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The first and second law of thermodynamics are the most fundamental equations of thermodynamics. They may be combined into what is known as fundamental thermodynamic relation which describes all of the changes of thermodynamic state functions of a system of uniform temperature and pressure. As a simple example, conside... | Thermodynamic equations | Wikipedia | 438 | 1515898 | https://en.wikipedia.org/wiki/Thermodynamic%20equations | Physical sciences | Thermodynamics | Physics |
By the principle of minimum energy, there are a number of other state functions which may be defined which have the dimensions of energy and which are minimized according to the second law under certain conditions other than constant entropy. These are called thermodynamic potentials. For each such potential, the relev... | Thermodynamic equations | Wikipedia | 512 | 1515898 | https://en.wikipedia.org/wiki/Thermodynamic%20equations | Physical sciences | Thermodynamics | Physics |
which is known as the Gibbs-Duhem relationship. The Gibbs-Duhem is a relationship among the intensive parameters of the system. It follows that for a simple system with r components, there will be r+1 independent parameters, or degrees of freedom. For example, a simple system with a single component will have two degre... | Thermodynamic equations | Wikipedia | 478 | 1515898 | https://en.wikipedia.org/wiki/Thermodynamic%20equations | Physical sciences | Thermodynamics | Physics |
Maxwell relations in thermodynamics are critical because they provide a means of simply measuring the change in properties of pressure, temperature, and specific volume, to determine a change in entropy. Entropy cannot be measured directly. The change in entropy with respect to pressure at a constant temperature is the... | Thermodynamic equations | Wikipedia | 323 | 1515898 | https://en.wikipedia.org/wiki/Thermodynamic%20equations | Physical sciences | Thermodynamics | Physics |
Tapejaridae (from a Tupi word meaning "the lord of the ways") is a family of azhdarchoid pterosaurs from the Cretaceous period. Members are currently known from Brazil, England, Hungary, Morocco, Spain, the United States, and China. The most primitive genera were found in China, indicating that the family has an Asian ... | Tapejaridae | Wikipedia | 456 | 7131739 | https://en.wikipedia.org/wiki/Tapejaridae | Biology and health sciences | Pterosaurs | Animals |
Tapejaridae was named and defined by Brazilian paleontologist Alexander Kellner in 1989 as the clade containing both Tapejara and Tupuxuara, plus all descendants of their most recent common ancestor. In 2007, Kellner divided the family: Tapejarinae, consisting of Tapejara and its close relatives, and Thalassodrominae, ... | Tapejaridae | Wikipedia | 372 | 7131739 | https://en.wikipedia.org/wiki/Tapejaridae | Biology and health sciences | Pterosaurs | Animals |
Regarding the core tapejarid clade, American paleontologist Brian Andres and colleagues formally defined Tapejaridae as the clade containing Tapejara and Sinopterus in 2014. They also re-defined the subfamily Tapejarinae as all species closer to Tapejara than to Sinopterus, and added a new clade, Tapejarini, to include... | Tapejaridae | Wikipedia | 499 | 7131739 | https://en.wikipedia.org/wiki/Tapejaridae | Biology and health sciences | Pterosaurs | Animals |
The cladogram below shows the phylogenetic analysis conducted by paleontologist Gabriela Cerqueira and colleagues in 2021, which uses Kellner's nomenclature of Tapejaridae.
Below are two cladograms representing different concepts of Tapejaridae. The first one shows the phylogenetic analysis conducted by Andres in 2021... | Tapejaridae | Wikipedia | 324 | 7131739 | https://en.wikipedia.org/wiki/Tapejaridae | Biology and health sciences | Pterosaurs | Animals |
Telluric contamination is contamination of the astronomical spectra by the Earth's atmosphere.
Interference with astronomical observations
Most astronomical observations are conducted by measuring photons (electromagnetic waves) which originate beyond the sky. The molecules in the Earth's atmosphere, however, absorb ... | Telluric contamination | Wikipedia | 341 | 5464288 | https://en.wikipedia.org/wiki/Telluric%20contamination | Physical sciences | Basics | Astronomy |
In the mathematical area of graph theory, Kőnig's theorem, proved by , describes an equivalence between the maximum matching problem and the minimum vertex cover problem in bipartite graphs. It was discovered independently, also in 1931, by Jenő Egerváry in the more general case of weighted graphs.
Setting
A vertex c... | Kőnig's theorem (graph theory) | Wikipedia | 488 | 5465118 | https://en.wikipedia.org/wiki/K%C5%91nig%27s%20theorem%20%28graph%20theory%29 | Mathematics | Graph theory | null |
The size of the maximum matching in is the size of a maximum flow in , which, in turn, is the size of a minimum cut in the network , as follows from the max-flow min-cut theorem.
Let be a minimum cut. Let and , such that and . Then the minimum cut is composed only of edges going from to or from to , as any edg... | Kőnig's theorem (graph theory) | Wikipedia | 505 | 5465118 | https://en.wikipedia.org/wiki/K%C5%91nig%27s%20theorem%20%28graph%20theory%29 | Mathematics | Graph theory | null |
Proof using linear programming duality
To explain this proof, we first have to extend the notion of a matching to that of a fractional matching - an assignment of a weight in [0,1] to each edge, such that the sum of weights near each vertex is at most 1 (an integral matching is a special case of a fractional matching ... | Kőnig's theorem (graph theory) | Wikipedia | 415 | 5465118 | https://en.wikipedia.org/wiki/K%C5%91nig%27s%20theorem%20%28graph%20theory%29 | Mathematics | Graph theory | null |
Now, the minimum fractional cover LP is exactly the dual linear program of the maximum fractional matching LP. Therefore, by the LP duality theorem, both programs have the same solution. This fact is true not only in bipartite graphs but in arbitrary graphs:In any graph, the largest size of a fractional matching equals... | Kőnig's theorem (graph theory) | Wikipedia | 485 | 5465118 | https://en.wikipedia.org/wiki/K%C5%91nig%27s%20theorem%20%28graph%20theory%29 | Mathematics | Graph theory | null |
Non-bipartite graphs
For graphs that are not bipartite, the minimum vertex cover may be larger than the maximum matching. Moreover, the two problems are very different in complexity: maximum matchings can be found in polynomial time for any graph, while minimum vertex cover is NP-complete.
The complement of a vertex ... | Kőnig's theorem (graph theory) | Wikipedia | 482 | 5465118 | https://en.wikipedia.org/wiki/K%C5%91nig%27s%20theorem%20%28graph%20theory%29 | Mathematics | Graph theory | null |
A graph is perfect if and only if its complement is perfect, and Kőnig's theorem can be seen as equivalent to the statement that the complement of a bipartite graph is perfect. For, each color class in a coloring of the complement of a bipartite graph is of size at most 2 and the classes of size 2 form a matching, a cl... | Kőnig's theorem (graph theory) | Wikipedia | 501 | 5465118 | https://en.wikipedia.org/wiki/K%C5%91nig%27s%20theorem%20%28graph%20theory%29 | Mathematics | Graph theory | null |
Weighted variants
Konig's theorem can be extended to weighted graphs.
Egerváry's theorem for edge-weighted graphs
Jenő Egerváry (1931) considered graphs in which each edge e has a non-negative integer weight we. The weight vector is denoted by w. The w-weight of a matching is the sum of weights of the edges particip... | Kőnig's theorem (graph theory) | Wikipedia | 451 | 5465118 | https://en.wikipedia.org/wiki/K%C5%91nig%27s%20theorem%20%28graph%20theory%29 | Mathematics | Graph theory | null |
Europasaurus (meaning 'Europe lizard') is a basal macronarian sauropod, a form of quadrupedal herbivorous dinosaur. It lived during the Late Jurassic (middle Kimmeridgian, from about 154 to 151 million years ago) of northern Germany, and has been identified as an example of insular dwarfism resulting from the isolation... | Europasaurus | Wikipedia | 323 | 5468113 | https://en.wikipedia.org/wiki/Europasaurus | Biology and health sciences | Sauropods | Animals |
After more fossil material was found, including bones, excavation of the bone-bearing layer commenced in April 1999, conducted by a local association of private fossil collectors. Although the quarry operator was cooperative, excavation was complicated by the near-vertical orientation of the layers that limited access,... | Europasaurus | Wikipedia | 460 | 5468113 | https://en.wikipedia.org/wiki/Europasaurus | Biology and health sciences | Sauropods | Animals |
In 2006, the new sauropod taxon was formally described as Europasaurus holgeri. The given etymology for the genus name is "reptile from Europe", and the specific name honours Holger Lüdtke, the discoverer of the first fossils. Given the comparatively small size of the bones, it was initially assumed that they stem from... | Europasaurus | Wikipedia | 466 | 5468113 | https://en.wikipedia.org/wiki/Europasaurus | Biology and health sciences | Sauropods | Animals |
Aside from being a very small neosauropod, Europasaurus was thought to have multiple unique morphological features to distinguish it from close relatives by its original describers, Sander et al. (2006). The nasal process of the premaxilla was thought to curve anteriorly while projecting upwards (now known to be preser... | Europasaurus | Wikipedia | 329 | 5468113 | https://en.wikipedia.org/wiki/Europasaurus | Biology and health sciences | Sauropods | Animals |
Nearly all external skull bones have been preserved among Europasaurus specimens, except the prefrontals. Some additional bones are only represented by very fragmented and uninformative fossils, such as the lacrimals. Eight premaxillae are known, with a generally rectangular snout shape as found in Camarasaurus. The an... | Europasaurus | Wikipedia | 301 | 5468113 | https://en.wikipedia.org/wiki/Europasaurus | Biology and health sciences | Sauropods | Animals |
A single maxilla is present in the well-preserved material of Europasaurus, DFMMh/FV 291.17. This maxilla has a long body, with two elongate processes, a nasal and a posterior process. There is only a weak lacrimal process, like in most sauropods except Rapetosaurus. The nasal process is elongate and covers the anterio... | Europasaurus | Wikipedia | 343 | 5468113 | https://en.wikipedia.org/wiki/Europasaurus | Biology and health sciences | Sauropods | Animals |
Among the nasal bones of Europasaurus, several are known, but few are complete or undistorted. The nasals are overlapped posteriorly by the frontal bones, and towards the side, they articulate bluntly with the prefrontals. Unlike the nasals of Giraffatitan, those in Europasaurus project horizontally forwards, forming a... | Europasaurus | Wikipedia | 479 | 5468113 | https://en.wikipedia.org/wiki/Europasaurus | Biology and health sciences | Sauropods | Animals |
A triradiate postorbital bone is present in Europasaurus, which evolved as the fusion of the postfrontal and postosbital bone of more basal taxa. Between the anterior and ventrally projecting processes the postorbital forms the margin of the orbit, and between the posterior and ventral processes it borders the infratem... | Europasaurus | Wikipedia | 73 | 5468113 | https://en.wikipedia.org/wiki/Europasaurus | Biology and health sciences | Sauropods | Animals |
Multiple jugals are known from Europasaurus, which are more similar in morphology to basal sauropodomorphs than other macronarians. It forms part of the border of the orbit, infratemporal fenestra and the bottom edge of the skull, but does not reach the antorbital fenestra. The posterior process of the jugal are very f... | Europasaurus | Wikipedia | 444 | 5468113 | https://en.wikipedia.org/wiki/Europasaurus | Biology and health sciences | Sauropods | Animals |
There are thirteen preserved elements of the palate of Europasaurus, including the quadrate, pterygoid and ectopterygoid. The quadrates articulate with the palate and braincase bones, as well as the external skull bones. They are similar in shape to those of Giraffatitan and Camarasaurus, and have well-developed articu... | Europasaurus | Wikipedia | 257 | 5468113 | https://en.wikipedia.org/wiki/Europasaurus | Biology and health sciences | Sauropods | Animals |
The cervical vertebrae of Europasaurus are the best preserved and most represented of the vertebral column. However, not the entire neck is known, so the cervical number could be between Camarasaurus (12 vertebrae) and Rapetosaurus (17 vertebrae). Additionally, the multiple cervical vertebrae come from different-aged i... | Europasaurus | Wikipedia | 499 | 5468113 | https://en.wikipedia.org/wiki/Europasaurus | Biology and health sciences | Sauropods | Animals |
A series of all complete is only known from a single specimen, DFMMh/FV 100, which was destroyed in a fire in 2003. All five vertebrae, the characteristic number of more basal neosauropods, are incorporated into the . The third and fourth sacrals represent the primordial sacrals, present in all dinosaur groups. The se... | Europasaurus | Wikipedia | 499 | 5468113 | https://en.wikipedia.org/wiki/Europasaurus | Biology and health sciences | Sauropods | Animals |
Classification
When it was first named, Europasaurus was considered to be a taxon within Macronaria that didn't fall within the family Brachiosauridae or the clade Titanosauromorpha. This indicated that the dwarfism of the taxon was a result of evolution, instead of being a characteristic of a group. Three matrices wer... | Europasaurus | Wikipedia | 339 | 5468113 | https://en.wikipedia.org/wiki/Europasaurus | Biology and health sciences | Sauropods | Animals |
Placement as a brachiosaurid
In a 2012 analysis of the phylogeny of Titanosauriformes, D'Emic (2012) considered Europasaurus to belong to Brachiosauridae, instead of being basal to the earliest brachiosaurids. The phylogeny resolved the most true brachiosaurids to date, although several potential brachiosaurids were i... | Europasaurus | Wikipedia | 354 | 5468113 | https://en.wikipedia.org/wiki/Europasaurus | Biology and health sciences | Sauropods | Animals |
A further phylogenetic analysis was performed on Brachiosauridae, based on that of D'Emic (2012). This phylogeny, conducted by D'Emic et al. (2016), resolved a very similar placement of Europasaurus within Brachiosauridae, although Sonorasaurus was placed in a clade with Giraffatitan, and Lusotitan was placed in a poly... | Europasaurus | Wikipedia | 398 | 5468113 | https://en.wikipedia.org/wiki/Europasaurus | Biology and health sciences | Sauropods | Animals |
Growth
It was identified that Europasaurus was a unique dwarf species, and not a juvenile of an existing taxon like Camarasaurus, by a histology analysis of multiple specimens of Europasaurus. The youngest specimen (DFMMh/FV 009) was shown by this analysis to lack signs of aging such as growth marks or laminar bone ... | Europasaurus | Wikipedia | 438 | 5468113 | https://en.wikipedia.org/wiki/Europasaurus | Biology and health sciences | Sauropods | Animals |
Examinations of the inner ears of infant Europasaurus suggest they were precocial, and it is suggested that they would have been reliant on the protections of adults in a herd to some degree, something not seen in larger sauropods due to the massive size difference in parent and offspring. The structure and long length... | Europasaurus | Wikipedia | 380 | 5468113 | https://en.wikipedia.org/wiki/Europasaurus | Biology and health sciences | Sauropods | Animals |
The Langenberg locality in Germany, from the early Oxfordian to late Kimmeridgian, displays the variety of plant and animal life from an island ecosystem from the late Jurassic. During the Kimmeridgian the locality would have been marine, being located between the Rhenisch, Bohemian, and London-Brabant Massifs. This do... | Europasaurus | Wikipedia | 379 | 5468113 | https://en.wikipedia.org/wiki/Europasaurus | Biology and health sciences | Sauropods | Animals |
Conifer cones and twigs can be identified as the araucarian Brachyphyllum, from the terrestrial fossils of the site. Deposited in the locality are many taxa, including a large accumulation of Europasaurus bones and individuals. At least 450 bones from Europasaurus were recovered from the Langenberg Quarry, with about 1... | Europasaurus | Wikipedia | 450 | 5468113 | https://en.wikipedia.org/wiki/Europasaurus | Biology and health sciences | Sauropods | Animals |
Dinosaur footprints preserved at the Langenberg Quarry display a possible reason for the extinction of Europasaurus, and potentially other insular dwarfs present on the islands of the region. The footprints are located above the deposit of Europasaurus individuals, which shows that at least 35,000 years after that dep... | Europasaurus | Wikipedia | 184 | 5468113 | https://en.wikipedia.org/wiki/Europasaurus | Biology and health sciences | Sauropods | Animals |
The Iberian pig, also known in Portugal as the Alentejo Pig, is a traditional breed of the domestic pig (Sus scrofa domesticus) that is native to the Iberian Peninsula. The Iberian pig, whose origins can probably be traced back to the Neolithic, when animal domestication started, is currently found in herds clustered i... | Black Iberian pig | Wikipedia | 510 | 15939419 | https://en.wikipedia.org/wiki/Black%20Iberian%20pig | Biology and health sciences | Pigs | Animals |
At least a hectare of healthy dehesa is needed to raise a single pig, and since the trees may be several hundred years old, the prospects for reforesting lost dehesa are slim at best. True dehesa is a richly diverse habitat with four different types of oak that are crucial in the production of prime-quality ham. The b... | Black Iberian pig | Wikipedia | 163 | 15939419 | https://en.wikipedia.org/wiki/Black%20Iberian%20pig | Biology and health sciences | Pigs | Animals |
Red bananas are a group of varieties of bananas with reddish-purple skin. Some are smaller and plumper than the common Cavendish banana, others much larger. Ripe, raw red bananas have a flesh that is creamy to light pink. They are also softer and sweeter than the yellow Cavendish varieties, some with a slight tangy ras... | Red banana | Wikipedia | 506 | 13261606 | https://en.wikipedia.org/wiki/Red%20banana | Biology and health sciences | Tropical and tropical-like fruit | Plants |
History
The first bananas to appear on the market in Toronto (in the 1870s and 1880s) were red bananas. Red bananas are available year-round at specialty markets and larger supermarkets in the United States.
Uses
Culinary
Red bananas are eaten in the same way as yellow bananas, by peeling the fruit before eating. T... | Red banana | Wikipedia | 151 | 13261606 | https://en.wikipedia.org/wiki/Red%20banana | Biology and health sciences | Tropical and tropical-like fruit | Plants |
A security bug or security defect is a software bug that can be exploited to gain unauthorized access or privileges on a computer system. Security bugs introduce security vulnerabilities by compromising one or more of:
Authentication of users and other entities
Authorization of access rights and privileges
Data co... | Security bug | Wikipedia | 215 | 8632926 | https://en.wikipedia.org/wiki/Security%20bug | Technology | Computer security | null |
Ceroxylon quindiuense, often called Quindío wax palm, is a palm native to the humid montane forests of the Andes in Colombia and Peru.
Description
This palm species can grow to a height of —or rarely, even as high as . It is the tallest recorded monocot in the world. The trunk is cylindrical, smooth, light colored, ... | Ceroxylon quindiuense | Wikipedia | 494 | 8634017 | https://en.wikipedia.org/wiki/Ceroxylon%20quindiuense | Biology and health sciences | Arecales (inc. Palms) | Plants |
Cultivation and uses
The wax of the trunk was used to make candles, especially in the 19th century. The outer part of the stem of the palm has been used locally for building houses, and was used to build water supply systems for impoverished farmers. It is cultivated as an ornamental plant in Colombia and California.
... | Ceroxylon quindiuense | Wikipedia | 62 | 8634017 | https://en.wikipedia.org/wiki/Ceroxylon%20quindiuense | Biology and health sciences | Arecales (inc. Palms) | Plants |
Celestial cartography, uranography,
astrography or star cartography is the aspect of astronomy and branch of cartography concerned with mapping stars, galaxies, and other astronomical objects on the celestial sphere. Measuring the position and light of charted objects requires a variety of instruments and techniques. ... | Celestial cartography | Wikipedia | 485 | 10780895 | https://en.wikipedia.org/wiki/Celestial%20cartography | Physical sciences | Celestial sphere: General | Astronomy |
Important historical star tables
c:AD 150, Almagest – contains the last known star table from antiquity, prepared by Ptolemy, 1,028 stars.
c.964, Book of the Fixed Stars, Arabic version of the Almagest by al-Sufi.
1627, Rudolphine Tables – contains the first West Enlightenment star table, based on measurements of Ty... | Celestial cartography | Wikipedia | 498 | 10780895 | https://en.wikipedia.org/wiki/Celestial%20cartography | Physical sciences | Celestial sphere: General | Astronomy |
Telescopic
1729 Atlas Coelestis by John Flamsteed
1801 Uranographia by Johann Elert Bode
1843 Uranometria Nova by Friedrich Wilhelm Argelander
Photographic
1914 Franklin-Adams Charts, by John Franklin-Adams, a very early photographic atlas.
The Falkau Atlas (Hans Vehrenberg). Stars to magnitude 13.
Atlas Stellar... | Celestial cartography | Wikipedia | 140 | 10780895 | https://en.wikipedia.org/wiki/Celestial%20cartography | Physical sciences | Celestial sphere: General | Astronomy |
Bright Star Atlas – Wil Tirion (stars to magnitude 6.5)
Cambridge Star Atlas – Wil Tirion (Stars to magnitude 6.5)
Norton's Star Atlas and Reference Handbook – Ed. Ian Ridpath (stars to magnitude 6.5)
Stars & Planets Guide – Ian Ridpath and Wil Tirion (stars to magnitude 6.0)
Cambridge Double Star Atlas – James Mul... | Celestial cartography | Wikipedia | 506 | 10780895 | https://en.wikipedia.org/wiki/Celestial%20cartography | Physical sciences | Celestial sphere: General | Astronomy |
Free and printable from files
The TriAtlas Project
Toshimi Taki Star Atlases
DeepSky Hunter Star Atlas
Andrew Johnson mag 7 | Celestial cartography | Wikipedia | 28 | 10780895 | https://en.wikipedia.org/wiki/Celestial%20cartography | Physical sciences | Celestial sphere: General | Astronomy |
Wattieza was a genus of prehistoric trees that existed in the mid-Devonian that belong to the cladoxylopsids, close relatives of the modern ferns and horsetails. The 2005 discovery (publicly revealed in 2007) in Schoharie County, New York, of fossils from the Middle Devonian about 385 million years ago united the crown... | Wattieza | Wikipedia | 481 | 10780959 | https://en.wikipedia.org/wiki/Wattieza | Biology and health sciences | Pteridophytes | Plants |
E–Z configuration, or the E–Z convention, is the IUPAC preferred method of describing the absolute stereochemistry of double bonds in organic chemistry. It is an extension of cis–trans isomer notation (which only describes relative stereochemistry) that can be used to describe double bonds having two, three or four sub... | E–Z notation | Wikipedia | 479 | 10791959 | https://en.wikipedia.org/wiki/E%E2%80%93Z%20notation | Physical sciences | Stereochemistry | Chemistry |
Undefined ene stereochemistry
The prefix 'E/Z-' can be used to indicate uncertainty in the E or Z isomers for an ene bond. For graphical representations, wavy single bonds are the standard way to represent unknown or unspecified stereochemistry or a mixture of isomers (as with tetrahedral stereocenters). A crossed dou... | E–Z notation | Wikipedia | 104 | 10791959 | https://en.wikipedia.org/wiki/E%E2%80%93Z%20notation | Physical sciences | Stereochemistry | Chemistry |
Coastal geography is the study of the constantly changing region between the ocean and the land, incorporating both the physical geography (i.e. coastal geomorphology, climatology and oceanography) and the human geography (sociology and history) of the coast. It includes understanding coastal weathering processes, par... | Coastal geography | Wikipedia | 333 | 2160676 | https://en.wikipedia.org/wiki/Coastal%20geography | Physical sciences | Oceanic and coastal landforms | Earth science |
Probably the most important effect is longshore drift (LSD)(Also known as Littoral Drift), the process by which sediment is continuously moved along beaches by wave action. LSD occurs because waves hit the shore at an angle, pick up sediment (sand) on the shore and carry it down the beach at an angle (this is called sw... | Coastal geography | Wikipedia | 495 | 2160676 | https://en.wikipedia.org/wiki/Coastal%20geography | Physical sciences | Oceanic and coastal landforms | Earth science |
Physical processes
The main physical Weathering process on beaches is salt-crystal growth. Wind carries salt spray onto rocks, where it is absorbed into small pores and cracks within the rocks. There the water evaporates and the salt crystallises, creating pressure and often breaking down the rock. In some beaches calc... | Coastal geography | Wikipedia | 445 | 2160676 | https://en.wikipedia.org/wiki/Coastal%20geography | Physical sciences | Oceanic and coastal landforms | Earth science |
If the coast suddenly changes direction, especially around an estuary, spits are likely to form. Long shore drift pushes the sediment along the beach but when it reaches a turn as in the diagram, the long shore drift does not always easily turn with it, especially near an estuary where the outward flow from a river may... | Coastal geography | Wikipedia | 450 | 2160676 | https://en.wikipedia.org/wiki/Coastal%20geography | Physical sciences | Oceanic and coastal landforms | Earth science |
In linear algebra, an eigenvector ( ) or characteristic vector is a vector that has its direction unchanged (or reversed) by a given linear transformation. More precisely, an eigenvector, , of a linear transformation, , is scaled by a constant factor, , when the linear transformation is applied to it: . The correspondi... | Eigenvalues and eigenvectors | Wikipedia | 463 | 2161429 | https://en.wikipedia.org/wiki/Eigenvalues%20and%20eigenvectors | Mathematics | Algebra | null |
Overview
Eigenvalues and eigenvectors feature prominently in the analysis of linear transformations. The prefix eigen- is adopted from the German word eigen (cognate with the English word own) for 'proper', 'characteristic', 'own'. Originally used to study principal axes of the rotational motion of rigid bodies, eigen... | Eigenvalues and eigenvectors | Wikipedia | 500 | 2161429 | https://en.wikipedia.org/wiki/Eigenvalues%20and%20eigenvectors | Mathematics | Algebra | null |
Alternatively, the linear transformation could take the form of an n by n matrix, in which case the eigenvectors are n by 1 matrices. If the linear transformation is expressed in the form of an n by n matrix A, then the eigenvalue equation for a linear transformation above can be rewritten as the matrix multiplication
... | Eigenvalues and eigenvectors | Wikipedia | 340 | 2161429 | https://en.wikipedia.org/wiki/Eigenvalues%20and%20eigenvectors | Mathematics | Algebra | null |
In the early 19th century, Augustin-Louis Cauchy saw how their work could be used to classify the quadric surfaces, and generalized it to arbitrary dimensions. Cauchy also coined the term racine caractéristique (characteristic root), for what is now called eigenvalue; his term survives in characteristic equation.{{efn|... | Eigenvalues and eigenvectors | Wikipedia | 450 | 2161429 | https://en.wikipedia.org/wiki/Eigenvalues%20and%20eigenvectors | Mathematics | Algebra | null |
Around the same time, Francesco Brioschi proved that the eigenvalues of orthogonal matrices lie on the unit circle, and Alfred Clebsch found the corresponding result for skew-symmetric matrices. Finally, Karl Weierstrass clarified an important aspect in the stability theory started by Laplace, by realizing that defecti... | Eigenvalues and eigenvectors | Wikipedia | 489 | 2161429 | https://en.wikipedia.org/wiki/Eigenvalues%20and%20eigenvectors | Mathematics | Algebra | null |
then is an eigenvector of the linear transformation and the scale factor is the eigenvalue corresponding to that eigenvector. Equation () is the eigenvalue equation for the matrix .
Equation () can be stated equivalently as
where is the by identity matrix and 0 is the zero vector.
Eigenvalues and the characte... | Eigenvalues and eigenvectors | Wikipedia | 403 | 2161429 | https://en.wikipedia.org/wiki/Eigenvalues%20and%20eigenvectors | Mathematics | Algebra | null |
If the entries of the matrix A are all real numbers, then the coefficients of the characteristic polynomial will also be real numbers, but the eigenvalues may still have nonzero imaginary parts. The entries of the corresponding eigenvectors therefore may also have nonzero imaginary parts. Similarly, the eigenvalues may... | Eigenvalues and eigenvectors | Wikipedia | 506 | 2161429 | https://en.wikipedia.org/wiki/Eigenvalues%20and%20eigenvectors | Mathematics | Algebra | null |
If μA(λi) = 1, then λi is said to be a simple eigenvalue. If μA(λi) equals the geometric multiplicity of λi, γA(λi), defined in the next section, then λi is said to be a semisimple eigenvalue.
Eigenspaces, geometric multiplicity, and the eigenbasis for matrices
Given a particular eigenvalue λ of the n by n matrix A... | Eigenvalues and eigenvectors | Wikipedia | 406 | 2161429 | https://en.wikipedia.org/wiki/Eigenvalues%20and%20eigenvectors | Mathematics | Algebra | null |
The dimension of the eigenspace E associated with λ, or equivalently the maximum number of linearly independent eigenvectors associated with λ, is referred to as the eigenvalue's geometric multiplicity . Because E is also the nullspace of (A − λI), the geometric multiplicity of λ is the dimension of the nullspace of (A... | Eigenvalues and eigenvectors | Wikipedia | 497 | 2161429 | https://en.wikipedia.org/wiki/Eigenvalues%20and%20eigenvectors | Mathematics | Algebra | null |
Additional properties
Let be an arbitrary matrix of complex numbers with eigenvalues . Each eigenvalue appears times in this list, where is the eigenvalue's algebraic multiplicity. The following are properties of this matrix and its eigenvalues:
The trace of , defined as the sum of its diagonal elements, is also ... | Eigenvalues and eigenvectors | Wikipedia | 510 | 2161429 | https://en.wikipedia.org/wiki/Eigenvalues%20and%20eigenvectors | Mathematics | Algebra | null |
Comparing this equation to equation (), it follows immediately that a left eigenvector of is the same as the transpose of a right eigenvector of , with the same eigenvalue. Furthermore, since the characteristic polynomial of is the same as the characteristic polynomial of , the left and right eigenvectors of are ass... | Eigenvalues and eigenvectors | Wikipedia | 409 | 2161429 | https://en.wikipedia.org/wiki/Eigenvalues%20and%20eigenvectors | Mathematics | Algebra | null |
Conversely, suppose a matrix A is diagonalizable. Let P be a non-singular square matrix such that P−1AP is some diagonal matrix D. Left multiplying both by P, . Each column of P must therefore be an eigenvector of A whose eigenvalue is the corresponding diagonal element of D. Since the columns of P must be linearly ind... | Eigenvalues and eigenvectors | Wikipedia | 511 | 2161429 | https://en.wikipedia.org/wiki/Eigenvalues%20and%20eigenvectors | Mathematics | Algebra | null |
The roots of the characteristic polynomial are 2, 1, and 11, which are the only three eigenvalues of A. These eigenvalues correspond to the eigenvectors and or any nonzero multiple thereof.
Three-dimensional matrix example with complex eigenvalues
Consider the cyclic permutation matrix
This matrix shifts the coo... | Eigenvalues and eigenvectors | Wikipedia | 473 | 2161429 | https://en.wikipedia.org/wiki/Eigenvalues%20and%20eigenvectors | Mathematics | Algebra | null |
The roots of this polynomial, and hence the eigenvalues, are 2 and 3. The algebraic multiplicity of each eigenvalue is 2; in other words they are both double roots. The sum of the algebraic multiplicities of all distinct eigenvalues is μA = 4 = n, the order of the characteristic polynomial and the dimension of A.
On t... | Eigenvalues and eigenvectors | Wikipedia | 509 | 2161429 | https://en.wikipedia.org/wiki/Eigenvalues%20and%20eigenvectors | Mathematics | Algebra | null |
General definition
The concept of eigenvalues and eigenvectors extends naturally to arbitrary linear transformations on arbitrary vector spaces. Let V be any vector space over some field K of scalars, and let T be a linear transformation mapping V into V,
We say that a nonzero vector v ∈ V is an eigenvector of T if a... | Eigenvalues and eigenvectors | Wikipedia | 499 | 2161429 | https://en.wikipedia.org/wiki/Eigenvalues%20and%20eigenvectors | Mathematics | Algebra | null |
Any subspace spanned by eigenvectors of T is an invariant subspace of T, and the restriction of T to such a subspace is diagonalizable. Moreover, if the entire vector space V can be spanned by the eigenvectors of T, or equivalently if the direct sum of the eigenspaces associated with all the eigenvalues of T is the ent... | Eigenvalues and eigenvectors | Wikipedia | 506 | 2161429 | https://en.wikipedia.org/wiki/Eigenvalues%20and%20eigenvectors | Mathematics | Algebra | null |
A similar procedure is used for solving a differential equation of the form
Calculation
The calculation of eigenvalues and eigenvectors is a topic where theory, as presented in elementary linear algebra textbooks, is often very far from practice.
Classical method
The classical method is to first find the eigen... | Eigenvalues and eigenvectors | Wikipedia | 500 | 2161429 | https://en.wikipedia.org/wiki/Eigenvalues%20and%20eigenvectors | Mathematics | Algebra | null |
Both equations reduce to the single linear equation . Therefore, any vector of the form , for any nonzero real number , is an eigenvector of with eigenvalue .
The matrix above has another eigenvalue . A similar calculation shows that the corresponding eigenvectors are the nonzero solutions of , that is, any vector o... | Eigenvalues and eigenvectors | Wikipedia | 440 | 2161429 | https://en.wikipedia.org/wiki/Eigenvalues%20and%20eigenvectors | Mathematics | Algebra | null |
The characteristic equation for a rotation is a quadratic equation with discriminant , which is a negative number whenever is not an integer multiple of 180°. Therefore, except for these special cases, the two eigenvalues are complex numbers, ; and all eigenvectors have non-real entries. Indeed, except for those speci... | Eigenvalues and eigenvectors | Wikipedia | 421 | 2161429 | https://en.wikipedia.org/wiki/Eigenvalues%20and%20eigenvectors | Mathematics | Algebra | null |
Graphs
In spectral graph theory, an eigenvalue of a graph is defined as an eigenvalue of the graph's adjacency matrix , or (increasingly) of the graph's Laplacian matrix due to its discrete Laplace operator, which is either (sometimes called the combinatorial Laplacian) or (sometimes called the normalized Laplacian)... | Eigenvalues and eigenvectors | Wikipedia | 490 | 2161429 | https://en.wikipedia.org/wiki/Eigenvalues%20and%20eigenvectors | Mathematics | Algebra | null |
In dimensions, becomes a mass matrix and a stiffness matrix. Admissible solutions are then a linear combination of solutions to the generalized eigenvalue problem
where is the eigenvalue and is the (imaginary) angular frequency. The principal vibration modes are different from the principal compliance modes, whic... | Eigenvalues and eigenvectors | Wikipedia | 489 | 2161429 | https://en.wikipedia.org/wiki/Eigenvalues%20and%20eigenvectors | Mathematics | Algebra | null |
The bra–ket notation is often used in this context. A vector, which represents a state of the system, in the Hilbert space of square integrable functions is represented by . In this notation, the Schrödinger equation is:
where is an eigenstate of and represents the eigenvalue. is an observable self-adjoint operat... | Eigenvalues and eigenvectors | Wikipedia | 489 | 2161429 | https://en.wikipedia.org/wiki/Eigenvalues%20and%20eigenvectors | Mathematics | Algebra | null |
In geology, especially in the study of glacial till, eigenvectors and eigenvalues are used as a method by which a mass of information of a clast's fabric can be summarized in a 3-D space by six numbers. In the field, a geologist may collect such data for hundreds or thousands of clasts in a soil sample, which can be co... | Eigenvalues and eigenvectors | Wikipedia | 439 | 2161429 | https://en.wikipedia.org/wiki/Eigenvalues%20and%20eigenvectors | Mathematics | Algebra | null |
In image processing, processed images of faces can be seen as vectors whose components are the brightnesses of each pixel. The dimension of this vector space is the number of pixels. The eigenvectors of the covariance matrix associated with a large set of normalized pictures of faces are called eigenfaces; this is an e... | Eigenvalues and eigenvectors | Wikipedia | 207 | 2161429 | https://en.wikipedia.org/wiki/Eigenvalues%20and%20eigenvectors | Mathematics | Algebra | null |
Ceprano Man, Argil, and Ceprano Calvarium, is a Middle Pleistocene archaic human fossil, a single skull cap (calvarium), accidentally unearthed in a highway construction project in 1994 near Ceprano in the Province of Frosinone, Italy. It was initially considered Homo cepranensis, Homo erectus, or possibly Homo anteces... | Ceprano Man | Wikipedia | 498 | 2165266 | https://en.wikipedia.org/wiki/Ceprano%20Man | Biology and health sciences | Homo | Biology |
Segre and Mallegni (2012) strongly retain their beliefs that the skull is 900-800 ka and is not the same age as the clay it was found in situ. Di Vincenzo et al. (2017) explain that this thought process is based on the belief of secondary deposition into younger strata, though they believe otherwise based on renewed an... | Ceprano Man | Wikipedia | 387 | 2165266 | https://en.wikipedia.org/wiki/Ceprano%20Man | Biology and health sciences | Homo | Biology |
Paleopathology
The specimen preserves several injuries, The first is a deep, wide recess infiltrating deep the left greater wing of the sphenotemporal suture on the sphenoidal sinus, which was found but not reported early in Ceprano's literature history. Second, a healed depression on the right brow. This was probably... | Ceprano Man | Wikipedia | 483 | 2165266 | https://en.wikipedia.org/wiki/Ceprano%20Man | Biology and health sciences | Homo | Biology |
Manzi et al. (2001) pose the possibility that it may be an adult Homo antecessor, but do not make the referral based on the reasoning that no elements from Gran Dolina match in age or completeness to directly compare with Ceprano. They also state that a less parsimonious explanation would be the accommodation of two co... | Ceprano Man | Wikipedia | 430 | 2165266 | https://en.wikipedia.org/wiki/Ceprano%20Man | Biology and health sciences | Homo | Biology |
Segre and Mallegni (2012) retain use of Homo cepranensis and dispute redating of the site. Freidline et al. (2012) follow suit with the opinion of Guipert (2005). Guipert (2005) digitally reconstructed several hominin fossils exhibiting extreme degrees of distortion, including the cranial remains from Arago. In their r... | Ceprano Man | Wikipedia | 488 | 2165266 | https://en.wikipedia.org/wiki/Ceprano%20Man | Biology and health sciences | Homo | Biology |
Technology
Lithics at Ceprano tend to be located higher up and in volcanic sediment. Choppers are more common than lithics at the Castro dei Volschi facie, and overlie the choppers found at Arce Fontana Liri. They are 458-385 ka (as low as 200 ka) in age, which was, at the time, much younger than the cranium. It is su... | Ceprano Man | Wikipedia | 479 | 2165266 | https://en.wikipedia.org/wiki/Ceprano%20Man | Biology and health sciences | Homo | Biology |
Paranthropus robustus is a species of robust australopithecine from the Early and possibly Middle Pleistocene of the Cradle of Humankind, South Africa, about 2.27 to 0.87 (or, more conservatively, 2 to 1) million years ago. It has been identified in Kromdraai, Swartkrans, Sterkfontein, Gondolin, Cooper's, and Drimolen ... | Paranthropus robustus | Wikipedia | 406 | 2165269 | https://en.wikipedia.org/wiki/Paranthropus%20robustus | Biology and health sciences | Australopithecines | Biology |
P. robustus seems to have consumed a high proportion of C4 savanna plants. In addition, it may have also eaten fruits, underground storage organs (such as roots and tubers), and perhaps honey and termites. P. robustus may have used bones as tools to extract and process food. It is unclear if P. robustus lived in a hare... | Paranthropus robustus | Wikipedia | 473 | 2165269 | https://en.wikipedia.org/wiki/Paranthropus%20robustus | Biology and health sciences | Australopithecines | Biology |
At this point in time, Australian anthropologist Raymond Dart had made the very first claim (quite controversially at the time) of an early ape-like human ancestor in 1924 from South Africa, Australopithecus africanus, based on the Taung child. In 1936, Broom had described "Plesianthropus transvaalensis" (now synonymis... | Paranthropus robustus | Wikipedia | 260 | 2165269 | https://en.wikipedia.org/wiki/Paranthropus%20robustus | Biology and health sciences | Australopithecines | Biology |
"P. crassidens"
In 1948, at the nearby Swartkrans Cave, Broom described "P. crassidens" (distinct from P. robustus) based on a subadult jaw, SK 6, because Swartkrans and Kromdraai clearly dated to different time intervals based on the diverging animal assemblages in these caves. At this point in time, humans and allies... | Paranthropus robustus | Wikipedia | 454 | 2165269 | https://en.wikipedia.org/wiki/Paranthropus%20robustus | Biology and health sciences | Australopithecines | Biology |
In 1949, also in Swartkrans Cave, Broom and Robinson found a mandible which they preliminary described as "intermediate between one of the ape-men and true man," classifying it as a new genus and species "Telanthropus capensis". Most immediate reactions favoured synonymising "T. capensis" with "P. crassidens", whose re... | Paranthropus robustus | Wikipedia | 366 | 2165269 | https://en.wikipedia.org/wiki/Paranthropus%20robustus | Biology and health sciences | Australopithecines | Biology |
In 1972, Robinson suggested including Gigantopithecus in "Paranthropinae", with the Miocene Pakistani "G. bilaspurensis" (now Indopithecus) as the ancestor of Paranthropus and the Chinese G. blacki. He also believed that they both had a massive build. In contrast, he reported a very small build for A. africanus (which ... | Paranthropus robustus | Wikipedia | 453 | 2165269 | https://en.wikipedia.org/wiki/Paranthropus%20robustus | Biology and health sciences | Australopithecines | Biology |
It was long assumed that if Paranthropus is a valid genus then P. robustus was the ancestor of P. boisei, but in 1985, anthropologists Alan Walker and Richard Leakey found that the 2.5-million-year-old East African skull KNM WT 17000—which they assigned to a new species A. aethiopicus|A. aethiopicus—was ancestral to A.... | Paranthropus robustus | Wikipedia | 470 | 2165269 | https://en.wikipedia.org/wiki/Paranthropus%20robustus | Biology and health sciences | Australopithecines | Biology |
Typical of Paranthropus, P. robustus exhibits post-canine megadontia with enormous cheek teeth but human-sized incisors and canines. The premolars are shaped like molars. The enamel thickness on the cheek teeth is relatively on par with that of modern humans, though australopithecine cheek tooth enamel thickens especia... | Paranthropus robustus | Wikipedia | 442 | 2165269 | https://en.wikipedia.org/wiki/Paranthropus%20robustus | Biology and health sciences | Australopithecines | Biology |
The posterior semicircular canals in the inner ear of SK 46 and SK 47 are unlike those of the apelike Australopithecus or Homo, suggesting different locomotory and head movement patterns, since inner ear anatomy affects the vestibular system (sense of balance). The posterior semicircular canals of modern humans are tho... | Paranthropus robustus | Wikipedia | 425 | 2165269 | https://en.wikipedia.org/wiki/Paranthropus%20robustus | Biology and health sciences | Australopithecines | Biology |
Blood vessels
In 1983, while studying SK 1585 (P. robustus) and KNM-ER 407 (P. boisei, which he referred to as robustus), French anthropologist Roger Saban stated that the parietal branch of the middle meningeal artery originated from the posterior branch in P. robustus and P. boisei instead of the anterior branch as i... | Paranthropus robustus | Wikipedia | 327 | 2165269 | https://en.wikipedia.org/wiki/Paranthropus%20robustus | Biology and health sciences | Australopithecines | Biology |
Torso
Few vertebrae are assigned to P. robustus. The only thoracolumbar series (thoracic and lumbar series) preserved belongs to the juvenile SKW 14002, and either represents the 1st to the 4th lumbar vertebrae, or the 2nd to the 5th. SK 3981 preserves a 12th thoracic vertebra (the last in the series), and a lower lumb... | Paranthropus robustus | Wikipedia | 399 | 2165269 | https://en.wikipedia.org/wiki/Paranthropus%20robustus | Biology and health sciences | Australopithecines | Biology |
The pelvis is similar to the pelvises of A. africanus and A. afarensis, but it has a wider iliac blade and smaller acetabulum and hip joint. Like modern humans, the ilium of P. robustus features development of the surface and thickening of the posterior superior iliac spine, which are important in stabilising the sacru... | Paranthropus robustus | Wikipedia | 234 | 2165269 | https://en.wikipedia.org/wiki/Paranthropus%20robustus | Biology and health sciences | Australopithecines | Biology |
Limbs
The distal (lower) humerus of P. robustus falls within the variation of both modern humans and chimps, as the distal humerus is quite similar between humans and chimpanzees. The radius of P. robustus is comparable in form to Australopithecus species. The wrist joint had the same maneuverability as that of modern ... | Paranthropus robustus | Wikipedia | 309 | 2165269 | https://en.wikipedia.org/wiki/Paranthropus%20robustus | Biology and health sciences | Australopithecines | Biology |
The femur, as in P. boisei and H. habilis, is flattened anteroposteriorly (on the front and back side). This may indicate a walking gait more similar to early hominins than to modern humans (less efficient gait). Four femora assigned to P. robustus—SK 19, SK 82, SK 97, and SK 3121—exhibit an apparently high anisotropic... | Paranthropus robustus | Wikipedia | 301 | 2165269 | https://en.wikipedia.org/wiki/Paranthropus%20robustus | Biology and health sciences | Australopithecines | Biology |
Size
Broom had noted that the ankle bone and humerus of the holotype TM 1517 were about the same dimensions as that of a modern San woman, and so assumed humanlike proportions in P. robustus. In 1972, Robinson estimated Paranthropus as having been massive. He calculated the humerus-to-femur ratio of P. robustus by usin... | Paranthropus robustus | Wikipedia | 224 | 2165269 | https://en.wikipedia.org/wiki/Paranthropus%20robustus | Biology and health sciences | Australopithecines | Biology |
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