research animals.
Threats
Of the 31 species assessed, 24 (77%) are categorized by the IUCN Red List as Least Concern, 1 species as Data Deficient, 4 species as Vulnerable, and 2 species as Endangered.
The conservation status of many tenrec species is of concern due to an increase of threats within the last 50 years. The main threats facing tenrecs include habitat loss due to deforestation, fragmentation and degradation, hunting, incidental capture, and climate change. Slash-and-burn agriculture, as well as commercial logging and mining of metals is negatively affecting tenrec species that inhabit forests. Five of the six threatened Tenrec species are dependent on forest habitats.
Conservation
As of 2022, conservation of the tenrec population is not being prioritized. Because most tenrecs are dependent on forest habitats, conservation efforts would need to include a focus on reduction in deforestation on Madagascar as well as habitat restoration. Current conservation efforts include that of the Madagascar Ankizy Fund, started by a paleontological team from Stony Brook University to improve access to health care and education facilities for villagers in remote areas of Madagascar. A healthy and educated local human population will, in the long term, benefit the Malagasy fauna, such as tenrecs.
Species
The three subfamilies, eight genera, and 31 extant species of tenrecs are | Tenrec | Wikipedia | 510 | 312933 | https://en.wikipedia.org/wiki/Tenrec | Biology and health sciences | Other afrotheres | Animals |
FAMILY TENRECIDAE
Subfamily Geogalinae
Genus Geogale
Large-eared tenrec (Geogale aurita)
Subfamily Oryzorictinae
Genus Microgale
Short-tailed shrew tenrec (Microgale brevicaudata)
Cowan's shrew tenrec (Microgale cowani)
Drouhard's shrew tenrec (Microgale drouhardi)
Dryad shrew tenrec (Microgale dryas)
Pale shrew tenrec (Microgale fotsifotsy)
Gracile shrew tenrec (Microgale gracilis)
Grandidier's shrew tenrec (Microgale grandidieri)
Naked-nosed shrew tenrec (Microgale gymnorhyncha)
Jenkins's shrew tenrec (Microgale jenkinsae)
Northern shrew tenrec (Microgale jobihely)
Lesser long-tailed shrew tenrec (Microgale longicaudata)
Microgale macpheei (extinct)
Major's long-tailed tenrec (Microgale majori)
Web-footed tenrec (Microgale mergulus)
Montane shrew tenrec (Microgale monticola)
Nasolo's shrew tenrec (Microgale nasoloi)
Pygmy shrew tenrec (Microgale parvula)
Greater long-tailed shrew tenrec (Microgale principula)
Least shrew tenrec (Microgale pusilla)
Shrew-toothed shrew tenrec (Microgale soricoides)
Taiva shrew tenrec (Microgale taiva)
Thomas's shrew tenrec (Microgale thomasi)
Genus Nesogale
Dobson's shrew tenrec (Nesogale dobsoni)
Talazac's shrew tenrec (Nesogale talazaci)
Genus Oryzorictes
Mole-like rice tenrec (Oryzorictes hova)
Four-toed rice tenrec (Oryzorictes tetradactylus)
Subfamily Tenrecinae
Tribe Setiferini
Genus Echinops
Lesser hedgehog tenrec (Echinops telfairi)
Genus Setifer
Greater hedgehog tenrec (Setifer setosus)
Tribe Tenrecini | Tenrec | Wikipedia | 512 | 312933 | https://en.wikipedia.org/wiki/Tenrec | Biology and health sciences | Other afrotheres | Animals |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.