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Contact mechanics-Contact mechanics-Frictional contact mechanics emphasizes the effect of friction forces. | milkshake721/2.1M-wiki-STEM |
Contact mechanics-Contact mechanics-Contact mechanics is part of mechanical engineering. The physical and mathematical formulation of the subject is built upon the mechanics of materials and continuum mechanics and focuses on computations involving elastic, viscoelastic, and plastic bodies in static or dynamic contact.... | milkshake721/2.1M-wiki-STEM |
Contact mechanics-Contact mechanics-The original work in contact mechanics dates back to 1881 with the publication of the paper "On the contact of elastic solids" ("Ueber die Berührung fester elastischer Körper") by Heinrich Hertz. Hertz was attempting to understand how the optical properties of multiple, stacked lense... | milkshake721/2.1M-wiki-STEM |
Contact mechanics-History-Classical contact mechanics is most notably associated with Heinrich Hertz. In 1882, Hertz solved the contact problem of two elastic bodies with curved surfaces. This still-relevant classical solution provides a foundation for modern problems in contact mechanics. For example, in mechanical en... | milkshake721/2.1M-wiki-STEM |
Contact mechanics-History-It was not until nearly one hundred years later that Johnson, Kendall, and Roberts found a similar solution for the case of adhesive contact. This theory was rejected by Boris Derjaguin and co-workers who proposed a different theory of adhesion in the 1970s. The Derjaguin model came to be know... | milkshake721/2.1M-wiki-STEM |
Contact mechanics-History-Further advancement in the field of contact mechanics in the mid-twentieth century may be attributed to names such as Bowden and Tabor. Bowden and Tabor were the first to emphasize the importance of surface roughness for bodies in contact. Through investigation of the surface roughness, the tr... | milkshake721/2.1M-wiki-STEM |
Contact mechanics-History-The contributions of Archard (1957) must also be mentioned in discussion of pioneering works in this field. Archard concluded that, even for rough elastic surfaces, the contact area is approximately proportional to the normal force. Further important insights along these lines were provided by... | milkshake721/2.1M-wiki-STEM |
Contact mechanics-Classical solutions for non-adhesive elastic contact-The theory of contact between elastic bodies can be used to find contact areas and indentation depths for simple geometries. Some commonly used solutions are listed below. The theory used to compute these solutions is discussed later in the article.... | milkshake721/2.1M-wiki-STEM |
Contact mechanics-Classical solutions for non-adhesive elastic contact-The distribution of normal pressure in the contact area as a function of distance from the center of the circle is p(r)=p0(1−r2a2)12 where p0 is the maximum contact pressure given by p0=3F2πa2=1π(6FE∗2R2)13 The radius of the circle is related to th... | milkshake721/2.1M-wiki-STEM |
Contact mechanics-Classical solutions for non-adhesive elastic contact-Contact between a rigid cylinder with flat end and an elastic half-space If a rigid cylinder is pressed into an elastic half-space, it creates a pressure distribution described by p(r)=p0(1−r2R2)−12 where R is the radius of the cylinder and p0=1πE∗... | milkshake721/2.1M-wiki-STEM |
Contact mechanics-Classical solutions for non-adhesive elastic contact-Contact between two cylinders with parallel axes In contact between two cylinders with parallel axes, the force is linearly proportional to the length of cylinders L and to the indentation depth d: F≈π4E∗Ld The radii of curvature are entirely absent... | milkshake721/2.1M-wiki-STEM |
Contact mechanics-Classical solutions for non-adhesive elastic contact-The Method of Dimensionality Reduction Some contact problems can be solved with the Method of Dimensionality Reduction (MDR). In this method, the initial three-dimensional system is replaced with a contact of a body with a linear elastic or viscoela... | milkshake721/2.1M-wiki-STEM |
Contact mechanics-Hertzian theory of non-adhesive elastic contact-The classical theory of contact focused primarily on non-adhesive contact where no tension force is allowed to occur within the contact area, i.e., contacting bodies can be separated without adhesion forces. Several analytical and numerical approaches ha... | milkshake721/2.1M-wiki-STEM |
Contact mechanics-Hertzian theory of non-adhesive elastic contact-As an example, consider two objects which meet at some surface S in the ( x ,y )-plane with the z -axis assumed normal to the surface. One of the bodies will experience a normally-directed pressure distribution pz=p(x,y)=qz(x,y) and in-plane surface t... | milkshake721/2.1M-wiki-STEM |
Contact mechanics-Hertzian theory of non-adhesive elastic contact-Assumptions in Hertzian theory The following assumptions are made in determining the solutions of Hertzian contact problems: The strains are small and within the elastic limit.
The surfaces are continuous and non-conforming (implying that the area of con... | milkshake721/2.1M-wiki-STEM |
Contact mechanics-Hertzian theory of non-adhesive elastic contact-Analytical solution techniques Analytical solution methods for non-adhesive contact problem can be classified into two types based on the geometry of the area of contact. A conforming contact is one in which the two bodies touch at multiple points before... | milkshake721/2.1M-wiki-STEM |
Contact mechanics-Hertzian theory of non-adhesive elastic contact-A common approach in linear elasticity is to superpose a number of solutions each of which corresponds to a point load acting over the area of contact. For example, in the case of loading of a half-plane, the Flamant solution is often used as a starting ... | milkshake721/2.1M-wiki-STEM |
Contact mechanics-Hertzian theory of non-adhesive elastic contact-Point contact on a (2D) half-plane A starting point for solving contact problems is to understand the effect of a "point-load" applied to an isotropic, homogeneous, and linear elastic half-plane, shown in the figure to the right. The problem may be eithe... | milkshake721/2.1M-wiki-STEM |
Contact mechanics-Hertzian theory of non-adhesive elastic contact-Line contact on a (2D) half-plane Normal loading over a region Suppose, rather than a point load P , a distributed load p(x) is applied to the surface instead, over the range a<x<b . The principle of linear superposition can be applied to determine th... | milkshake721/2.1M-wiki-STEM |
Contact mechanics-Hertzian theory of non-adhesive elastic contact-Point contact on a (3D) half-space Analogously to the Flamant solution for the 2D half-plane, fundamental solutions are known for the linearly elastic 3D half-space as well. These were found by Boussinesq for a concentrated normal load and by Cerruti for... | milkshake721/2.1M-wiki-STEM |
Contact mechanics-Hertzian theory of non-adhesive elastic contact-Numerical solution techniques Distinctions between conforming and non-conforming contact do not have to be made when numerical solution schemes are employed to solve contact problems. These methods do not rely on further assumptions within the solution p... | milkshake721/2.1M-wiki-STEM |
Contact mechanics-Hertzian theory of non-adhesive elastic contact-These conditions are valid in a general way. The mathematical formulation of the gap depends upon the kinematics of the underlying theory of the solid (e.g., linear or nonlinear solid in two- or three dimensions, beam or shell model). By restating the no... | milkshake721/2.1M-wiki-STEM |
Contact mechanics-Hertzian theory of non-adhesive elastic contact-h=h0+g+Cp,h⋅p=0,p≥0,h≥0, where C is a matrix, whose elements are so called influence coefficients relating the contact pressure and the deformation. The strict LCP formulation of the CM problem presented above, allows for direct application of well-esta... | milkshake721/2.1M-wiki-STEM |
Contact mechanics-Contact between rough surfaces-When two bodies with rough surfaces are pressed against each other, the true contact area formed between the two bodies, A , is much smaller than the apparent or nominal contact area A0 . The mechanics of contacting rough surfaces are discussed in terms of normal conta... | milkshake721/2.1M-wiki-STEM |
Contact mechanics-Contact between rough surfaces-Fn(h)=∫h∞(s−h)nϕ∗(s)dsn=ηAnF0(h)Aa=πηARσF1(h)P=43ηAErRσ32F32(h) where: d is the separation, A is the nominal contact area, η is the surface density of asperities, E∗ is the effective Young modulus. | milkshake721/2.1M-wiki-STEM |
Contact mechanics-Contact between rough surfaces-A and P can be determined when the Fn(h) terms are calculated for the given surfaces using the convolution of the surface roughness ϕ∗(s) . Several studies have followed the suggested curve fits for Fn(h) assuming a Gaussian surface high distribution with curve fits ... | milkshake721/2.1M-wiki-STEM |
Contact mechanics-Contact between rough surfaces-Recently the exact approximants to Ar and P were published by Jedynak. They are given by the following rational formulas, which are approximants to the integrals Fn(h) . They are calculated for the Gaussian distribution of asperities, which have been shown to be unrea... | milkshake721/2.1M-wiki-STEM |
Contact mechanics-Contact between rough surfaces-exp (−h22) For F1(h) the coefficients are 0.398942280401 0.159773702775 0.0389687688311 0.00364356495452 1.653807476138 1.170419428529 0.448892964428 0.0951971709160 0.00931642803836 6.383774657279 10 −6] The maximum relative error is 9.93 10 −8% For F32(h) the coeff... | milkshake721/2.1M-wiki-STEM |
Contact mechanics-Contact between rough surfaces-For the situation where the asperities on the two surfaces have a Gaussian height distribution and the peaks can be assumed to be spherical, the average contact pressure is sufficient to cause yield when av 1.1 0.39 σ0 where σy is the uniaxial yield stress and σ0 is t... | milkshake721/2.1M-wiki-STEM |
Contact mechanics-Contact between rough surfaces-The Greenwood-Williamson model requires knowledge of two statistically dependent quantities; the standard deviation of the surface roughness and the curvature of the asperity peaks. An alternative definition of the plasticity index has been given by Mikic. Yield occurs w... | milkshake721/2.1M-wiki-STEM |
Contact mechanics-Contact between rough surfaces-In this definition Ψ represents the micro-roughness in a state of complete plasticity and only one statistical quantity, the rms slope, is needed which can be calculated from surface measurements. For Ψ<23 , the surface behaves elastically during contact.
In both the G... | milkshake721/2.1M-wiki-STEM |
Contact mechanics-Contact between rough surfaces-The most frequently cited equations given by the GT model are for the asperity contact area Aa=π2(ηβσ)2AF2(λ), and load carried by asperities 15 π(ηβσ)2σβE′AF52(λ), where: ηβσ , roughness parameter, A , nominal contact area, λ , Stribeck oil film parameter, first defined... | milkshake721/2.1M-wiki-STEM |
Contact mechanics-Contact between rough surfaces-The exact solutions for Aa and P are firstly presented by Jedynak. They are expressed by Fn as follows. They are calculated for the Gaussian distribution of asperities, which have been shown to be unrealistic for engineering surface but can be assumed where friction, ... | milkshake721/2.1M-wiki-STEM |
Contact mechanics-Contact between rough surfaces-In paper one can find comprehensive review of existing approximants to F52 . New proposals give the most accurate approximants to F52 and F2 , which are reported in the literature. They are given by the following rational formulas, which are very exact approximants to... | milkshake721/2.1M-wiki-STEM |
Contact mechanics-Adhesive contact between elastic bodies-When two solid surfaces are brought into close proximity, they experience attractive van der Waals forces. Bradley's van der Waals model provides a means of calculating the tensile force between two rigid spheres with perfectly smooth surfaces. The Hertzian mode... | milkshake721/2.1M-wiki-STEM |
Contact mechanics-Adhesive contact between elastic bodies-It was observed that, though Hertz theory applied at large loads, at low loads the area of contact was larger than that predicted by Hertz theory, the area of contact had a non-zero value even when the load was removed, and there was even strong adhesion if the ... | milkshake721/2.1M-wiki-STEM |
Contact mechanics-Adhesive contact between elastic bodies-Bradley model of rigid contact It is commonly assumed that the surface force between two atomic planes at a distance z from each other can be derived from the Lennard-Jones potential. With this assumption 16 γ3z0[(zz0)−9−(zz0)−3] where F is the force (positive... | milkshake721/2.1M-wiki-STEM |
Contact mechanics-Adhesive contact between elastic bodies-The Bradley model applied the Lennard-Jones potential to find the force of adhesion between two rigid spheres. The total force between the spheres is found to be 16 γπR3[14(zz0)−8−(zz0)−2];1R=1R1+1R2 where R1,R2 are the radii of the two spheres.
The two spheres... | milkshake721/2.1M-wiki-STEM |
Contact mechanics-Adhesive contact between elastic bodies-Johnson-Kendall-Roberts (JKR) model of elastic contact To incorporate the effect of adhesion in Hertzian contact, Johnson, Kendall, and Roberts formulated the JKR theory of adhesive contact using a balance between the stored elastic energy and the loss in surfac... | milkshake721/2.1M-wiki-STEM |
Contact mechanics-Adhesive contact between elastic bodies-The radius of contact between two spheres from DMT theory is a3=3R4E∗(F+4γπR) and the pull-off force is Fc=−4γπR When the pull-off force is achieved the contact area becomes zero and there is no singularity in the contact stresses at the edge of the contact area... | milkshake721/2.1M-wiki-STEM |
Contact mechanics-Adhesive contact between elastic bodies-In terms of the work of adhesion Δγ a3=3R4E∗(F+2ΔγπR) and Fc=−2ΔγπR Tabor parameter In 1977, Tabor showed that the apparent contradiction between the JKR and DMT theories could be resolved by noting that the two theories were the extreme limits of a single theo... | milkshake721/2.1M-wiki-STEM |
Contact mechanics-Adhesive contact between elastic bodies-Non-dimensionalized values of a,c,F,d are introduced at this stage that are defied as := := := := πc2;F¯=FπΔγR In addition, Maugis proposed a parameter λ which is equivalent to the Tabor parameter μ . This parameter is defined as := 1.16 μ where the step cohe... | milkshake721/2.1M-wiki-STEM |
Contact mechanics-Adhesive contact between elastic bodies-Carpick-Ogletree-Salmeron (COS) model The Maugis-Dugdale model can only be solved iteratively if the value of λ is not known a-priori. The Carpick-Ogletree-Salmeron approximate solution simplifies the process by using the following relation to determine the con... | milkshake721/2.1M-wiki-STEM |
Swayback-Swayback-Swayback, also known clinically as lordosis, refers to abnormal bent-back postures in humans and in quadrupeds, especially horses. Extreme lordosis can cause physical damage to the spinal cord and associated ligaments and tendons which can lead to severe pain. Moderate lordosis does not generally impa... | milkshake721/2.1M-wiki-STEM |
Swayback-Humans-Swayback posture in humans is characterised by the posterior displacement of the rib cage in comparison to the pelvis. It looks like the person has a hyperextension of the lower back, however this is not necessarily the case. Most sway-back exhibits a posteriorly tilted pelvis; the lumbar region is usua... | milkshake721/2.1M-wiki-STEM |
Swayback-Horses-Usually called "swayback", soft back, or low back, an excessive downward bend in the back is an undesirable conformation trait. Swayback is caused in part from a loss of muscle tone in both the back and abdominal muscles, plus a weakening and stretching of the ligaments. As in humans, it may be influenc... | milkshake721/2.1M-wiki-STEM |
Collagen, type IX, alpha 2-Collagen, type IX, alpha 2-Collagen alpha-2(IX) chain is a protein that in humans is encoded by the COL9A2 gene.This gene encodes one of the three alpha chains of type IX collagen, the major collagen component of hyaline cartilage. Type IX collagen, a heterotrimeric molecule, is usually found... | milkshake721/2.1M-wiki-STEM |
Lower oceanic crust-Lower oceanic crust-The lower oceanic crust is the lower part of the oceanic crust and represents the major part of it (volumetrically biggest part). It is generally located 4–8 km below the ocean floor and the major lithologies are mafic (ultramafic and gabbroic rocks) which derive from melts risin... | milkshake721/2.1M-wiki-STEM |
Lower oceanic crust-Processes-The lower oceanic crust connects the earth's mantle with the MORB, where around 60% of the total magma production of the Earth happens. The three main processes happening in this region of the oceanic crust are partial melting of the earth's mantle, melt accumulation at various depths and ... | milkshake721/2.1M-wiki-STEM |
Lower oceanic crust-Spreading rates-The most important parameter controlling the processes operating in the lower oceanic crust is the magma supply, this is further controlled by the spreading rate, and therefore, spreading rate is a critical variable in models for the formation of the lower oceanic crust. The rate at ... | milkshake721/2.1M-wiki-STEM |
Lower oceanic crust-Spreading rates-Modally and compositionally layered gabbroic rock is often found (or abundant) in the lower crustal sections of ophiolite. The layered lower crust is thus one of the key features of all models of fast-spreading lower crust. Nevertheless, distinct modal layering as observed in major o... | milkshake721/2.1M-wiki-STEM |
Lower oceanic crust-Spreading rates-Slow-spreading ridges Slow- and intermediate-spreading ridges form typically valleys about 30 to 50 km (19 to 31 mi) wide and 1 to 5 km (0.62 to 3.11 mi) deep, with step-like inward-facing scarps, similar to rift valleys on land. Compared to fast spreading-ridges, the magma supply an... | milkshake721/2.1M-wiki-STEM |
Vesmír-Vesmír-Vesmír is a Czech science magazine that has been published since 1871. As of 2012, it is produced by the Czech Academy of Sciences and published by Academia. | milkshake721/2.1M-wiki-STEM |
Checksum-Checksum-A checksum is a small-sized block of data derived from another block of digital data for the purpose of detecting errors that may have been introduced during its transmission or storage. By themselves, checksums are often used to verify data integrity but are not relied upon to verify data authenticit... | milkshake721/2.1M-wiki-STEM |
Checksum-Checksum-Checksum functions are related to hash functions, fingerprints, randomization functions, and cryptographic hash functions. However, each of those concepts has different applications and therefore different design goals. For instance, a function returning the start of a string can provide a hash approp... | milkshake721/2.1M-wiki-STEM |
Checksum-Checksum-Check digits and parity bits are special cases of checksums, appropriate for small blocks of data (such as Social Security numbers, bank account numbers, computer words, single bytes, etc.). Some error-correcting codes are based on special checksums which not only detect common errors but also allow t... | milkshake721/2.1M-wiki-STEM |
Checksum-Algorithms-Parity byte or parity word The simplest checksum algorithm is the so-called longitudinal parity check, which breaks the data into "words" with a fixed number n of bits, and then computes the bitwise exclusive or (XOR) of all those words. The result is appended to the message as an extra word. In sim... | milkshake721/2.1M-wiki-STEM |
Checksum-Algorithms-Sum complement A variant of the previous algorithm is to add all the "words" as unsigned binary numbers, discarding any overflow bits, and append the two's complement of the total as the checksum. To validate a message, the receiver adds all the words in the same manner, including the checksum; if t... | milkshake721/2.1M-wiki-STEM |
Checksum-Algorithms-Position-dependent The simple checksums described above fail to detect some common errors which affect many bits at once, such as changing the order of data words, or inserting or deleting words with all bits set to zero. The checksum algorithms most used in practice, such as Fletcher's checksum, Ad... | milkshake721/2.1M-wiki-STEM |
Checksum-Algorithms-Fuzzy checksum The idea of fuzzy checksum was developed for detection of email spam by building up cooperative databases from multiple ISPs of email suspected to be spam. The content of such spam may often vary in its details, which would render normal checksumming ineffective. By contrast, a "fuzzy... | milkshake721/2.1M-wiki-STEM |
Checksum-Algorithms-General considerations A message that is m bits long can be viewed as a corner of the m-dimensional hypercube. The effect of a checksum algorithm that yields an n-bit checksum is to map each m-bit message to a corner of a larger hypercube, with dimension m + n. The 2m + n corners of this hypercube r... | milkshake721/2.1M-wiki-STEM |
Checksum-Algorithms-A single-bit transmission error then corresponds to a displacement from a valid corner (the correct message and checksum) to one of the m adjacent corners. An error which affects k bits moves the message to a corner which is k steps removed from its correct corner. The goal of a good checksum algori... | milkshake721/2.1M-wiki-STEM |
Horizontal situation indicator-Horizontal situation indicator-The horizontal situation indicator (commonly called the HSI) is an aircraft flight instrument normally mounted below the artificial horizon in place of a conventional heading indicator. It combines a heading indicator with a VHF omnidirectional range-instrum... | milkshake721/2.1M-wiki-STEM |
Horizontal situation indicator-Horizontal situation indicator-The most modern HSI displays are electronic and often integrated with electronic flight instrument systems into so-called "glass cockpit" systems. | milkshake721/2.1M-wiki-STEM |
Horizontal situation indicator-Horizontal situation indicator-HSI is part of a remote indicating compass system, which was developed to compensate for the errors and limitations of the older type of heading indicators. The two panel-mounted components of a typical system include the HSI and a slaving control and compen... | milkshake721/2.1M-wiki-STEM |
Xerox Character Code Standard-Xerox Character Code Standard-The Xerox Character Code Standard (XCCS) is a historical 16-bit character encoding that was created by Xerox in 1980 for the exchange of information between elements of the Xerox Network Systems Architecture. It encodes the characters required for languages us... | milkshake721/2.1M-wiki-STEM |
Xerox Character Code Standard-Code charts-Character sets overview Character set 0x00 Character set 0x21 Character set 0x22 Character set 0x23 Character set 0x24 Character set 0x25 Character set 0x26 Character set 0x27 Character set 0x28 Character set 0x30 Character set 0x31 Character set 0xE0 Character set 0xE1 Charact... | milkshake721/2.1M-wiki-STEM |
Fischer group Fi22-Fischer group Fi22-In the area of modern algebra known as group theory, the Fischer group Fi22 is a sporadic simple group of order 217 · 39 · 52 · 7 · 11 · 13 = 64561751654400 ≈ 6×1013. | milkshake721/2.1M-wiki-STEM |
Fischer group Fi22-History-Fi22 is one of the 26 sporadic groups and is the smallest of the three Fischer groups. It was introduced by Bernd Fischer (1971, 1976) while investigating 3-transposition groups.
The outer automorphism group has order 2, and the Schur multiplier has order 6. | milkshake721/2.1M-wiki-STEM |
Fischer group Fi22-Representations-The Fischer group Fi22 has a rank 3 action on a graph of 3510 vertices corresponding to its 3-transpositions, with point stabilizer the double cover of the group PSU6(2). It also has two rank 3 actions on 14080 points, exchanged by an outer automorphism.
Fi22 has an irreducible real r... | milkshake721/2.1M-wiki-STEM |
Fischer group Fi22-Generalized Monstrous Moonshine-Conway and Norton suggested in their 1979 paper that monstrous moonshine is not limited to the monster, but that similar phenomena may be found for other groups. Larissa Queen and others subsequently found that one can construct the expansions of many Hauptmoduln from ... | milkshake721/2.1M-wiki-STEM |
Fischer group Fi22-Maximal subgroups-Wilson (1984) found the 12 conjugacy classes of maximal subgroups of Fi22 as follows: 2·U6(2) O7(3) (Two classes, fused by an outer automorphism) O+8(2):S3 210:M22 26:S6(2) (2 × 21+8):(U4(2):2) U4(3):2 × S3 2F4(2)' (This is the Tits group) 25+8:(S3 × A6) 31+6:23+4:32:2 S10 (Two clas... | milkshake721/2.1M-wiki-STEM |
Hagedorn temperature-Hagedorn temperature-The Hagedorn temperature, TH, is the temperature in theoretical physics where hadronic matter (i.e. ordinary matter) is no longer stable, and must either "evaporate" or convert into quark matter; as such, it can be thought of as the "boiling point" of hadronic matter. It was di... | milkshake721/2.1M-wiki-STEM |
Hagedorn temperature-Hagedorn temperature-The Hagedorn temperature, TH, is about 150 MeV/kB or about 1.7×1012 K, little above the mass–energy of the lightest hadrons, the pion. Matter at Hagedorn temperature or above will spew out fireballs of new particles, which can again produce new fireballs, and the ejected partic... | milkshake721/2.1M-wiki-STEM |
Hagedorn temperature-Hagedorn temperature-In string theory, a separate Hagedorn temperature can be defined for strings rather than hadrons. This temperature is extremely high (1030 K) and thus of mainly theoretical interest. | milkshake721/2.1M-wiki-STEM |
Hagedorn temperature-History-The Hagedorn temperature was discovered by German physicist Rolf Hagedorn in the 1960s while working at CERN. His work on the statistical bootstrap model of hadron production showed that because increases in energy in a system will cause new particles to be produced, an increase of collisio... | milkshake721/2.1M-wiki-STEM |
Hagedorn temperature-Technical explanation-Hagedorn temperature is the temperature TH above which the partition sum diverges in a system with exponential growth in the density of states. | milkshake721/2.1M-wiki-STEM |
Hagedorn temperature-Technical explanation-lim Tr [e−βH]=∞ Because of the divergence, people may come to the incorrect conclusion that it is impossible to have temperatures above the Hagedorn temperature, which would make it the absolute hot temperature, because it would require an infinite amount of energy. In equati... | milkshake721/2.1M-wiki-STEM |
Hagedorn temperature-Technical explanation-The concept of exponential growth in the number of states was originally proposed in the context of condensed matter physics. It was incorporated into high-energy physics in the early 1970s by Steven Frautschi and Hagedorn. In hadronic physics, the Hagedorn temperature is the ... | milkshake721/2.1M-wiki-STEM |
Hagedorn temperature-In string theory-In string theory, it indicates a phase transition: the transition at which very long strings are copiously produced. It is controlled by the size of the string tension, which is smaller than the Planck scale by some power of the coupling constant. By adjusting the tension to be sma... | milkshake721/2.1M-wiki-STEM |
Mean curvature flow-Mean curvature flow-In the field of differential geometry in mathematics, mean curvature flow is an example of a geometric flow of hypersurfaces in a Riemannian manifold (for example, smooth surfaces in 3-dimensional Euclidean space). Intuitively, a family of surfaces evolves under mean curvature fl... | milkshake721/2.1M-wiki-STEM |
Mean curvature flow-Mean curvature flow-Under the constraint that volume enclosed is constant, this is called surface tension flow.
It is a parabolic partial differential equation, and can be interpreted as "smoothing". | milkshake721/2.1M-wiki-STEM |
Mean curvature flow-Existence and uniqueness-The following was shown by Michael Gage and Richard S. Hamilton as an application of Hamilton's general existence theorem for parabolic geometric flows.Let M be a compact smooth manifold, let (M′,g) be a complete smooth Riemannian manifold, and let f:M→M′ be a smooth imme... | milkshake721/2.1M-wiki-STEM |
Mean curvature flow-Existence and uniqueness-Necessarily, the restriction of F to (0,T)×M is C∞ One refers to F as the (maximally extended) mean curvature flow with initial data f | milkshake721/2.1M-wiki-STEM |
Mean curvature flow-Convergence theorems-Following Hamilton's epochal 1982 work on the Ricci flow, in 1984 Gerhard Huisken employed the same methods for the mean curvature flow to produce the following analogous result: If (M′,g) is the Euclidean space Rn+1 , where n≥2 denotes the dimension of M , then T is necess... | milkshake721/2.1M-wiki-STEM |
Mean curvature flow-Convergence theorems-Gage and Hamilton extended Huisken's result to the case n=1 . Matthew Grayson (1987) showed that if f:S1→R2 is any smooth embedding, then the mean curvature flow with initial data f eventually consists exclusively of embeddings with strictly positive curvature, at which point... | milkshake721/2.1M-wiki-STEM |
Mean curvature flow-Properties-The mean curvature flow extremalizes surface area, and minimal surfaces are the critical points for the mean curvature flow; minima solve the isoperimetric problem.
For manifolds embedded in a Kähler–Einstein manifold, if the surface is a Lagrangian submanifold, the mean curvature flow is... | milkshake721/2.1M-wiki-STEM |
Mean curvature flow-Mean curvature flow of a three-dimensional surface-The differential equation for mean-curvature flow of a surface given by z=S(x,y) is given by ∂S∂t=2DH(x,y)1+(∂S∂x)2+(∂S∂y)2 with D being a constant relating the curvature and the speed of the surface normal, and the mean curvature being H(x,y)=12(... | milkshake721/2.1M-wiki-STEM |
Mean curvature flow-Mean curvature flow of a three-dimensional surface-In the limits |∂S∂x|≪1 and |∂S∂y|≪1 , so that the surface is nearly planar with its normal nearly parallel to the z axis, this reduces to a diffusion equation ∂S∂t=D∇2S While the conventional diffusion equation is a linear parabolic partial differ... | milkshake721/2.1M-wiki-STEM |
Mean curvature flow-Mean curvature flow of a three-dimensional surface-Every smooth convex surface collapses to a point under the mean-curvature flow, without other singularities, and converges to the shape of a sphere as it does so. For surfaces of dimension two or more this is a theorem of Gerhard Huisken; for the on... | milkshake721/2.1M-wiki-STEM |
Mean curvature flow-Example: mean curvature flow of m-dimensional spheres-A simple example of mean curvature flow is given by a family of concentric round hyperspheres in Rm+1 . The mean curvature of an m -dimensional sphere of radius R is H=m/R Due to the rotational symmetry of the sphere (or in general, due to th... | milkshake721/2.1M-wiki-STEM |
Mean curvature flow-Example: mean curvature flow of m-dimensional spheres-The solution of this ODE (obtained, e.g., by separation of variables) is R(t)=R02−2mt ,which exists for t∈(−∞,R02/2m) | milkshake721/2.1M-wiki-STEM |
Gastrointestinal neuroectodermal tumor-Gastrointestinal neuroectodermal tumor-A gastrointestinal neuroectodermal tumor is a neuroectodermal tumor that appears in the gastrointestinal system. | milkshake721/2.1M-wiki-STEM |
UQCRC2-UQCRC2-Cytochrome b-c1 complex subunit 2, mitochondrial (UQCRC2), also known as QCR2, UQCR2, or MC3DN5 is a protein that in humans is encoded by the UQCRC2 gene. The product of UQCRC2 is a subunit of the respiratory chain protein Ubiquinol Cytochrome c Reductase (UQCR, Complex III or Cytochrome bc1 complex), whi... | milkshake721/2.1M-wiki-STEM |
UQCRC2-Structure-UQCRC2 is located on the p arm of chromosome 16 in position 12.2 and has 14 exons. The UQCRC2 gene produces a 48.4 kDa protein composed of 453 amino acids. UQCRC2 belongs to the peptidase M16 family and UQCRC2/QCR2 subfamily. UQCRC2 has a transit peptide domain. Ubiquinol Cytochrome c Reductase (b-c1 c... | milkshake721/2.1M-wiki-STEM |
UQCRC2-Function-The protein encoded by this gene is located in the mitochondrion, where it is part of the ubiquinol-cytochrome c reductase complex (also known as complex III). This complex constitutes a part of the mitochondrial respiratory chain. The core protein UQCRC2 is required for the assembly and stabilization o... | milkshake721/2.1M-wiki-STEM |
UQCRC2-Clinical Significance-Variants of UQCRC2 have been associated with mitochondrial complex III deficiency, nuclear, type 5. Mitochondrial complex III deficiency nuclear type 5 is a disorder of the mitochondrial respiratory chain resulting in a highly variable phenotype depending on which tissues are affected. Clin... | milkshake721/2.1M-wiki-STEM |
UQCRC2-Interactions-UQCRC2 has 98 protein-protein interactions with 90 of them being co-complex interactions. CAC1A, QCR1, UQCRC1, CACNA1A, STOM, a8k1f4, HLA-B, ARF6, and Mapk3 have been found to interact with UQCRC2. | milkshake721/2.1M-wiki-STEM |
Transversal (combinatorics)-Transversal (combinatorics)-In mathematics, particularly in combinatorics, given a family of sets, here called a collection C, a transversal (also called a cross-section) is a set containing exactly one element from each member of the collection. When the sets of the collection are mutually ... | milkshake721/2.1M-wiki-STEM |
Transversal (combinatorics)-Existence and number-A fundamental question in the study of SDR is whether or not an SDR exists. Hall's marriage theorem gives necessary and sufficient conditions for a finite collection of sets, some possibly overlapping, to have a transversal. The condition is that, for every integer k, ev... | milkshake721/2.1M-wiki-STEM |
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