"Schur-convex function In mathematics, a Schur-convex function, also known as S-convex, isotonic function and order-preserving function is a function formula_1 that for all formula_2 such that formula_3 is majorized by formula_4, one has that formula_5. Named after Issai Schur, Schur-convex functions are used in the study of majorization. Every function that is convex and symmetric is also Schur-convex. The opposite implication is not true, but all Schur-convex functions are symmetric (under permutations of the arguments). A function ""f"" is 'Schur-concave' if its negative, -""f"", is Schur-convex. If ""f"" is symmetric and all first partial derivatives exist, then ""f"" is Schur-convex"
"to the focus, until it is reduced to a point in the focus when the object is at an infinite distance. These features make convex mirrors very useful: since everything appears smaller in the mirror, they cover a wider field of view than a normal plane mirror does. A concave mirror, or converging mirror, has a reflecting surface that bulges inward (away from the incident light). Concave mirrors reflect light inward to one focal point.They are used to focus light. Unlike convex mirrors, concave mirrors show different image types depending on the distance between the object and the mirror. These"
"the lens's spherical aberration corrects the opposite aberration in the mirror (such as Schmidt and meniscus correctors). Convex lenses produce an image of an object at infinity at their focus; if the sun is imaged, much of the visible and infrared light incident on the lens is concentrated into the small image. A large lens creates enough intensity to burn a flammable object at the focal point. Since ignition can be achieved even with a poorly made lens, lenses have been used as burning-glasses for at least 2400 years. A modern application is the use of relatively large lenses to"
"biconvex, and there is a biconcave element between them. The Plasmat lens is made in many variants, e.g. departing from exact symmetry, adding a lens to one or both groups, or separating the innermost or outermost element from the rest of the group. Standard lenses for large-format cameras are generally of the Plasmat type, as are many macro lenses. Convertible lenses for large-format photography often consist of Plasmat cells. Plasmat lens The Plasmat lens is a widely used and long-established lens type invented by Paul Rudolph in 1918, especially common in large-format photography. It provides high correction of aberrations with"
"inner pair are biconcave. The symmetrical structure provides good correction for many aberrations. The Aviar type of lens (Taylor Hobson) is similar but is considered to have a different origin, from the splitting of the central biconcave element of the Cooke triplet. The resulting two biconcave elements are closer together than in the Dialyte/Celor design. Dialyte lens A dialyte lens (sometimes called a dialyt) is a compound lens design that corrects optical aberrations where the lens elements are widely air-spaced. The design is used to save on the amount of glass used for specific elements or where elements can not"
"can be drawn: Acceptable lens shapes are most bi-convex, plano-convex and mild meniscus shapes. Not unacceptable but hard to mould are bi-concave lenses, steep meniscus lenses, and lenses with severe features (e.g. a bump on a convex surface). In general, plano-curved lenses are easier to mould than lenses with both sides curved since matching of flat faces is easier. Moulding concave forms with small centre thickness is difficult due to sticking of the moulded part to the mould occurring as a result of the different thermal expansion coefficients. Furthermore it is recommended to avoid undercuts and sharp edges. For the"
"from a large area and focus it into a small spot, as in concentrated solar power. Concave mirrors are used to form optical cavities, which are important in laser construction. Some dental mirrors use a concave surface to provide a magnified image. The mirror landing aid system of modern aircraft carriers also uses a concave mirror. Most curved mirrors have a spherical profile. These are the simplest to make, and it is the best shape for general-purpose use. Spherical mirrors, however, suffer from spherical aberration—parallel rays reflected from such mirrors do not focus to a single point. For parallel rays,"
"of equal radii to the cavity length (""R"" = ""R"" = ""L""). This design produces the smallest possible beam diameter at the cavity mirrors for a given cavity length, and is often used in lasers where the purity of the transverse mode pattern is important. A concave-convex cavity has one convex mirror with a negative radius of curvature. This design produces no intracavity focus of the beam, and is thus useful in very high-power lasers where the intensity of the intracavity light might be damaging to the intracavity medium if brought to a focus. A transparent dielectric sphere, such as"
"The Complan lens has a curved film plane, hence in these cameras the negative must be held in an arc to improve the edge-to-edge sharpness of the image. The Minox enlarger also holds the negative in this same curve. Later models, beginning with late model Minox B, to the current model TLX, using the 15 mm f/3.5 four-element, three-group flat-field Minox lens, holds the negative flat. The advance was attributed by Kasiemeir (Small MINOX Big Pictures 1971 edition) to new rare-earth elements. Note that lens performance between old and new complan/minox lenses was rated by MINOX themselves as identical. At"
"of a convex objective lens and a concave eye lens, and this design is now called a Galilean telescope. Johannes Kepler proposed an improvement on the design that used a convex eyepiece, often called the Keplerian Telescope. The next big step in the development of refractors was the advent of the Achromatic lens in the early 18th century, which corrected the chromatic aberration in Keplerian telescopes up to that time—allowing for much shorter instruments with much larger objectives. For reflecting telescopes, which use a curved mirror in place of the objective lens, theory preceded practice. The theoretical basis for curved"
"mirrors are called ""converging mirrors"" because they tend to collect light that falls on them, refocusing parallel incoming rays toward a focus. This is because the light is reflected at different angles, since the normal to the surface differs with each spot on the mirror. Concave mirrors are used in reflecting telescopes. They are also used to provide a magnified image of the face for applying make-up or shaving. In illumination applications, concave mirrors are used to gather light from a small source and direct it outward in a beam as in torches, headlamps and spotlights, or to collect light"
"Condenser (optics) A condenser is an optical lens which renders a divergent beam from a point source into a parallel or converging beam to illuminate an object. Condensers are an essential part of any imaging device, such as microscopes, enlargers, slide projectors, and telescopes. The concept is applicable to all kinds of radiation undergoing optical transformation, such as electrons in electron microscopy, neutron radiation and synchrotron radiation optics. Condensers are located above the light source and under the sample in an upright microscope, and above the stage and below the light source in an inverted microscope. They act to gather"
"this affects the focusing of the eye. Most animals' eyes are adapted to either underwater or air vision, and do not focus properly when in the other environment. The crystalline lenses of fishes' eyes are extremely convex, almost spherical, and their refractive indices are the highest of all the animals. These properties enable proper focusing of the light rays and in turn proper image formation on the retina. This convex lens gives the name to the fisheye lens in photography. By wearing a flat diving mask, humans can see clearly underwater. The scuba mask's flat window separates the eyes from"
"that one of the mirrors can be involved in the light path twice — each light path reflects along a different meridional path. Stevick-Paul telescopes are off-axis versions of Paul 3-mirror systems with an added flat diagonal mirror. A convex secondary mirror is placed just to the side of the light entering the telescope, and positioned afocally so as to send parallel light on to the tertiary. The concave tertiary mirror is positioned exactly twice as far to the side of the entering beam as was the convex secondary, and its own radius of curvature distant from the secondary. Because"
"condensers that can nonetheless focus light with the same degree of precision even without oil between the top lens and the slide. Microscopy Resource Center"", 2006. Condenser (optics) A condenser is an optical lens which renders a divergent beam from a point source into a parallel or converging beam to illuminate an object. Condensers are an essential part of any imaging device, such as microscopes, enlargers, slide projectors, and telescopes. The concept is applicable to all kinds of radiation undergoing optical transformation, such as electrons in electron microscopy, neutron radiation and synchrotron radiation optics. Condensers are located above the light"
"with angle β. The ratio β/α is called the angular magnification. It equals the ratio between the retinal image sizes obtained with and without the telescope. Refracting telescopes can come in many different configurations to correct for image orientation and types of aberration. Because the image was formed by the bending of light, or refraction, these telescopes are called ""refracting telescopes"" or ""refractors"". The design Galileo Galilei used in 1609 is commonly called a Galilean telescope. It used a convergent (plano-convex) objective lens and a divergent (plano-concave) eyepiece lens (Galileo, 1610). A Galilean telescope, because the design has no intermediary"
"without comparing 20×30 inch (about 50×75 cm) enlargements of exactly the same scene side by side. Typical lens elements have spherically curved surfaces. However, this causes off-axis light to be focused closer to the lens than axial rays (spherical aberration); especially severe in wide angle or wide aperture lenses. This can be prevented by using elements with convoluted aspheric curves. Although this was theoretically proven by René Descartes in 1637, the grinding and polishing of aspheric glass surfaces was extremely difficult and expensive. The first camera lens with an inexpensive mass-produced molded glass aspheric element was the unnamed 12.5mm f/2.8"
"Axicon An axicon is a specialized type of lens which has a conical surface. An axicon transforms a laser beam into a ring shaped distribution. They can be convex or concave and be made of any optical material. The combination with other axicons or lenses allows a wide variety of beam patterns to be generated. It can be used to turn a Gaussian beam into non-diffractive Bessel-like beams. Axicons were first proposed in 1954 by John McLeod. Axicons are used in atomic traps and for generating plasma in linear accelerators. They are used in eye surgery in cases where a"
"four curved vanes; each vane forms a convex and a concave surface. The light mill is uniformly coated by gold nanocrystals, which are a strong light absorber. Upon exposure, due to geometric effect, the convex side of the vane receives more photon energy than the concave side does, and subsequently the gas molecules receive more heat from the convex side than from the concave side. At rough vacuum, this asymmetric heating effect generates a net gas movement across each vane, from the concave side to the convex side, as shown by the researchers' Direct Simulation Monte Carlo (DSMC) modeling. The"
"have large eye lenses, good eye relief and can be very comfortable to use. The König eyepiece has a concave-convex positive doublet and a plano-convex singlet. The strongly convex surfaces of the doublet and singlet face and (nearly) touch each other. The doublet has its concave surface facing the light source and the singlet has its almost flat (slightly convex) surface facing the eye. It was designed in 1915 by German optician Albert König (1871−1946) as a simplified Abbe. The design allows for high magnification with remarkably high eye relief — the highest eye relief proportional to focal length of"
"a converging lens (for example a convex lens), the focal length is positive, and is the distance at which a beam of collimated light will be focused to a single spot. For a diverging lens (for example a concave lens), the focal length is negative, and is the distance to the point from which a collimated beam appears to be diverging after passing through the lens. When a lens is used to form an image of some object, the distance from the object to the lens ""u"", the distance from the lens to the image ""v"", and the focal length"
"principles. The combination of an objective lens 1 and some type of eyepiece 2 is used to gather more light than the human eye is able to collect on its own, focus it 5, and present the viewer with a brighter, clearer, and magnified virtual image 6. The objective in a refracting telescope refracts or bends light. This refraction causes parallel light rays to converge at a focal point; while those not parallel converge upon a focal plane. The telescope converts a bundle of parallel rays to make an angle α, with the optical axis to a second parallel bundle"
"the curvature of the lens is carried out by the ciliary muscles surrounding the lens; this process is called ""accommodation"". Accommodation narrows the inner diameter of the ciliary body, which actually relaxes the fibers of the suspensory ligament attached to the periphery of the lens, and allows the lens to relax into a more convex, or globular, shape. A more convex lens refracts light more strongly and focuses divergent light rays from near objects onto the retina, allowing closer objects to be brought into better focus. The human eye contains enough complexity to warrant specialized attention and care beyond the"
"evaluation of the optical properties of liquid crystalline phases (i.e., orientation of the optical axes) is in 1911 when it was used by Mauging to investigate the alignment of nematic and chiral-nematic phases. A beam of convergent (or divergent) light is known to be a linear superposition of many plane waves over a cone of solid angles. The raytracing of Figure 1 illustrates the basic concept of ""conoscopy"": transformation of a directional distribution of rays of light in the front focal plane into a lateral distribution (""directions image"") appearing in the back focal plane (which is more or less curved)."
"Plasmat lens The Plasmat lens is a widely used and long-established lens type invented by Paul Rudolph in 1918, especially common in large-format photography. It provides high correction of aberrations with a moderate maximum aperture (e.g. ). It is a specific instance of the Dagor type double-meniscus anastigmat. Double-meniscus anastigmats use widely separated positive and negative surfaces, generally thick meniscus lenses, to achieve a flat field. The most basic form is two sharply curved meniscus elements located symmetrically about a stop. Further refinement of the form replaces the two simple meniscus lenses with achromats for chromatic correction. The Dagor type"
"relay to have higher NA for a given diameter. These are known as Hopkins rod lenses. Karl Storz GmbH licensed the patent for the Hopkins relay lens and introduced endoscopes including such lenses in 1965. Relay lens In optics, a relay lens is a lens or lens group that inverts an image and extends the optical tube. Relay lenses are found in refracting telescopes, endoscopes and periscopes for the purpose of extending the length of the system, and before eyepieces for the purpose of inverting an image. They may be made of one or more conventional lenses or achromatic doublets,"
"Conoscopy Conoscopy (from Ancient Greek ""κῶνος"" (konos) ""cone, spinning top, pine cone"" and ""σκοπέω"" (skopeo) ""examine, inspect, look to or into, consider"") is an optical technique to make observations of a transparent specimen in a cone of converging rays of light. The various directions of light propagation are observable simultaneously . A conoscope is an apparatus to carry out ""conoscopic observations"" and measurements, often realized by a microscope with a Bertrand lens for observation of the ""direction's image"". The earliest reference to the use of ""conoscopy"" (i.e., observation in convergent light with a polarization microscope with a Bertrand lens) for"
"binoculars and in opera glasses. A simple convex lens placed after the focus of the objective lens presents the viewer with a magnified inverted image. This configuration may have been used in the first refracting telescopes from the Netherlands and was proposed as a way to have a much wider field of view and higher magnification in telescopes in Johannes Kepler's 1611 book ""Dioptrice"". Since the lens is placed after the focal plane of the objective it also allowed for use of a micrometer at the focal plane (used for determining the angular size and/or distance between objects observed). Huygens"
"researchers. the superlens has not yet been demonstrated at visible or near-infrared wavelengths. A prototype flat ultrathin lens, with no curvature has been developed. A single convex lens mounted in a frame with a handle or stand is a magnifying glass. Lenses are used as prosthetics for the correction of visual impairments such as myopia, hypermetropia, presbyopia, and astigmatism. (See corrective lens, contact lens, eyeglasses.) Most lenses used for other purposes have strict axial symmetry; eyeglass lenses are only approximately symmetric. They are usually shaped to fit in a roughly oval, not circular, frame; the optical centres are placed over"
"the order of meters, by using many lenses in series. To minimize the absorption the lens must be made from a material such as aluminium, beryllium, or lithium. CRLs were first demonstrated in the mid-1990s by a group of scientists at the ESRF. They drilled holes in an aluminium block, and achieved focusing in two dimensions. For X-rays a concave lens focuses the X-rays because the index of refraction is slightly below unity. In a CRL of this type the walls between the cylindrical holes act as concave lenses for X-rays traveling perpendicular to the axis of the drilled cylinders."
"The ""Huashu"" article ""Sijing"" 四鏡 ""Four Lenses"" (1.10), with the earliest known reference to the basic types of simple lenses, metaphorically illustrates relativism and, in modern terms, subjective idealism. I have always by me four lenses. The first is called [""gui""] 璧 (the 'sceptre', a diverging bi-concave lens). The second is called [""zhu""] 珠 (the 'pearl', biconvex). The third is called [""zhi""] 砥 (the 'whetstone', plano-concave). The fourth is called [""yu""] 盂 (the 'bowl', plano-convex). With [""gui""] the object is larger (than the image). With [""zhu""] the object is smaller (than the image). With [""zhi""] the image appears upright. With"
"condensers are difficult to use for magnifications of above 400X, as the aplanatic cone is only representative of a numerical aperture (NA) of 0.6. This condenser is composed of two lenses, a plano-convex lens somewhat larger than a hemisphere and a large bi-convex lens serving as a collecting lens to the first. The focus of the first lens is traditionally about 2mm away from the plane face coinciding with the sample plane. A pinhole cap can be used to align the optical axis of the condenser with that of the microscope. The Abbe condenser is still the basis for most"
"Lens (anatomy) The lens is a transparent, biconvex structure in the eye that, along with the cornea, helps to refract light to be focused on the retina. The lens, by changing shape, functions to change the focal distance of the eye so that it can focus on objects at various distances, thus allowing a sharp real image of the object of interest to be formed on the retina. This adjustment of the lens is known as accommodation (see also ). Accommodation is similar to the focusing of a photographic camera via movement of its lenses. The lens is also known"