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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, consider a system composed of a number of k different types of particles and has the volume as its only external variable. The fundamental thermodynamic relation may then be expressed in terms of the internal energy as:
Some important aspects of this equation should be noted: , ,
The thermodynamic space has k+2 dimensions
The differential quantities (U, S, V, Ni) are all extensive quantities. The coefficients of the differential quantities are intensive quantities (temperature, pressure, chemical potential). Each pair in the equation are known as a conjugate pair with respect to the internal energy. The intensive variables may be viewed as a generalized "force". An imbalance in the intensive variable will cause a "flow" of the extensive variable in a direction to counter the imbalance.
The equation may be seen as a particular case of the chain rule. In other words: from which the following identifications can be made: These equations are known as "equations of state" with respect to the internal energy. (Note - the relation between pressure, volume, temperature, and particle number which is commonly called "the equation of state" is just one of many possible equations of state.) If we know all k+2 of the above equations of state, we may reconstitute the fundamental equation and recover all thermodynamic properties of the system.
The fundamental equation can be solved for any other differential and similar expressions can be found. For example, we may solve for and find that
Thermodynamic potentials
By the principle of minimum energy, the second law can be restated by saying that for a fixed entropy, when the constraints on the system are relaxed, the internal energy assumes a minimum value. This will require that the system be connected to its surroundings, since otherwise the energy would remain constant. | 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 relevant fundamental equation results from the same Second-Law principle that gives rise to energy minimization under restricted conditions: that the total entropy of the system and its environment is maximized in equilibrium. The intensive parameters give the derivatives of the environment entropy with respect to the extensive properties of the system.
The four most common thermodynamic potentials are:
After each potential is shown its "natural variables". These variables are important because if the thermodynamic potential is expressed in terms of its natural variables, then it will contain all of the thermodynamic relationships necessary to derive any other relationship. In other words, it too will be a fundamental equation. For the above four potentials, the fundamental equations are expressed as:
The thermodynamic square can be used as a tool to recall and derive these potentials.
First order equations
Just as with the internal energy version of the fundamental equation, the chain rule can be used on the above equations to find k+2 equations of state with respect to the particular potential. If Φ is a thermodynamic potential, then the fundamental equation may be expressed as:
where the are the natural variables of the potential. If is conjugate to then we have the equations of state for that potential, one for each set of conjugate variables.
Only one equation of state will not be sufficient to reconstitute the fundamental equation. All equations of state will be needed to fully characterize the thermodynamic system. Note that what is commonly called "the equation of state" is just the "mechanical" equation of state involving the Helmholtz potential and the volume:
For an ideal gas, this becomes the familiar PV=NkBT.
Euler integrals
Because all of the natural variables of the internal energy U are extensive quantities, it follows from Euler's homogeneous function theorem that
Substituting into the expressions for the other main potentials we have the following expressions for the thermodynamic potentials:
Note that the Euler integrals are sometimes also referred to as fundamental equations.
Gibbs–Duhem relationship
Differentiating the Euler equation for the internal energy and combining with the fundamental equation for internal energy, it follows that: | 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 degrees of freedom, and may be specified by only two parameters, such as pressure and volume for example. The law is named after Willard Gibbs and Pierre Duhem.
Second order equations
There are many relationships that follow mathematically from the above basic equations. See Exact differential for a list of mathematical relationships. Many equations are expressed as second derivatives of the thermodynamic potentials (see Bridgman equations).
Maxwell relations
Maxwell relations are equalities involving the second derivatives of thermodynamic potentials with respect to their natural variables. They follow directly from the fact that the order of differentiation does not matter when taking the second derivative. The four most common Maxwell relations are:
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The thermodynamic square can be used as a tool to recall and derive these relations.
Material properties
Second derivatives of thermodynamic potentials generally describe the response of the system to small changes. The number of second derivatives which are independent of each other is relatively small, which means that most material properties can be described in terms of just a few "standard" properties. For the case of a single component system, there are three properties generally considered "standard" from which all others may be derived:
Compressibility at constant temperature or constant entropy
Specific heat (per-particle) at constant pressure or constant volume
Coefficient of thermal expansion
These properties are seen to be the three possible second derivative of the Gibbs free energy with respect to temperature and pressure.
Thermodynamic property relations
Properties such as pressure, volume, temperature, unit cell volume, bulk modulus and mass are easily measured. Other properties are measured through simple relations, such as density, specific volume, specific weight. Properties such as internal energy, entropy, enthalpy, and heat transfer are not so easily measured or determined through simple relations. Thus, we use more complex relations such as Maxwell relations, the Clapeyron equation, and the Mayer relation. | 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 same as the negative change in specific volume with respect to temperature at a constant pressure, for a simple compressible system. Maxwell relations in thermodynamics are often used to derive thermodynamic relations.
The Clapeyron equation allows us to use pressure, temperature, and specific volume to determine an enthalpy change that is connected to a phase change. It is significant to any phase change process that happens at a constant pressure and temperature. One of the relations it resolved to is the enthalpy of vaporization at a provided temperature by measuring the slope of a saturation curve on a pressure vs. temperature graph. It also allows us to determine the specific volume of a saturated vapor and liquid at that provided temperature. In the equation below, represents the specific latent heat, represents temperature, and represents the change in specific volume.
The Mayer relation states that the specific heat capacity of a gas at constant volume is slightly less than at constant pressure. This relation was built on the reasoning that energy must be supplied to raise the temperature of the gas and for the gas to do work in a volume changing case. According to this relation, the difference between the specific heat capacities is the same as the universal gas constant. This relation is represented by the difference between Cp and Cv:
Cp – Cv = R | 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 origin.
Description
Tapejarids were small to medium-sized pterosaurs with several unique, shared characteristics, mainly relating to the skull. Most tapejarids possessed a bony crest arising from the snout (formed mostly by the premaxillary bones of the upper jaw tip). In some species, this bony crest is known to have supported an even larger crest of softer, fibrous tissue that extends back along the skull. Tapejarids are also characterized by their large nasoantorbital fenestra, the main opening in the skull in front of the eyes, which spans at least half the length of the entire skull in this family. Their eye sockets were small and pear-shaped. Studies of tapejarid brain cases show that they had extremely good vision, more so than in other pterosaur groups, and probably relied nearly exclusively on vision when hunting or interacting with other members of their species. Tapejarids had unusually reduced shoulder girdles that would have been slung low on the torso, resulting in wings that protruded from near the belly rather than near the back, a "bottom decker" arrangement reminiscent of some planes.
Biology
Tapejarids appear to have been arboreal, having more curved claws than other azhdarchoid pterosaurs and occurring more commonly in fossil sites with other arboreal flying vertebrates such as early birds. Tapejarids have long been speculated as having been frugivores or omnivores, based on their parrot-like beaks. Direct evidence for plant-eating is known in a specimen of Sinopterus that preserves seeds in the abdominal cavity. The Barremian-
Aptian distribution of some tapejarids may even be partially associated with the first radiation phase of the angiosperms, especially of the genus Klitzschophyllites which represents a more basal angiosperm.
Classification | 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, consisting of Thalassodromeus and Tupuxuara. A 2011 subsumed the family Chaoyangopterinae in as the subfamily Chaoyangopterinae, something not followed by future authors. Kellner's concept of a Tapejaridae consisting of Tapejarinae and Thalassodrominae would be the basis for numerous subsequent phylogenetic analyses.
Various opposing studies have arose challenging Kellner's concept of Tapejaridae. The 2003 model of paleontologist David Unwin found Tupuxara and Thalassodromeus to be more distantly related to Tapejara and therefore outside of Tapejaridae, instead being related to Azhdarchidae. Later, in 2006, British paleontologists David Martill and Darren Naish followed Unwin's concept, and provided a revised definition for Tapejaridae was also proposed: the clade containing all species more closely related to Tapejara than to Quetzalcoatlus. A 2008 study by Lü Junchang and colleagues also corroborated this model, and used the term "Tupuxuaridae" to include both genera. In 2009, British paleontologist Mark Witton also agreed with the Unwin model. However, he noted that the term Thalassodrominae was created before Tupuxuaridae, meaning it had naming priority. He elevated Thalassodrominae to family level, thus creating the denomination Thalassodromidae. | 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 all descendants of the last common ancestor of Tapejara and Tupandactylus. In 2020, in the description of the genus Wightia, an opposing subfamily was named, Sinopterinae, consisting of tapejarids more closely related to Sinopterus than Tapejara. These studies follow the Unwin model, opposing Kellner's model of Tapejaridae while corroborating the close relationship between thalassodromids, azhdarchids, rather than tapejarids.
In 2023, paleontologist Rodrigo Pêgas and colleagues argued that despite the disagreements about the position of Thalassodromeus and its relatives, the species in question were consistently related. Therefore, they favored the term Thalassodromidae to have consistency with other studies that used the same name, despite finding them to form a natural grouping with Tapejaridae in their phylogenetic analysis (per the Kellner model). Thus, Thalassodromidae and Tapejaridae would be separate families within Tapejaromorpha.In their 2023 study, Pêgas and colleagues redefined Tapejaridae to be the most recent common ancestor of Sinopterus, Tapejara, and Caupedactylus in order to preserve the scope of the family in light of finding Caupedactylus, traditionally a tapejarine, outside of the Andres definition of Tapejaridae. They divided this redefined Tapejaridae into the groups Eutapejaria, containing the subfamilies Sinopterinae and Tapejarinae, and Caupedactylia, containing the pterosaurs Caupedactylus and Aymberedactylus. In 2024, Pêgas rejected this redefinition of Tapejaridae in light of non-compliance with phylocode rules, applying the Tapejara and Sinopterus definition and deeming Eutapejaria a synonym. Instead, he created the larger group contain Tapejaridae and Caupedactylia, removing Caupedactylus and Aymberedactylus from the family itself. | 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, in which Tapejaridae consists of the subfamilies Tapejarinae and Sinopterinae. He found the pterosaurs Lacusovagus and Keresdrakon as tapejarines, an arrangement that had never been recovered in previous analyses. Regarding the interrelationships of Tapejaridae, Andres follows Unwin's concept. The second cladogram shows the phylogenetic analysis conducted by Pêgas in 2024. He also found Tapejaridae to consist of both Tapejarinae and Sinopterinae, but differed from Andres in recovering the tapejarid Bakonydraco as a sinopterine instead of tapejarine. He created the new subtribe Caiuajarina within Tapejarini to include Caiuajara and Torukjara. Additionally, his analysis further differs from that of Andres in finding both Tapejaridae and Thalassodromidae within Tapejaromorpha, which corroborates the close relationship between thalassodromids and tapejarids, similar to Kellner.
Topology 1: Andres (2021).
Topology 2: Pêgas (2024).
Subclades
Summary of the phylogenetic definitons of tapejarid subclades as discussed in the classification section. | 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 and emit their own light, especially in the visible and near-IR portion of the spectrum, and any ground-based observation is subject to contamination from these telluric (earth-originating) sources. Water vapor and oxygen are two of the more important molecules in telluric contamination. Contamination by water vapor was particularly pronounced in the Mount Wilson solar Doppler measurements.
Many scientific telescopes have spectrographs, which measure photons as a function of wavelength or frequency, with typical resolution on the order of a nanometer of visible light. Spectroscopic observations can be used in myriad contexts, including measuring the chemical composition and physical properties of astronomical objects as well as measuring object velocities from the Doppler shift of spectral lines. Unless they are corrected for, telluric contamination can produce errors or reduce precision in such data.
Telluric contamination can also be important for photometric measurements.
Telluric correction
It is possible to correct for the effects of telluric contamination in an astronomical spectrum. This is done by preparing a telluric correction function, made by dividing a model spectrum of a star by an observation of an astronomical photometric standard star. This function can then be multiplied by an astronomical observation at each wavelength point.
While this method can restore the original shape of the spectrum, the regions affected can be prone to high levels noise due to the low number of counts in that area of the spectrum. | 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 cover in a graph is a set of vertices that includes at least one endpoint of every edge, and a vertex cover is minimum if no other vertex cover has fewer vertices. A matching in a graph is a set of edges no two of which share an endpoint, and a matching is maximum if no other matching has more edges.
It is obvious from the definition that any vertex-cover set must be at least as large as any matching set (since for every edge in the matching, at least one vertex is needed in the cover). In particular, the minimum vertex cover set is at least as large as the maximum matching set. Kőnig's theorem states that, in any bipartite graph, the minimum vertex cover set and the maximum matching set have in fact the same size.
Statement of the theorem
In any bipartite graph, the number of edges in a maximum matching equals the number of vertices in a minimum vertex cover.
Example
The bipartite graph shown in the above illustration has 14 vertices; a matching with six edges is shown in blue, and a vertex cover with six vertices is shown in red. There can be no smaller vertex cover, because any vertex cover has to include at least one endpoint of each matched edge (as well as of every other edge), so this is a minimum vertex cover. Similarly, there can be no larger matching, because any matched edge has to include at least one endpoint in the vertex cover, so this is a maximum matching. Kőnig's theorem states that the equality between the sizes of the matching and the cover (in this example, both numbers are six) applies more generally to any bipartite graph.
Proofs
Constructive proof
The following proof provides a way of constructing a minimum vertex cover from a maximum matching. Let be a bipartite graph and let be the two parts of the vertex set . Suppose that is a maximum matching for .
Construct the flow network derived from in such way that there are edges of capacity from the source to every vertex and from every vertex to the sink , and of capacity from to for any . | 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 edge from to would make the size of the cut infinite.
Therefore, the size of the minimum cut is equal to . On the other hand, is a vertex cover, as any edge that is not incident to vertices from and must be incident to a pair of vertices from and , which would contradict the fact that there are no edges between and .
Thus, is a minimum vertex cover of .
Constructive proof without flow concepts
No vertex in a vertex cover can cover more than one edge of (because the edge half-overlap would prevent from being a matching in the first place), so if a vertex cover with vertices can be constructed, it must be a minimum cover.
To construct such a cover, let be the set of unmatched vertices in (possibly empty), and let be the set of vertices that are either in or are connected to by alternating paths (paths that alternate between edges that are in the matching and edges that are not in the matching). Let
Every edge in either belongs to an alternating path (and has a right endpoint in ), or it has a left endpoint in . For, if is matched but not in an alternating path, then its left endpoint cannot be in an alternating path (because two matched edges can not share a vertex) and thus belongs to . Alternatively, if is unmatched but not in an alternating path, then its left endpoint cannot be in an alternating path, for such a path could be extended by adding to it. Thus, forms a vertex cover.
Additionally, every vertex in is an endpoint of a matched edge.
For, every vertex in is matched because is a superset of , the set of unmatched left vertices.
And every vertex in must also be matched, for if there existed an alternating path to an unmatched vertex then changing the matching by removing the matched edges from this path and adding the unmatched edges in their place would increase the size of the matching. However, no matched edge can have both of its endpoints in . Thus, is a vertex cover of cardinality equal to , and must be a minimum vertex cover. | 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 in which the weights are in {0,1}). Similarly we define a fractional vertex-cover - an assignment of a non-negative weight to each vertex, such that the sum of weights in each edge is at least 1 (an integral vertex-cover is a special case of a fractional vertex-cover in which the weights are in {0,1}).
The maximum fractional matching size in a graph is the solution of the following linear program:Maximize 1E · x
Subject to: x ≥ 0E
__ AG · x ≤ 1V.where x is a vector of size |E| in which each element represents the weight of an edge in the fractional matching. 1E is a vector of |E| ones, so the first line indicates the size of the matching. 0E is a vector of |E| zeros, so the second line indicates the constraint that the weights are non-negative. 1V is a vector of |V| ones and AG is the incidence matrix of G, so the third line indicates the constraint that the sum of weights near each vertex is at most 1.
Similarly, the minimum fractional vertex-cover size in is the solution of the following LP:Minimize 1V · y
Subject to: y ≥ 0V
__ AGT · y ≥ 1E.where y is a vector of size |V| in which each element represents the weight of a vertex in the fractional cover. Here, the first line is the size of the cover, the second line represents the non-negativity of the weights, and the third line represents the requirement that the sum of weights near each edge must be at least 1. | 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 the smallest size of a fractional vertex cover.What makes bipartite graphs special is that, in bipartite graphs, both these linear programs have optimal solutions in which all variable values are integers. This follows from the fact that in the fractional matching polytope of a bipartite graph, all extreme points have only integer coordinates, and the same is true for the fractional vertex-cover polytope. Therefore the above theorem implies:
In any bipartite graph, the largest size of a matching equals the smallest size of a vertex cover.
Algorithm
The constructive proof described above provides an algorithm for producing a minimum vertex cover given a maximum matching. Thus, the Hopcroft–Karp algorithm for finding maximum matchings in bipartite graphs may also be used to solve the vertex cover problem efficiently in these graphs.
Despite the equivalence of the two problems from the point of view of exact solutions, they are not equivalent for approximation algorithms. Bipartite maximum matchings can be approximated arbitrarily accurately in constant time by distributed algorithms; in contrast, approximating the minimum vertex cover of a bipartite graph requires at least logarithmic time.
Example
In the graph shown in the introduction take to be the set of vertices in the bottom layer of the diagram and to be the set of vertices in the top layer of the diagram. From left to right label the vertices in the bottom layer with the numbers 1, …, 7 and label the vertices in the top layer with the numbers 8, …, 14. The set of unmatched vertices from is {1}. The alternating paths starting from are 1–10–3–13–7, 1–10–3–11–5–13–7, 1–11–5–13–7, 1–11–5–10–3–13–7, and all subpaths of these starting from 1. The set is therefore {1,3,5,7,10,11,13}, resulting in , and the minimum vertex cover . | 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 cover in any graph is an independent set, so a minimum vertex cover is complementary to a maximum independent set; finding maximum independent sets is another NP-complete problem. The equivalence between matching and covering articulated in Kőnig's theorem allows minimum vertex covers and maximum independent sets to be computed in polynomial time for bipartite graphs, despite the NP-completeness of these problems for more general graph families.
History
Kőnig's theorem is named after the Hungarian mathematician Dénes Kőnig. Kőnig had announced in 1914 and published in 1916 the results that every regular bipartite graph has a perfect matching, and more generally that the chromatic index of any bipartite graph (that is, the minimum number of matchings into which it can be partitioned) equals its maximum degree – the latter statement is known as Kőnig's line coloring theorem. However, attribute Kőnig's theorem itself to a later paper of Kőnig (1931).
According to , Kőnig attributed the idea of studying matchings in bipartite graphs to his father, mathematician Gyula Kőnig. In Hungarian, Kőnig's name has a double acute accent, but his theorem is sometimes spelled (incorrectly) in German characters, with an umlaut.
Related theorems
Kőnig's theorem is equivalent to many other min-max theorems in graph theory and combinatorics, such as Hall's marriage theorem and Dilworth's theorem. Since bipartite matching is a special case of maximum flow, the theorem also results from the max-flow min-cut theorem.
Connections with perfect graphs
A graph is said to be perfect if, in every induced subgraph, the chromatic number equals the size of the largest clique. Any bipartite graph is perfect, because each of its subgraphs is either bipartite or independent; in a bipartite graph that is not independent the chromatic number and the size of the largest clique are both two while in an independent set the chromatic number and clique number are both one. | 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 clique in the complement of a graph G is an independent set in G, and as we have already described an independent set in a bipartite graph G is a complement of a vertex cover in G. Thus, any matching M in a bipartite graph G with n vertices corresponds to a coloring of the complement of G with n-|M| colors, which by the perfection of complements of bipartite graphs corresponds to an independent set in G with n-|M| vertices, which corresponds to a vertex cover of G with M vertices. Conversely, Kőnig's theorem proves the perfection of the complements of bipartite graphs, a result proven in a more explicit form by .
One can also connect Kőnig's line coloring theorem to a different class of perfect graphs, the line graphs of bipartite graphs. If G is a graph, the line graph L(G) has a vertex for each edge of G, and an edge for each pair of adjacent edges in G. Thus, the chromatic number of L(G) equals the chromatic index of G. If G is bipartite, the cliques in L(G) are exactly the sets of edges in G sharing a common endpoint. Now Kőnig's line coloring theorem, stating that the chromatic index equals the maximum vertex degree in any bipartite graph, can be interpreted as stating that the line graph of a bipartite graph is perfect.
Since line graphs of bipartite graphs are perfect, the complements of line graphs of bipartite graphs are also perfect. A clique in the complement of the line graph of G is just a matching in G. And a coloring in the complement of the line graph of G, when G is bipartite, is a partition of the edges of G into subsets of edges sharing a common endpoint; the endpoints shared by each of these subsets form a vertex cover for G. Therefore, Kőnig's theorem itself can also be interpreted as stating that the complements of line graphs of bipartite graphs are perfect. | 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 participating in the matching. A w-vertex-cover is a multiset of vertices ("multiset" means that each vertex may appear several times), in which each edge e is adjacent to at least we vertices. Egerváry's theorem says:In any edge-weighted bipartite graph, the maximum w-weight of a matching equals the smallest number of vertices in a w-vertex-cover.The maximum w-weight of a fractional matching is given by the LP:
Maximize w · x
Subject to: x ≥ 0E
__ AG · x ≤ 1V.And the minimum number of vertices in a fractional w-vertex-cover is given by the dual LP:Minimize 1V · y
Subject to: y ≥ 0V
__ AGT · y ≥ w.As in the proof of Konig's theorem, the LP duality theorem implies that the optimal values are equal (for any graph), and the fact that the graph is bipartite implies that these programs have optimal solutions in which all values are integers.
Theorem for vertex-weighted graphs
One can consider a graph in which each vertex v has a non-negative integer weight bv. The weight vector is denoted by b. The b-weight of a vertex-cover is the sum of bv for all v in the cover. A b-matching is an assignment of a non-negative integral weight to each edge, such that the sum of weights of edges adjacent to any vertex v is at most bv. Egerváry's theorem can be extended, using a similar argument, to graphs that have both edge-weights w and vertex-weights b:
In any edge-weighted vertex-weighted bipartite graph, the maximum w-weight of a b-matching equals the minimum b-weight of vertices in a w-vertex-cover. | 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 of a sauropod population on an island within the Lower Saxony basin.
Discovery and naming
In 1998, a single sauropod tooth was discovered by private fossil collector Holger Lüdtke in an active quarry at Langenberg Mountain, between the communities of Oker, Harlingerode and Göttingerode in Germany. The Langenberg chalk quarry had been active for more than a century; rocks are quarried using blasting and are mostly processed into fertilisers. The quarry exposes a nearly continuous, thick succession of carbonate rocks belonging to the Süntel Formation, that ranges in age from the early Oxfordian to late Kimmeridgian stages and have been deposited in a shallow sea with a water depth of less than . The layers exposed in the quarry are oriented nearly vertically and slightly overturned, which is a result of the ascent of the adjacent Harz mountains during the Lower Cretaceous. Widely known as a classical exposure among geologists, the quarry had been extensively studied, and visited by students of geology for decades. Although rich in fossils of marine invertebrates, fossils of land-living animals had been rare. The sauropod tooth was the first specimen of a sauropod dinosaur from the Jurassic of northern Germany. | 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, as well as by the ongoing quarrying. The sauropod material could not be excavated directly from the layer but had to be collected from lose blocks that resulting from blasting. The exact origin of the bone material was therefore unclear, but could later be traced to a single bed (bed 83). An excavation conducted between July 20–28 of 2000 rescued ca. of bone-bearing blocks containing vertebrate remains. Fossils were prepared and stored in the Dinosaur Park Münchehagen (DFMMh), a private dinosaur open-air museum located close to Hanover. Due to the very good preservation of the bones, consolidating agents had to be applied only occasionally, and preparation could be conducted comparatively quickly as bone would separate easily from the surrounding rock. Bones of simple shape could sometimes be prepared in less than an hour, while the preparation of a sacrum required a workload of three weeks. By January 2001, 200 single vertebrate bones had already been prepared. At this point, the highest bone density was found in a block measuring 70 x 70 cm, which contained ca. 20 bones. By January 2002, preparation of an even larger block had revealed a partial sauropod skull – the first to be discovered in Europe. Before complete removal of the bones from the block, a silicon cast was made of the block to document the precise three-dimensional position of the individual bones.
Part of the Europasaurus fossil material got damaged or destroyed by arson fire in the night from the 4th to the 5th of October, 2003. The fire destroyed the laboratory and exhibition hall of the Dinosaur Park Münchehagen, resulting in the loss of 106 bones, which account for 15% of the bones prepared at the time. Furthermore, the fire affected most of the still unprepared blocks, with firefighting water hitting the hot stone causing additional crumbling. Destroyed specimens include DFMMh/FV 100, which included the best preserved vertebral series and the only complete pelvis. | 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 juvenile individuals. The 2006 publication, however, established that the majority of specimens were adult, and that Europasaurus was an island dwarf. The number of individual sauropod bones had increased to 650 and include variously articulated individuals; the material was found within an area of squared. From these specimens, the holotype was selected, a disarticulated but associated individual (DFMMh/FV 291). The holotype includes multiple cranial bones (premaxilla, maxilla and quadratojugal), a partial braincase, multiple mandible bones (dentary, surangular and angular), large amounts of teeth, cervical vertebrae, sacral vertebrae and ribs from the neck and torso. At least 10 other individuals were referred to the same taxon based on overlap in material.
A large-scale excavation campaign commenced in the summer of 2012, with the goal to excavate Europasaurus bones not only from lose blocks but directly from the rock layer. Access to the bone-bearing layer required the removal of some 600 tons of rock using excavators and wheel loaders, and the constant pumping out of water from the base of the quarry. Excavations continued in spring and summer 2013. The campaign resulted in the discovery of new fish, turtle, and crocodile remains, as well as valuable information of the bone-bearing layer; additional Europasaurus bones, however, could not be located. By 2014, around 1300 vertebrate bones had been prepared from bed 83, the majority of which stemming from Europasaurus; an estimated 3000 additional bones await preparation. A minimum number of 20 individuals was identified based on jaw bones.
Description
Europasaurus is a very small sauropod, measuring only long and weighed as an adult. This length was estimated based on a partial femur, scaled to the size of a nearly complete Camarasaurus specimen. Younger individuals are known, from sizes of to the youngest juvenile at .
Distinguishing characteristics | 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 preservational), there is a notch on the upper surface of the centra of cervical vertebrae, the scapula has a prominent process on the posterior surface of its body, and the astragalus (an ankle bone) is twice as wide as tall.
When compared to Camarasaurus, Europasaurus has a different morphology of the postorbital where the posterior flange is not as short, a short contact between the nasal and frontal bones of the skull, the shape of its parietal (rectangular in Europasaurus), and the neural spines of its vertebrae in front of the pelvis are unsplit. Comparisons with Brachiosaurus (now named Giraffatitan) were also mentioned, and it was identified that Europasaurus has a shorter snout, a contact between the quadratojugal and squamosal, and a humerus (upper forelimb bone) that has flattened and aligned proximal and distal surfaces. There were finally quick comparisons to the potential brachiosaurid Lusotitan, which has a different ilium and astragalus shape, and Cetiosaurus humerocristatus (named Duriatitan), which has a deltopectoral crest that is less prominent and extends across less of the humerus.
Skull | 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 anterior projection of the premaxilla identified in Sander et al. (2006) was re-identified as a preservational artifact in Marpmann et al. (2014), similar to the anatomy found in Camarasaurus and Euhelopus to a lesser degree. The dorsal projection of the premaxilla, the one which contacts the nasal bone, begins as a postero-dorsal projection, before becoming straight vertical at the point of the subnarial foramen, until it reaches the nasal. This weak "step" is seen in Camarasaurus and Euhelopus, and is present more strongly in Abydosaurus, Giraffatitan and a possible skull of Brachiosaurus. These latter taxa also have a longer snout, with more distance from the first tooth until the nasal process of the premaxilla. As well, Europasaurus shares with the basal camarasauromorphs (brachiosaurids, Camarasaurus, Euhelopus and Malawisaurus) a similarly sized orbit and nasal fenestra, whereas the nasal opening is significantly reduced in derived titanosaurs (Rapetosaurus, Tapuiasaurus and Nemegtosaurus). | 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 anterior and ventral rim of the antorbital fenestra. This process extends about 120º from the horizontal tooth row. The base of the nasal process also forms the body of the lacrimal process, and at their divergence is the antorbital fenestra, similar in shape to those of Camarasaurus, Euhelopus, Abydosaurus and Giraffatitan, but about 1/2 taller proportionally. The pre-antorbital fenestra, a small opening in front of or beneath the antorbital opening, is well developed in taxa like Diplodocus and Tapuiasaurus, is nearly absent, like in Camarasaurus and Euhelopus. There were about 12–13 total teeth in the maxilla of Europasaurus, fewer than in more basal taxa (16 teeth in Jobaria and 14–25 in Atlasaurus), but falling within the range of variation in Brachiosauridae (15 in Brachiosaurus to 10 in Abydosaurus). All of the unworn teeth preserved display up to four small denticles on their mesial edges. A small amount of the posterior tooth crowns are slightly twisted (~15º), but much less than in brachiosaurids (30–45º). | 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 small portion of the skull roof over the antorbital fenestrae. Four frontals are known from Europasaurus, three being from the left and one being from the right. Because of their disarticulation, it is likely that the frontals never fused during growth, unlike in Camarasaurus. The frontals form a portion of the skull roof, articulating with other bones such as the nasals, parietals, prefrontals and postorbitals, and they are longer antero-posteriorly than they are wide, a unique character among a eusauropodan. Like in diplodocoids (Amargasaurus, Dicraeosaurus and Diplodocus), as well as Camarasaurus, the frontals are excluded from the frontoparietal fenestra (or parietal fenestra when frontals are excluded). The frontals are also excluded from the supratemporal fenestra margin (a widespread character in sauropods more derived than Shunosaurus), and they only have a small, unornamented participation in the orbit. Several parietal bones are known in Europasaurus, which show a rectangular shape much wider than long. the parietals are also wide when viewed from the back of the skull, being slightly taller than the foramen magnum (spinal cord opening). The parietals contribute to about half the post-temporal fenestra (opening above the very back of the skull) border, with the other region enclosed by the squamosal bones and some braincase bones. Parietals also form part of the edge of the supratemporal fenestra, which is wider than long in Europasaurus, like in Giraffatitan, Camarasaurus and Spinophorosaurus. Besides the before mentioned fenestra, the parietals also have a "postparietal fenestra", something rarely seen outside of Dicraeosauridae | 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 infratemporal fenestra | 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 fragile and narrow, showing a bone scar from the articulation with the quadratojugal. There are two prominences projecting from the back of the jugal body, which diverge at 75º and form the bottom and front edges of the infratemporal fenestra. Like in Riojasaurus and Massospondylus, two non-sauropod sauropodomorphs, the jugal forms a large part of the orbit edge, from the back to the front bottom corner. This feature has been seen in embryos of titanosaurs, but no adult individuals. The quadratojugal bone is an elongate element that has two projecting arms, one anterior and one dorsal. Like in other sauropods, the anterior process is longer than the dorsal, but in Europasaurus the arms are more similar lengths. The horizontal process is parallel to the tooth row of Europasaurus, similar to in Camarasaurus but unlike in Giraffatitan and Abydosaurus. There is a prominent ventral flange on the anterior arm of the bone, which is a possibly synapomorphy of Brachiosauridae, although it is also found in some Camarasaurus individuals. The two quadratojugal processes diverge at a nearly right angle (90º), although the dorsal process curves as it follows the shape of the quadrate. Squamosals found from Europasaurus show the same approximate shape in lateral view as Camarasaurus, that of a question mark. The squamosals articulate with many skull bones, including those of the skull roof, those of the ventral skull, and those of the braincase. Like the postorbitals, the squamosals are triradiate, with a ventral, anterior and medial
process. | 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 articular surfaces. A single shaft is present for a majority of the quadrates length, with a pterygoid wing along the medial side. Pterygoids are the largest of the sauropod palate bones, and it has a triradiate shape, like the postorbitals. An anterior projection contacts the opposite pterygoid, while a lateral wing contacts the ectopterygoid, and a posterior wing supports the quadrate and basipterygoid (a bone that provides connection between the palate and the braincase). The ectopterygoid is a small palate bone, which articulates the central palate bones (pterygoid and palatine) with the maxilla. Ectopterygoids are L-shaped, with an anterior process attaching to the maxilla, and a dorsal process that meets the pterygoid.
Vertebrae | 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 individuals, and the centrum length and internal structure are known to change throughout development. The adult cervical centra are elongated and (anterior end is ball-shaped), with a notch in the top of the rear end of the centrum. This feature was described as characteristic of Europasaurus but is also known in Euhelopus and Giraffatitan. In the side of the centra of Europasaurus there is an excavation which opens into the internal of the vertebrae. Unlike in Giraffatitan and brachiosaurids, Europasaurus does not have thin ridges () dividing this opening. Europasaurus shares laminae features on the upper vertebrae with basal macronarians and brachiosaurids. Differing from the anterior and middle cervicals, the posterior cervical vertebrae are less elongate, and taller proportionally, like in other macronarians, with significant changes in the positions of articular surfaces.
Front dorsal vertebrae are strongly opisthocoelous like the cervicals, and can be placed in the series based on the absence of the and low placement. The internal structures are open and like Camarasaurus, Giraffatitan and Galvesaurus, but unlike these taxa this pneumaticity does not extend into the middle and posterior dorsal vertebrae. The arrangement and presence of anterior laminae in Europasaurus is similar to other basal macronarians, but unlike more basal taxa (e.g. Mamenchisaurus, Haplocanthosaurus) and more derived taxa (e.g. Giraffatitan). The middle dorsals possess a pneumatic cavity that extends upwards into the , like in Barapasaurus, Cetiosaurus, Tehuelchesaurus, and Camarasaurus. The ventral edge of this opening is rhomboidal and well-defined. In the posterior vertebrae, the lateral pneumatic cavity has shifted higher on the centrum, a change seen in other basal macronarians. These are wide anteriorly, and narrow to become acutely angled posteriorly. The of Europasaurus stands vertically, a basal feature not seen in Brachiosaurus or more derived sauropods. | 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 second, S2, is the ancestral sauropodomorph sacral that was added in basal sauropodomorphs, who all share three sacrals to the exception of Plateosaurus. The fifth sacral, fused behind the primordial pair, is a caudosacral, migrated from the tail into the pelvis in taxa around Leonerasaurus. The first sacral, articulated with the ilium but not fused to the other vertebrae, represents the dorsosacral, bringing the count to five vertebrae found in all neosauropods. The level of fusion of the dorsosacral confirms the evolutionary history of the sauropod sacral count: the primordial pair incorporating first a dorsal (total of three), then a caudal (total of four), then another dorsal to make a total of five vertebrae.
Skin
Among macronarians, fossilized skin impressions are only known from Haestasaurus, Tehuelchesaurus and Saltasaurus. Both Tehuelchesaurus and Haestasaurus may be closely related to Europasaurus, and the characteristics of all sauropod skin impressions are similar. Haestasaurus, the first dinosaur known from skin impressions, preserved integument over a portion of the arm around the elbow joint. Dermal impressions are more widespread in the material of Tehuelchesaurus, where they are known from the areas of the forelimb, scapula and torso. There are no bony plates or nodules, to indicate armour, but there are several types of scales. The skin types of Tehuelchesaurus are overall more similar to those found in diplodocids and Haestasaurus than in the titanosaur embryos of Auca Mahuevo. As the shape and articulation of the preserved tubercles in these basal macronarians are similar in other taxa where skin is preserved, including specimens of Brontosaurus excelsus and intermediate diplodocoids, such dermal structures are probably widespread throughout Neosauropoda. | 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 were analysed with the inclusion of Europasaurus, that of Wilson (2002) and Upchurch (1998) and Upchurch et al. (2004). All analyses resulted in similar phylogenetics, where Europasaurus placed more derived than Camarasaurus but outside a clade of Brachiosauridae and Titanosauromorpha (now named Titanosauriformes). The results of the favoured analysis of Sander et al. (2006) are shown below on the left:
During a description of the vertebrae of Europasaurus by Carballido & Sander (2013), another phylogenetic analysis was conducted (right column above). The cladistic matrix was expanded to include more sauropod taxa, such as Bellusaurus, Cedarosaurus and Tapuiasaurus. The taxon Brachiosaurus was also separated into true Brachiosaurus (B. altithorax) and Giraffatitan (B. brancai), based on Taylor (2009). Based on this newer and more expansive analysis, Europasaurus was found to be in a similar placement, as a basal camarasauromorph closer to titanosaurs than Camarasaurus. However, Euhelopus, Tehuelchesaurus, Tastavinsaurus and Galvesaurus were placed between Europasaurus and Brachiosauridae. | 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 instead determined to belong to Somphospondyli (Paluxysaurus, Sauroposeidon and Qiaowanlong). However, D'Emic was tentative in considering Europasaurus to be a confirmed brachiosaurid. While there was strong support in the phylogeny for its placement, Europasaurus, one of few basal macronarians with a skull, lacks multiple bones that display characteristic features of the group, such as caudal vertebrae. The cladogram below on the left illustrates the phylogenetic results of D'Emic (2012), with Euhelopodidae and Titanosauria collapsed.
A later analysis on titanosauriformes agreed with D'Emic (2012) in the placement of Europasaurus. It formed a polytomy with Brachiosaurus and the "French Bothriospondylus" (named Vouivria) as the basalmost brachiosaurids. Next most derived in the clade was Lusotitan, with Giraffatitan, Abydosaurus, Cedarosaurus and Venenosaurus forming a more derived clade of brachiosaurids. The "twisted" teeth of Europasaurus were found to be one of the unique features of Brachiosauridae, which could mean a confident referral of isolated sauropod teeth to the clade. | 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 polytomy with Abydosaurus and Cedarosaurus. The remaining tree was the same as in D'Emic (2012), although Brachiosaurus was collapsed into a polytomy with more derived brachiosaurids. Another phylogeny, Mannion et al. (2017) found similar results to D'Emic (2012) and D'Emic et al. (2016). Europasaurus was the basalmost brachiosaurid, with the "French Bothriospondylus", or Vouivria, as the next most basal brachiosaurid. Brachiosaurus was placed outside of a poltomy of all other brachiosaurids, Giraffatitan, Abydosaurus, Sonorasaurus, Cedarosaurus and Venenosaurus. A 2017 phylogeny, that of Royo-Torres et al. (2017), resolved more complex relations within Brachiosauridae. Besides Europasaurus as the basalmost brachiosaurid, there were two subgroups within the clade, one containing Giraffatitan, Sonorasaurus and Lusotitan, and another including almost all other brachiosaurids, as well as Tastavinsaurus. This second clade would be termed Laurasiformes under the group's definition. Brachiosaurus was in a polytomy with the two subclades of Brachiosauridae. The phylogeny of Royo-Torres et al. (2017) can be seen above, in the right column.
Paleobiology | 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 tissue, and is also the smallest specimen at in length. Such bone tissue is an indicator of rapid growth, so the specimen is probably a young juvenile. A larger specimen (DFMMh/FV 291.9) at shows large amounts of laminar tissue, with no growth marks present, so is likely a juvenile as well. The next smallest specimen (DFMMh/FV 001) has shows the presence of growth marks (specifically annuli), and at the length of is possibly a subadult. Further larger, DFMMh/FV 495 displays mature osteons as well as annuli, and is . The second largest analysed specimen (DFMMh/FV 153) also shows growth marks, but they are more frequent. This specimen is . A single partial femur represents the largest known Europasaurus individual, at a body length of . Unlike all other specimens, this one (DFMMh/FV 415) shows the presence of lines of arrested growth, indicating it died after reaching full body size. The internal bone is also partially lamellar, which shows it had stopped growing recently.
These combinations of growth factors show that Europasaurus developed its small size because of a largely reduced growth rate, gaining size slower than larger taxa such as Camarasaurus. This slowing growth rate is the opposite of the general trend of sauropods and theropods, who reached greater sizes with increased growth rates. Some of the close relatives of Europasaurus represent the largest dinosaurs known, including Brachiosaurus and Sauroposeidon. Marpmann et al. (2014) proposed that the small size and reduced growth rate of Europasaurus was an effect of pedomorphism, where the adults of taxa retain juvenile characteristics, such as size. | 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 of the inner ear in this genus also suggests that they had good senses of hearing, with Europasaurus. Intraspecific communication was also apparently important to this sauropod, based on these studies, suggesting this sauropod displayed clear, gregarious behavior.
Dwarfism
It has been suggested that an ancestor of Europasaurus would have quickly decreased in body size after emigrating to an island that existed at the time, as the largest of the islands in the region around northern Germany was smaller than squared, which may not have been able to support a community of large sauropods. Alternately, a macronarian may have shrunken concurrently with a larger landmass, until achieving the size of Europasaurus. Previous studies on insular (island) dwarfism are largely restricted to the Maastrichtian of Haţeg Island in Romania, home to the dwarf titanosaur Magyarosaurus and the dwarf hadrosaur Telmatosaurus. Telmatosaurus is known to be from a small adult, and although it is very small, Magyarosaurus specimens of small sizes are known to be from adult to old individuals. Magyarosaurus dacus adults were a similar body size to Europasaurus, but the largest of the latter had longer femora than the largest of the former, while Magyarosaurus hungaricus was significantly larger than either taxon. The dwarfism in Europasaurus represents the only significant rapid body mass change in derived Sauropodomorpha, with the general trend of taxa being a growth in overall size in other groups.
Palaeoecology | 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 does not indicate that the taxa present were marine, as the animals and plants may have been deposited allochthonously (albeit only by a short distance) from the surrounding islands. The sediments to show that there was an occasional influx of fresh or brackish water, and the fossils preserved display that. There are large numbers of marine bivalve fossils, as well as echinoderms and microfossils present in the limestone of the quarry, although many of the animals and plants were terrestrial.
Many marine taxa are preserved at Langenberg, although they would not have co-existed often with Europasaurus. There are at least three turtle genera, Plesiochelys, Thalassemys and up to two unnamed taxa. Actinopterygian fish are abundant, being represented by Lepidotes, Macromesodon, Proscinetes, Coelodus, Macrosemius, Notagogus, Histionotus, Ionoscopus, Callopterus, Caturus, Sauropsis, Belonostomus, and Thrissops. Also present are at least five distinct morphologies of hybodont sharks, the neoselachians Palaeoscyllium, Synechodus and Asterodermus. Two marine crocodyliforms are known from Langenberg, Machimosaurus and Steneosaurus, which likely fed off turtles and fish, and the amphibious crocodyliform Goniopholis has also been found. | 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 1/3 bearing tooth marks. Of these tooth marks, the sizes and shapes match well with the teeth of fish, crocodyliforms or other scavengers, but no confirmed theropod marks. The high number of individuals present suggests that a herd of Europasaurus was crossing a tidal zone and drowned. While the dominant large-bodied animal present is Europasaurus, there is also material from a diplodocid sauropod, a stegosaurian, and multiple theropods. Three cervicals of the diplodocid are preserved, and from their size it is possible that the taxon was also a dwarf. The stegosaurian and variety of theropods only preserve teeth, with the exception of a few bones possibly from a taxon in Ceratosauridae. Isolated teeth show that there were at least four different types of theropods present at the locality, including the megalosaurid Torvosaurus sp. as well as an additional megalosaurid and indeterminate members of the Allosauridae and Ceratosauria; and there are also the oldest teeth known from Velociraptorinae.
Besides the dinosaurs, many small-bodied terrestrial vertebrates are also preserved in the Langenberg quarry. Such animals include a well-preserved three-dimensional pterosaur skeleton from Dsungaripteridae, and isolated remains from Ornithocheiroidea and Ctenochasmatidae; a paramacellodid lizard; and partial skeletons and skulls from a relative of Theriosuchus now named as the genus Knoetschkesuchus. Teeth from dryolestid mammals are also preserved, as well as a docodont, a taxon in Eobaataridae, and a multituberculate with similarities to Proalbionbataar (now named Teutonodon).
Extinction | 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 deposit there was a drop in sea level which allowed for a faunal overturn. The inhabiting theropods of the island, that coexisted with Europasaurus, would have been about , but the theropods that arrived over the land bridge preserve footprints
up to , which indicates a body size between if reconstructed as an allosaurian. It was suggested by the describers of these tracks (Jens Lallensack and colleagues), that these theropod taxa likely made the specialized dwarf fauna extinct, and the bed from which the footprints originated (Langenberg bed 92) is probably the youngest in which Europasaurus is present. | 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 in Spain and the central and southern part of Portugal.
The most commonly accepted theory is that the pigs were first brought to the Iberian Peninsula by the Phoenicians from the Eastern Mediterranean coast (current-day Lebanon), where they interbred with wild boars. This cross gave rise to the ancestors of what are today Iberian pigs. The production of Iberian pig is deeply rooted to the Mediterranean ecosystem. It is a rare example in world swine production where the pig contributes so decisively to the preservation of the ecosystem. The Iberian breed is currently one of the few examples of a domesticated breed which has adapted to a pastoral setting where the land is particularly rich in natural resources, in this case acorns from the holm oak, gall oak and cork oak.
The numbers of the Iberian breed have been drastically reduced since 1960 due to several factors such as the outbreak of African swine fever and the lowered value of animal fats. In the past few years, however, the production of pigs of the Iberian type has increased to satisfy a renewed demand for top-quality meat and cured products. At the same time, breed specialisation has led to the disappearance of some ancestral varieties.
This traditional breed exhibits a good appetite and propensity to obesity, including a great capacity to accumulate intramuscular and epidermal fat. The high intramuscular fat is what produces the typical marbling; this, together with traditional feeding based on acorns, is what makes its ham taste so special. Iberian pigs are interesting from a human biomedical perspective because they present high feed intake and propensity to obesity, compatible with high values of serum leptin.
The Iberian pig can be either red or dark in colour, if black ranging from dark to grey, with little or no hair and a lean body, thus giving rise to the familiar name pata negra, or "black hoof". In traditional management, animals ranged freely in sparse oak forest (dehesa in Spain, montado in Portugal), they are constantly moving around and therefore burn more calories than confined pigs. This, in turn, produces the fine bones typical of this kind of jamón ibérico. | 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 bulk of the acorn harvest comes from the holm oak (Quercus rotundifolia) from November to February, but the season would be too short without the earlier harvests of Pyrenean oak (Quercus pyrenaica) and Portuguese or gall oak (Quercus lusitanica), and the late cork oak (Quercus suber) season, which between them extend the acorn-production period from September almost to April. | 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 raspberry flavor and others with an earthy one. Many red bananas are exported by producers in East Africa, Asia, South America, and the United Arab Emirates. They are a favorite in Central America, but are sold throughout the world.
Description
Red bananas should have a deep red or maroon rind when ripe and are best eaten when unbruised and slightly soft. This variety contains more beta-carotene and vitamin C than yellow bananas. It also contains potassium and iron. The redder the fruit, the more carotene and the higher the vitamin C level. As with yellow bananas, red bananas will ripen in a few days at room temperature and are best stored outside from refrigeration.
Compared with the most common banana, the Cavendish banana, they tend to be smaller, have a slightly thicker skin with a sweeter taste, but do have a longer shelf life than yellow bananas.
Nomenclature
It is known in English as Red dacca (Australia), Red banana, 'Red' banana (US), Claret banana, Cavendish banana "Cuban Red", Jamaican red banana, and Red Cavendish banana.
Taxonomy
The red banana is a triploid cultivar of the wild banana Musa acuminata, belonging to the AAA group.
Its official designation is Musa acuminata (AAA Group) 'Red Dacca'.
Synonyms include:
Musa acuminata Colla (AAA Group) cv. 'Red'
Musa sapientum L. f. rubra Bail.
Musa sapientum L. var. rubra (Firm.) Baker
Musa rubra Wall. ex Kurz.
Musa × paradisiaca L. ssp. sapientum (L.) Kuntze var. rubra
Musa acuminata Colla (AAA Group) cv. 'Cuban Red'
Musa acuminata Colla (Cavendish Group) cv. 'Cuban Red'
Musa acuminata Colla (AAA Group) cv. 'Red Jamaican'
Musa acuminata Colla (AAA Group) cv. 'Jamaican Red'
Musa acuminata Colla (AAA Group) cv. 'Spanish Red'. | 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. They are frequently eaten raw, whole, or chopped, and added to desserts and fruit salads, but can also be baked, fried, and toasted. Red bananas are also commonly sold dried in stores.
The red banana has more beta-carotene and vitamin C than the yellow banana varieties. All bananas contain natural sources of three sugars: sucrose, fructose, and glucose.
Cultivation
Pests and diseases
Panama disease | 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 confidentiality
Data integrity
Security bugs do not need be identified nor exploited to be qualified as such and are assumed to be much more common than known vulnerabilities in almost any system.
Causes
Security bugs, like all other software bugs, stem from root causes that can generally be traced to either absent or inadequate:
Software developer training
Use case analysis
Software engineering methodology
Quality assurance testing
and other best practices
Taxonomy
Security bugs generally fall into a fairly small number of broad categories that include:
Memory safety (e.g. buffer overflow and dangling pointer bugs)
Race condition
Secure input and output handling
Faulty use of an API
Improper use case handling
Improper exception handling
Resource leaks, often but not always due to improper exception handling
Preprocessing input strings before they are checked for being acceptable
Mitigation
See software security assurance. | 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, covered with wax; leaf scars forming dark rings around the trunk. The leaves are dark green and grayish, long, with a petiole up to . Fruits are globose and orange-red when ripe, in diameter.
Taxonomy
Ceroxylon quindiuense was described by Gustav Karl Wilhelm Hermann Karsten and published in Bonplandia (Hannover) 8: 70. (1860).
Etymology:
Ceroxylon: generic name composed of the Greek words: kèròs = "wax" and xγlon = "wood", in reference to the thick white wax found on the trunks.
quindiuense: geographical epithet alluding to its location in Quindío.
Synonymy:
Klopstockia quindiuensis H.Karst
Ceroxylon floccosum Burret
Ecology
It grows in large and dense populations along the central and eastern Andes of Colombia (rarely in the western Colombian Andes), with a disjunct distribution in the Andes of northern Peru. The elevational range of this species is between above sea level. It achieves a minimum reproductive age at 80 years. Wax palms provide habitats for many unique life forms, including endangered species such as the yellow-eared parrot (Ognorhynchus icterotis).
Vernacular names
Palma de cera, palma de ramo (both names in Colombia).
Conservation
Populations of Ceroxylon quindiuense are threatened by habitat disturbance, overharvesting and diseases. The fruit was used as feed for cattle and pigs. The leaves were extensively used in the Catholic celebrations of Palm Sunday; such leaves coming from young individuals which were damaged to death. That activity has been reduced severely in recent years due to law enforcement and widespread campaign. Felling of Ceroxylon quindiuense palms to obtain wax from the trunk also is an activity still going on in Colombia and Peru. The palm is recognized as the national tree of Colombia, and since the implementation of Law 61 of 1985, it is legally a protected species in that country. | 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.
Gallery | 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. These techniques have developed from angle measurements with quadrants and the unaided eye, through sextants combined with lenses for light magnification, up to current methods which include computer-automated space telescopes. Uranographers have historically produced planetary position tables, star tables, and star maps for use by both amateur and professional astronomers. More recently, computerized star maps have been compiled, and automated positioning of telescopes uses databases of stars and of other astronomical objects.
Etymology
The word "uranography" derived from the Greek "ουρανογραφια" (Koine Greek ουρανος "sky, heaven" + γραφειν "to write") through the Latin "uranographia". In Renaissance times, Uranographia was used as the book title of various celestial atlases. During the 19th century, "uranography" was defined as the "description of the heavens". Elijah H. Burritt re-defined it as the "geography of the heavens". The German word for uranography is "Uranographie", the French is "uranographie" and the Italian is "uranografia".
Astrometry
Astrometry, the science of spherical astronomy, is concerned with precise measurements of the location of celestial bodies in the celestial sphere and their kinematics relative to a reference frame on the celestial sphere. In principle, astrometry can involve such measurements of planets, stars, black holes and galaxies to any celestial body.
Throughout human history, astrometry played a significant role in shaping our understanding of the structure of the visible sky, which accompanies the location of bodies in it, hence making it a fundamental tool to celestial cartography.
Star catalogues
A determining fact source for drawing star charts is naturally a star table. This is apparent when comparing the imaginative "star maps" of Poeticon Astronomicon – illustrations beside a narrative text from the antiquity – to the star maps of Johann Bayer, based on precise star-position measurements from the Rudolphine Tables by Tycho Brahe. | 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 Tycho Brahe, 1,005 stars.
1690, Prodromus Astronomiae – by Johannes Hevelius for his Firmamentum Sobiescanum, 1,564 stars.
1729, Britannic Catalogue – by John Flamsteed for his Atlas Coelestis, position of more than 3,000 stars by accuracy of 10".
1903, Bonner Durchmusterung – by Friedrich Wilhelm Argelander and collaborators, circa 460,000 stars.
Star atlases
Naked-eye
15th century BC – The ceiling of the tomb TT71 for the Egyptian architect and minister Senenmut, who served Queen Hatshepsut, is adorned with a large and extensive star chart.
1 CE ? Poeticon astronomicon, allegedly by Gaius Julius Hyginus
1092 – Xin Yi Xiang Fa Yao (新儀 象法要), by Su Song, a horological treatise which had the earliest existent star maps in printed form. Su Song's star maps also featured the corrected position of the pole star which had been deciphered due to the efforts of astronomical observations by Su's peer, the polymath scientist Shen Kuo.
1515 – First European printed star charts published in Nuremberg, Germany, engraved by Albrecht Dürer.
1603 – Uranometria, by Johann Bayer, the first western modern star map based on Tycho Brahe's and Johannes Kepler's Tabulae Rudolphinae
1627 – Julius Schiller published the star atlas Coelum Stellatum Christianum, which replaced pagan constellations with biblical and early Christian figures.
1660 – Jan Janssonius' 11th volume of Atlas Major (not to be confused with the similarly named and scoped Atlas Maior) featured the Harmonia Macrocosmica by Andreas Cellarius
1693 – Firmamentum Sobiescanum sive Uranometria, by Johannes Hevelius, a star map updated with many new star positions based on Hevelius's Prodromus Astronomiae (1690) – 1564 stars. | 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 Stellarum (Hans Vehrenberg). Stars to magnitude 14.
True Visual Magnitude Photographic Star Atlas (Christos Papadopoulos). Stars to magnitude 13.5.
The Cambridge Photographic Star Atlas, Axel Mellinger and Ronald Stoyan, 2011. Stars to magnitude 14, natural color, 1°/cm.
Modern | 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 Mullaney and Wil Tirion (stars to magnitude 7.5)
Cambridge Atlas of Herschel Objects – James Mullaney and Wil Tirion (stars to magnitude 7.5)
Pocket Sky Atlas – Roger Sinnott (stars to magnitude 7.5)
Deep Sky Reiseatlas – Michael Feiler, Philip Noack (Telrad Finder Charts – stars to magnitude 7.5)
Atlas Coeli Skalnate Pleso (Atlas of the Heavens) 1950.0 – Antonín Bečvář (stars to magnitude 7.75 and about 12,000 clusters, galaxies and nebulae)
SkyAtlas 2000.0, second edition – Wil Tirion & Roger Sinnott (stars to magnitude 8.5)
1987, Uranometria 2000.0 Deep Sky Atlas – Wil Tirion, Barry Rappaport, Will Remaklus (stars to magnitude 9.7; 11.5 in selected close-ups)
Herald-Bobroff AstroAtlas – David Herald & Peter Bobroff (stars to magnitude 9 in main charts, 14 in selected sections)
Millennium Star Atlas – Roger Sinnott, Michael Perryman (stars to magnitude 11)
Field Guide to the Stars and Planets – Jay M. Pasachoff, Wil Tirion charts (stars to magnitude 7.5)
SkyGX (still in preparation) – Christopher Watson (stars to magnitude 12)
The Great Atlas of the Sky – Piotr Brych (2,400,000 stars to magnitude 12, galaxies to magnitude 18).
Interstellarum Deep Sky Atlas (2014) – Ronald Stoyan and Stephan Schurig (stars to magnitude 9.5)
Computerized
100,000 Stars
Cartes du Ciel
Celestia
Stars and Planets for Android
Stars and Planets for iOS
CyberSky
GoSkyWatch Planetarium
Google Sky
KStars
Stellarium
SKY-MAP.ORG
SkyMap Online
WorldWide Telescope
XEphem, for Unix-like systems
Stellarmap.com – online map of the stars
Star Walk and Kepler Explorer OpenLab: 2 celestial cartography apps for smartphones
SpaceEngine | 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 of Wattieza to a root and trunk known since 1870. The fossilized grove of "Gilboa stumps" discovered at Gilboa, New York, were described as Eospermatopteris, though the complete plant remained unknown. These fossils have been described as the earliest known trees, standing 8 m (26 ft) or more tall, resembling the unrelated modern tree fern.
Wattieza had fronds rather than leaves, and they reproduced with spores.
Belgian paleobotanist François Stockmans described the species Wattieza givetiana in 1968 from fossil fronds collected from Middle Devonian strata in the London-Brabant Massif in Belgium.
English geologist and palaeobotanist Chris Berry described Wattieza casasii in Review of Paleobotany and Palynology No. 112 in 2000, based on fossil branches (13 slabs) and numerous other fragments (Berry, 2000) collected from middle-Givetian strata from the lower member of the Campo Chico Formation (Casas et al, 2022). The lithology of the lower member consists of dark grey to green mudstones and shales, interbedded with medium-coarse-grained sandstones close to the base of the Campo Chico Formation, in outcrops of the road to the Rio Socuy (Casas et al, 2022; pag 24), close to the Cano Colorado river, Perija Range, Zulia, Venezuela (Casas et al, 2022). The fossil material of Wattieza casasii is held at the National Museum Cardiff, Cardiff, Wales, and the palaeontological section of the Museo de Biologia at the University of Zulia, Maracaibo, Venezuela (Berry, 2000; p. 127). The name Wattieza casasii was assigned to the species in honor of Jhonny Casas, one of the discoverers of the original material (Berry, 2000; pag. 144). | 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 substituents. E and Z notation are only used when a compound doesn't have two identical substituents.
Following the Cahn–Ingold–Prelog priority rules (CIP rules), each substituent on a double bond is assigned a priority, then positions of the higher of the two substituents on each carbon are compared to each other. If the two groups of higher priority are on opposite sides of the double bond (trans to each other), the bond is assigned the configuration E (from entgegen, , the German word for "opposite"). If the two groups of higher priority are on the same side of the double bond (cis to each other), the bond is assigned the configuration Z (from zusammen, , the German word for "together").
The letters E and Z are conventionally printed in italic type, within parentheses, and separated from the rest of the name with a hyphen. They are always printed as full capitals (not in lowercase or small capitals), but do not constitute the first letter of the name for English capitalization rules (as in the example above).
Another example: The CIP rules assign a higher priority to bromine than to chlorine, and a higher priority to chlorine than to hydrogen, hence the following (possibly counterintuitive) nomenclature.
For organic molecules with multiple double bonds, it is sometimes necessary to indicate the alkene location for each E or Z symbol. For example, the chemical name of alitretinoin is (2E,4E,6Z,8E)-3,7-dimethyl-9-(2,6,6-trimethyl-1-cyclohexenyl)nona-2,4,6,8-tetraenoic acid, indicating that the alkenes starting at positions 2, 4, and 8 are E while the one starting at position 6 is Z. | 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 double-bond has been used sometimes; it is no longer considered an acceptable style for general use by IUPAC but may still be required by computer software. | 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, particularly wave action, sediment movement and weather, and the ways in which humans interact with the coast.
Wave action and longshore drift
The waves of different strengths that constantly hit against the shoreline are the primary movers and shapers of the coastline. Despite the simplicity of this process, the differences between waves and the rocks they hit result in hugely varying shapes.
The effect that waves have depends on their strength. Strong waves, also called destructive waves, occur on high-energy beaches and are typical of winter. They reduce the quantity of sediment present on the beach by carrying it out to bars under the sea. Constructive, weak waves are typical of low-energy beaches and occur most during summer. They do the opposite to destructive waves and increase the size of the beach by piling sediment up onto the berm.
One of the most important transport mechanisms results from wave refraction. Since waves rarely break onto a shore at right angles, the upward movement of water onto the beach (swash) occurs at an oblique angle. However, the return of water (backwash) is at right angles to the beach, resulting in the net movement of beach material laterally. This movement is known as beach drift (Figure 3). The endless cycle of swash and backwash and resulting beach drift can be observed on all beaches. This may differ between coasts. | 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 swash). Due to gravity, the water then falls back perpendicular to the beach, dropping its sediment as it loses energy (this is called backwash). The sediment is then picked up by the next wave and pushed slightly further down the beach, resulting in a continual movement of sediment in one direction. This is the reason why long strips of coast are covered in sediment, not just the areas around river mouths, which are the main sources of beach sediment. LSD is reliant on a constant supply of sediment from rivers and if sediment supply is stopped or sediment falls into a submarine canals at any point along a beach, this can lead to bare beaches further along the shore.
LSD helps create many landforms including barrier islands, bay beaches and spits. In general LSD action serves to straighten the coast because the creation of barriers cuts off bays from the sea while sediment usually builds up in bays because the waves there are weaker (due to wave refraction), while sediment is carried away from the exposed headlands. The lack of sediment on headlands removes the protection of waves from them and makes them more vulnerable to weathering while the gathering of sediment in bays (where longshore drift is unable to remove it) protects the bays from further erosion and makes them pleasant recreational beaches.
Atmospheric processes
Onshore winds blowing "up" the beach, pick up sand and move it up the beach to form sand dunes.
Rain hits the shore and erodes rocks, and carries weathered material to the shoreline to form beaches.
Warm weather can encourage biological processes to occur more rapidly. In tropical areas some plants and animals protect stones from weathering, while other plants and animals actually eat away at the rocks.
Temperatures that vary from below to above freezing point result in frost weathering, whereas weather more than a few degrees below freezing point creates sea ice.
Biological processes
In tropical regions in particular, plants and animals not only affect the weathering of rocks but are a source of sediment themselves. The shells and skeletons of many organisms are of calcium carbonate and when this is broken down it forms sediment, limestone and clay. | 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 calcium carbonate is able to bind together other sediments to form beachrock and in warmer areas dunerock. Wind erosion is also a form of erosion, dust and sand is carried around in the air and slowly erodes rock, this happens in a similar way in the sea were the salt and sand is washed up onto the rocks.
Sea level changes (eustatic change)
The sea level on earth regularly rises and falls due to climatic changes. During cold periods more of the Earth's water is stored as ice in glaciers while during warm periods it is released and sea levels rise to cover more land. Sea levels are currently quite high, while just 18,000 years ago during the Pleistocene ice age they were quite low. Global warming may result in further rises in the future, which presents a risk to coastal cities as most would be flooded by only small rises. As sea levels rise, fjords and rias form. Fjords are flooded glacial valleys and rias are flooded river valleys. Fjords typically have steep rocky sides, while rias have dendritic drainage patterns typical of drainage zones. As tectonic plates move about the Earth they can rise and fall due to changing pressures and the presence of glaciers. If a beach is moving upwards relative to other plates this is known as isostatic change and raised beaches can be formed.
Land level changes (isostatic change)
This is found in the U.K. as above the line from the Wash to the Severn estuary, the land was covered in ice sheets during the last ice age. The weight of the ice caused northeast Scotland to sink, displacing the southeast and forcing it to rise. As the ice sheets receded the reverse process happened, as the land was released from the weight. At current estimates the southeast is sinking at a rate of about 2 mm per year, with northeast Scotland rising by the same amount.
Coastal landforms
Spits | 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 push sediment away from the coast. The area may also be shielded from wave action, preventing much long shore drift. On the side of the headland receiving weaker waves, shingle and other large sediments will build up under the water where waves are not strong enough to move them along. This provides a good place for smaller sediments to build up to sea level. The sediment, after passing the headland will accumulate on the other side and not continue down the beach, sheltered both by the headland and the shingle.
Slowly over time sediment simply builds on this area, extending the spit outwards, forming a barrier of sand. Once in a while, the wind direction will change and come from the other direction. During this period the sediment will be pushed along in the other direction. The spit will start to grow backwards, forming a 'hook'. After this process the spit will grow again in the original direction. Eventually the spit will not be able to grow any further because it is no longer sufficiently sheltered from erosion by waves, or because the estuary current prevents sediment resting. Usually in the salty but calm waters behind the spit there will form a salt marshland. Spits often form around the breakwater of artificial harbours requiring dredging.
Occasionally, if there is no estuary then it is possible for the spit to grow across to the other side of the bay and form what is called a bar, or barrier. Barriers come in several varieties, but all form in a manner similar to spits. They usually enclose a bay to form a lagoon. They can join two headlands or join a headland to the mainland. When an island is joined to the mainland with a bar or barrier it is known as a tombolo. This usually occurs due to wave refraction, but can also be caused by isostatic change, a change in the level of the land (e.g. Chesil Beach). | 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 corresponding eigenvalue, characteristic value, or characteristic root is the multiplying factor (possibly negative).
Geometrically, vectors are multi-dimensional quantities with magnitude and direction, often pictured as arrows. A linear transformation rotates, stretches, or shears the vectors upon which it acts. Its eigenvectors are those vectors that are only stretched, with neither rotation nor shear. The corresponding eigenvalue is the factor by which an eigenvector is stretched or squished. If the eigenvalue is negative, the eigenvector's direction is reversed.
The eigenvectors and eigenvalues of a linear transformation serve to characterize it, and so they play important roles in all the areas where linear algebra is applied, from geology to quantum mechanics. In particular, it is often the case that a system is represented by a linear transformation whose outputs are fed as inputs to the same transformation (feedback). In such an application, the largest eigenvalue is of particular importance, because it governs the long-term behavior of the system after many applications of the linear transformation, and the associated eigenvector is the steady state of the system.
Definition
Consider an matrix and a nonzero vector of length If multiplying with (denoted by ) simply scales by a factor of , where is a scalar, then is called an eigenvector of , and is the corresponding eigenvalue. This relationship can be expressed as: .
There is a direct correspondence between n-by-n square matrices and linear transformations from an n-dimensional vector space into itself, given any basis of the vector space. Hence, in a finite-dimensional vector space, it is equivalent to define eigenvalues and eigenvectors using either the language of matrices, or the language of linear transformations.
The following section gives a more general viewpoint that also covers infinite-dimensional vector spaces. | 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, eigenvalues and eigenvectors have a wide range of applications, for example in stability analysis, vibration analysis, atomic orbitals, facial recognition, and matrix diagonalization.
In essence, an eigenvector v of a linear transformation T is a nonzero vector that, when T is applied to it, does not change direction. Applying T to the eigenvector only scales the eigenvector by the scalar value λ, called an eigenvalue. This condition can be written as the equation
referred to as the eigenvalue equation or eigenequation. In general, λ may be any scalar. For example, λ may be negative, in which case the eigenvector reverses direction as part of the scaling, or it may be zero or complex.
The example here, based on the Mona Lisa, provides a simple illustration. Each point on the painting can be represented as a vector pointing from the center of the painting to that point. The linear transformation in this example is called a shear mapping. Points in the top half are moved to the right, and points in the bottom half are moved to the left, proportional to how far they are from the horizontal axis that goes through the middle of the painting. The vectors pointing to each point in the original image are therefore tilted right or left, and made longer or shorter by the transformation. Points along the horizontal axis do not move at all when this transformation is applied. Therefore, any vector that points directly to the right or left with no vertical component is an eigenvector of this transformation, because the mapping does not change its direction. Moreover, these eigenvectors all have an eigenvalue equal to one, because the mapping does not change their length either.
Linear transformations can take many different forms, mapping vectors in a variety of vector spaces, so the eigenvectors can also take many forms. For example, the linear transformation could be a differential operator like , in which case the eigenvectors are functions called eigenfunctions that are scaled by that differential operator, such as | 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
where the eigenvector v is an n by 1 matrix. For a matrix, eigenvalues and eigenvectors can be used to decompose the matrix—for example by diagonalizing it.
Eigenvalues and eigenvectors give rise to many closely related mathematical concepts, and the prefix eigen- is applied liberally when naming them:
The set of all eigenvectors of a linear transformation, each paired with its corresponding eigenvalue, is called the eigensystem of that transformation.
The set of all eigenvectors of T corresponding to the same eigenvalue, together with the zero vector, is called an eigenspace, or the characteristic space of T associated with that eigenvalue.
If a set of eigenvectors of T forms a basis of the domain of T, then this basis is called an eigenbasis.
History
Eigenvalues are often introduced in the context of linear algebra or matrix theory. Historically, however, they arose in the study of quadratic forms and differential equations.
In the 18th century, Leonhard Euler studied the rotational motion of a rigid body, and discovered the importance of the principal axes. Joseph-Louis Lagrange realized that the principal axes are the eigenvectors of the inertia matrix. | 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| Augustin Cauchy (1839) "Mémoire sur l'intégration des équations linéaires" (Memoir on the integration of linear equations), Comptes rendus, 8: 827–830, 845–865, 889–907, 931–937. From p. 827: ''"On sait d'ailleurs qu'en suivant la méthode de Lagrange, on obtient pour valeur générale de la variable prinicipale une fonction dans laquelle entrent avec la variable principale les racines d'une certaine équation que j'appellerai léquation caractéristique, le degré de cette équation étant précisément l'order de l'équation différentielle qu'il s'agit d'intégrer." (One knows, moreover, that by following Lagrange's method, one obtains for the general value of the principal variable a function in which there appear, together with the principal variable, the roots of a certain equation that I will call the "characteristic equation", the degree of this equation being precisely the order of the differential equation that must be integrated.)}}
Later, Joseph Fourier used the work of Lagrange and Pierre-Simon Laplace to solve the heat equation by separation of variables in his 1822 treatise The Analytic Theory of Heat (Théorie analytique de la chaleur). Charles-François Sturm elaborated on Fourier's ideas further, and brought them to the attention of Cauchy, who combined them with his own ideas and arrived at the fact that real symmetric matrices have real eigenvalues. This was extended by Charles Hermite in 1855 to what are now called Hermitian matrices. | 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 defective matrices can cause instability.
In the meantime, Joseph Liouville studied eigenvalue problems similar to those of Sturm; the discipline that grew out of their work is now called Sturm–Liouville theory. Schwarz studied the first eigenvalue of Laplace's equation on general domains towards the end of the 19th century, while Poincaré studied Poisson's equation a few years later.
At the start of the 20th century, David Hilbert studied the eigenvalues of integral operators by viewing the operators as infinite matrices. He was the first to use the German word eigen, which means "own", to denote eigenvalues and eigenvectors in 1904, though he may have been following a related usage by Hermann von Helmholtz. For some time, the standard term in English was "proper value", but the more distinctive term "eigenvalue" is the standard today.
The first numerical algorithm for computing eigenvalues and eigenvectors appeared in 1929, when Richard von Mises published the power method. One of the most popular methods today, the QR algorithm, was proposed independently by John G. F. Francis and Vera Kublanovskaya in 1961.
Eigenvalues and eigenvectors of matrices
Eigenvalues and eigenvectors are often introduced to students in the context of linear algebra courses focused on matrices.University of Michigan Mathematics (2016) Math Course Catalogue . Accessed on 2016-03-27.
Furthermore, linear transformations over a finite-dimensional vector space can be represented using matrices, which is especially common in numerical and computational applications.
Consider -dimensional vectors that are formed as a list of scalars, such as the three-dimensional vectors
These vectors are said to be scalar multiples of each other, or parallel or collinear, if there is a scalar such that
In this case, .
Now consider the linear transformation of -dimensional vectors defined by an by matrix ,
or
where, for each row,
If it occurs that and are scalar multiples, that is if | 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 characteristic polynomial
Equation () has a nonzero solution v if and only if the determinant of the matrix is zero. Therefore, the eigenvalues of A are values of λ that satisfy the equation
Using the Leibniz formula for determinants, the left-hand side of equation () is a polynomial function of the variable λ and the degree of this polynomial is n, the order of the matrix A. Its coefficients depend on the entries of A, except that its term of degree n is always (−1)nλn. This polynomial is called the characteristic polynomial of A. Equation () is called the characteristic equation or the secular equation of A.
The fundamental theorem of algebra implies that the characteristic polynomial of an n-by-n matrix A, being a polynomial of degree n, can be factored into the product of n linear terms,
where each λi may be real but in general is a complex number. The numbers λ1, λ2, ..., λn, which may not all have distinct values, are roots of the polynomial and are the eigenvalues of A.
As a brief example, which is described in more detail in the examples section later, consider the matrix
Taking the determinant of , the characteristic polynomial of A is
Setting the characteristic polynomial equal to zero, it has roots at and , which are the two eigenvalues of A. The eigenvectors corresponding to each eigenvalue can be found by solving for the components of v in the equation In this example, the eigenvectors are any nonzero scalar multiples of | 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 be irrational numbers even if all the entries of A are rational numbers or even if they are all integers. However, if the entries of A are all algebraic numbers, which include the rationals, the eigenvalues must also be algebraic numbers.
The non-real roots of a real polynomial with real coefficients can be grouped into pairs of complex conjugates, namely with the two members of each pair having imaginary parts that differ only in sign and the same real part. If the degree is odd, then by the intermediate value theorem at least one of the roots is real. Therefore, any real matrix with odd order has at least one real eigenvalue, whereas a real matrix with even order may not have any real eigenvalues. The eigenvectors associated with these complex eigenvalues are also complex and also appear in complex conjugate pairs.
Spectrum of a matrix
The spectrum of a matrix is the list of eigenvalues, repeated according to multiplicity; in an alternative notation the set of eigenvalues with their multiplicities.
An important quantity associated with the spectrum is the maximum absolute value of any eigenvalue. This is known as the spectral radius of the matrix.
Algebraic multiplicity
Let λi be an eigenvalue of an n by n matrix A. The algebraic multiplicity μA(λi) of the eigenvalue is its multiplicity as a root of the characteristic polynomial, that is, the largest integer k such that (λ − λi)k divides evenly that polynomial.
Suppose a matrix A has dimension n and d ≤ n distinct eigenvalues. Whereas equation () factors the characteristic polynomial of A into the product of n linear terms with some terms potentially repeating, the characteristic polynomial can also be written as the product of d terms each corresponding to a distinct eigenvalue and raised to the power of the algebraic multiplicity,
If d = n then the right-hand side is the product of n linear terms and this is the same as equation (). The size of each eigenvalue's algebraic multiplicity is related to the dimension n as | 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, define the set E to be all vectors v that satisfy equation (),
On one hand, this set is precisely the kernel or nullspace of the matrix (A − λI). On the other hand, by definition, any nonzero vector that satisfies this condition is an eigenvector of A associated with λ. So, the set E is the union of the zero vector with the set of all eigenvectors of A associated with λ, and E equals the nullspace of (A − λI). E is called the eigenspace or characteristic space of A associated with λ. In general λ is a complex number and the eigenvectors are complex n by 1 matrices. A property of the nullspace is that it is a linear subspace, so E is a linear subspace of .
Because the eigenspace E is a linear subspace, it is closed under addition. That is, if two vectors u and v belong to the set E, written , then or equivalently . This can be checked using the distributive property of matrix multiplication. Similarly, because E is a linear subspace, it is closed under scalar multiplication. That is, if and α is a complex number, or equivalently . This can be checked by noting that multiplication of complex matrices by complex numbers is commutative. As long as u + v and αv are not zero, they are also eigenvectors of A associated with λ. | 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 − λI), also called the nullity of (A − λI), which relates to the dimension and rank of (A − λI) as
Because of the definition of eigenvalues and eigenvectors, an eigenvalue's geometric multiplicity must be at least one, that is, each eigenvalue has at least one associated eigenvector. Furthermore, an eigenvalue's geometric multiplicity cannot exceed its algebraic multiplicity. Additionally, recall that an eigenvalue's algebraic multiplicity cannot exceed n.
To prove the inequality , consider how the definition of geometric multiplicity implies the existence of orthonormal eigenvectors , such that . We can therefore find a (unitary) matrix whose first columns are these eigenvectors, and whose remaining columns can be any orthonormal set of vectors orthogonal to these eigenvectors of . Then has full rank and is therefore invertible. Evaluating , we get a matrix whose top left block is the diagonal matrix . This can be seen by evaluating what the left-hand side does to the first column basis vectors. By reorganizing and adding on both sides, we get since commutes with . In other words, is similar to , and . But from the definition of , we know that contains a factor , which means that the algebraic multiplicity of must satisfy .
Suppose has distinct eigenvalues , where the geometric multiplicity of is . The total geometric multiplicity of ,
is the dimension of the sum of all the eigenspaces of 's eigenvalues, or equivalently the maximum number of linearly independent eigenvectors of . If , then
The direct sum of the eigenspaces of all of 's eigenvalues is the entire vector space .
A basis of can be formed from linearly independent eigenvectors of ; such a basis is called an eigenbasis Any vector in can be written as a linear combination of eigenvectors of . | 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 the sum of all eigenvalues,
The determinant of is the product of all its eigenvalues,
The eigenvalues of the th power of ; i.e., the eigenvalues of , for any positive integer , are .
The matrix is invertible if and only if every eigenvalue is nonzero.
If is invertible, then the eigenvalues of are and each eigenvalue's geometric multiplicity coincides. Moreover, since the characteristic polynomial of the inverse is the reciprocal polynomial of the original, the eigenvalues share the same algebraic multiplicity.
If is equal to its conjugate transpose , or equivalently if is Hermitian, then every eigenvalue is real. The same is true of any symmetric real matrix.
If is not only Hermitian but also positive-definite, positive-semidefinite, negative-definite, or negative-semidefinite, then every eigenvalue is positive, non-negative, negative, or non-positive, respectively.
If is unitary, every eigenvalue has absolute value .
If is a matrix and are its eigenvalues, then the eigenvalues of matrix (where is the identity matrix) are . Moreover, if , the eigenvalues of are . More generally, for a polynomial the eigenvalues of matrix are .
Left and right eigenvectors
Many disciplines traditionally represent vectors as matrices with a single column rather than as matrices with a single row. For that reason, the word "eigenvector" in the context of matrices almost always refers to a right eigenvector, namely a column vector that right multiplies the matrix in the defining equation, equation (),
The eigenvalue and eigenvector problem can also be defined for row vectors that left multiply matrix . In this formulation, the defining equation is
where is a scalar and is a matrix. Any row vector satisfying this equation is called a left eigenvector of and is its associated eigenvalue. Taking the transpose of this equation, | 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 associated with the same eigenvalues.
Diagonalization and the eigendecomposition
Suppose the eigenvectors of A form a basis, or equivalently A has n linearly independent eigenvectors v1, v2, ..., vn with associated eigenvalues λ1, λ2, ..., λn. The eigenvalues need not be distinct. Define a square matrix Q whose columns are the n linearly independent eigenvectors of A,
Since each column of Q is an eigenvector of A, right multiplying A by Q scales each column of Q by its associated eigenvalue,
With this in mind, define a diagonal matrix Λ where each diagonal element Λii is the eigenvalue associated with the ith column of Q. Then
Because the columns of Q are linearly independent, Q is invertible. Right multiplying both sides of the equation by Q−1,
or by instead left multiplying both sides by Q−1,
A can therefore be decomposed into a matrix composed of its eigenvectors, a diagonal matrix with its eigenvalues along the diagonal, and the inverse of the matrix of eigenvectors. This is called the eigendecomposition and it is a similarity transformation. Such a matrix A is said to be similar to the diagonal matrix Λ or diagonalizable. The matrix Q is the change of basis matrix of the similarity transformation. Essentially, the matrices A and Λ represent the same linear transformation expressed in two different bases. The eigenvectors are used as the basis when representing the linear transformation as Λ. | 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 independent for P to be invertible, there exist n linearly independent eigenvectors of A. It then follows that the eigenvectors of A form a basis if and only if A is diagonalizable.
A matrix that is not diagonalizable is said to be defective. For defective matrices, the notion of eigenvectors generalizes to generalized eigenvectors and the diagonal matrix of eigenvalues generalizes to the Jordan normal form. Over an algebraically closed field, any matrix A has a Jordan normal form and therefore admits a basis of generalized eigenvectors and a decomposition into generalized eigenspaces.
Variational characterization
In the Hermitian case, eigenvalues can be given a variational characterization. The largest eigenvalue of is the maximum value of the quadratic form . A value of that realizes that maximum is an eigenvector.
Matrix examples
Two-dimensional matrix example
Consider the matrix
The figure on the right shows the effect of this transformation on point coordinates in the plane. The eigenvectors v of this transformation satisfy equation (), and the values of λ for which the determinant of the matrix (A − λI) equals zero are the eigenvalues.
Taking the determinant to find characteristic polynomial of A,
Setting the characteristic polynomial equal to zero, it has roots at and , which are the two eigenvalues of A.
For , equation () becomes,
Any nonzero vector with v1 = −v2 solves this equation. Therefore,
is an eigenvector of A corresponding to λ = 1, as is any scalar multiple of this vector.
For , equation () becomes
Any nonzero vector with v1 = v2 solves this equation. Therefore,
is an eigenvector of A corresponding to λ = 3, as is any scalar multiple of this vector.
Thus, the vectors vλ=1 and vλ=3 are eigenvectors of A associated with the eigenvalues and , respectively.
Three-dimensional matrix example
Consider the matrix
The characteristic polynomial of A is | 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 coordinates of the vector up by one position and moves the first coordinate to the bottom. Its characteristic polynomial is 1 − λ3, whose roots are
where is an imaginary unit with
For the real eigenvalue λ1 = 1, any vector with three equal nonzero entries is an eigenvector. For example,
For the complex conjugate pair of imaginary eigenvalues,
Then
and
Therefore, the other two eigenvectors of A are complex and are and with eigenvalues λ2 and λ3, respectively. The two complex eigenvectors also appear in a complex conjugate pair,
Diagonal matrix example
Matrices with entries only along the main diagonal are called diagonal matrices. The eigenvalues of a diagonal matrix are the diagonal elements themselves. Consider the matrix
The characteristic polynomial of A is
which has the roots , , and . These roots are the diagonal elements as well as the eigenvalues of A.
Each diagonal element corresponds to an eigenvector whose only nonzero component is in the same row as that diagonal element. In the example, the eigenvalues correspond to the eigenvectors,
respectively, as well as scalar multiples of these vectors.
Triangular matrix example
A matrix whose elements above the main diagonal are all zero is called a lower triangular matrix, while a matrix whose elements below the main diagonal are all zero is called an upper triangular matrix. As with diagonal matrices, the eigenvalues of triangular matrices are the elements of the main diagonal.
Consider the lower triangular matrix,
The characteristic polynomial of A is
which has the roots , , and . These roots are the diagonal elements as well as the eigenvalues of A.
These eigenvalues correspond to the eigenvectors,
respectively, as well as scalar multiples of these vectors.
Matrix with repeated eigenvalues example
As in the previous example, the lower triangular matrix
has a characteristic polynomial that is the product of its diagonal elements, | 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 the other hand, the geometric multiplicity of the eigenvalue 2 is only 1, because its eigenspace is spanned by just one vector and is therefore 1-dimensional. Similarly, the geometric multiplicity of the eigenvalue 3 is 1 because its eigenspace is spanned by just one vector . The total geometric multiplicity γA is 2, which is the smallest it could be for a matrix with two distinct eigenvalues. Geometric multiplicities are defined in a later section.
Eigenvector-eigenvalue identity
For a Hermitian matrix, the norm squared of the jth component of a normalized eigenvector can be calculated using only the matrix eigenvalues and the eigenvalues of the corresponding minor matrix,
where is the submatrix formed by removing the jth row and column from the original matrix. This identity also extends to diagonalizable matrices, and has been rediscovered many times in the literature.
Eigenvalues and eigenfunctions of differential operators
The definitions of eigenvalue and eigenvectors of a linear transformation T remains valid even if the underlying vector space is an infinite-dimensional Hilbert or Banach space. A widely used class of linear transformations acting on infinite-dimensional spaces are the differential operators on function spaces. Let D be a linear differential operator on the space C∞ of infinitely differentiable real functions of a real argument t. The eigenvalue equation for D is the differential equation
The functions that satisfy this equation are eigenvectors of D and are commonly called eigenfunctions.
Derivative operator example
Consider the derivative operator with eigenvalue equation
This differential equation can be solved by multiplying both sides by dt/f(t) and integrating. Its solution, the exponential function
is the eigenfunction of the derivative operator. In this case the eigenfunction is itself a function of its associated eigenvalue. In particular, for λ = 0 the eigenfunction f(t) is a constant.
The main eigenfunction article gives other examples. | 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 and only if there exists a scalar λ ∈ K such that
This equation is called the eigenvalue equation for T, and the scalar λ is the eigenvalue of T corresponding to the eigenvector v. T(v) is the result of applying the transformation T to the vector v, while λv is the product of the scalar λ with v.
Eigenspaces, geometric multiplicity, and the eigenbasis
Given an eigenvalue λ, consider the set
which is the union of the zero vector with the set of all eigenvectors associated with λ. E is called the eigenspace or characteristic space of T associated with λ.
By definition of a linear transformation,
for x, y ∈ V and α ∈ K. Therefore, if u and v are eigenvectors of T associated with eigenvalue λ, namely u, v ∈ E, then
So, both u + v and αv are either zero or eigenvectors of T associated with λ, namely u + v, αv ∈ E, and E is closed under addition and scalar multiplication. The eigenspace E associated with λ is therefore a linear subspace of V.
If that subspace has dimension 1, it is sometimes called an eigenline.
The geometric multiplicity γT(λ) of an eigenvalue λ is the dimension of the eigenspace associated with λ, i.e., the maximum number of linearly independent eigenvectors associated with that eigenvalue. By the definition of eigenvalues and eigenvectors, γT(λ) ≥ 1 because every eigenvalue has at least one eigenvector.
The eigenspaces of T always form a direct sum. As a consequence, eigenvectors of different eigenvalues are always linearly independent. Therefore, the sum of the dimensions of the eigenspaces cannot exceed the dimension n of the vector space on which T operates, and there cannot be more than n distinct eigenvalues. | 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 entire vector space V, then a basis of V called an eigenbasis can be formed from linearly independent eigenvectors of T. When T admits an eigenbasis, T is diagonalizable.
Spectral theory
If λ is an eigenvalue of T, then the operator (T − λI) is not one-to-one, and therefore its inverse (T − λI)−1 does not exist. The converse is true for finite-dimensional vector spaces, but not for infinite-dimensional vector spaces. In general, the operator (T − λI) may not have an inverse even if λ is not an eigenvalue.
For this reason, in functional analysis eigenvalues can be generalized to the spectrum of a linear operator T as the set of all scalars λ for which the operator (T − λI) has no bounded inverse. The spectrum of an operator always contains all its eigenvalues but is not limited to them.
Associative algebras and representation theory
One can generalize the algebraic object that is acting on the vector space, replacing a single operator acting on a vector space with an algebra representation – an associative algebra acting on a module. The study of such actions is the field of representation theory.
The representation-theoretical concept of weight is an analog of eigenvalues, while weight vectors and weight spaces are the analogs of eigenvectors and eigenspaces, respectively.
Hecke eigensheaf is a tensor-multiple of itself and is considered in Langlands correspondence.
Dynamic equations
The simplest difference equations have the form
The solution of this equation for x in terms of t is found by using its characteristic equation
which can be found by stacking into matrix form a set of equations consisting of the above difference equation and the k – 1 equations giving a k-dimensional system of the first order in the stacked variable vector in terms of its once-lagged value, and taking the characteristic equation of this system's matrix. This equation gives k characteristic roots for use in the solution equation | 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 eigenvalues, and then calculate the eigenvectors for each eigenvalue. It is in several ways poorly suited for non-exact arithmetics such as floating-point.
Eigenvalues
The eigenvalues of a matrix can be determined by finding the roots of the characteristic polynomial. This is easy for matrices, but the difficulty increases rapidly with the size of the matrix.
In theory, the coefficients of the characteristic polynomial can be computed exactly, since they are sums of products of matrix elements; and there are algorithms that can find all the roots of a polynomial of arbitrary degree to any required accuracy. However, this approach is not viable in practice because the coefficients would be contaminated by unavoidable round-off errors, and the roots of a polynomial can be an extremely sensitive function of the coefficients (as exemplified by Wilkinson's polynomial). Even for matrices whose elements are integers the calculation becomes nontrivial, because the sums are very long; the constant term is the determinant, which for an matrix is a sum of different products.
Explicit algebraic formulas for the roots of a polynomial exist only if the degree is 4 or less. According to the Abel–Ruffini theorem there is no general, explicit and exact algebraic formula for the roots of a polynomial with degree 5 or more. (Generality matters because any polynomial with degree is the characteristic polynomial of some companion matrix of order .) Therefore, for matrices of order 5 or more, the eigenvalues and eigenvectors cannot be obtained by an explicit algebraic formula, and must therefore be computed by approximate numerical methods. Even the exact formula for the roots of a degree 3 polynomial is numerically impractical.
Eigenvectors
Once the (exact) value of an eigenvalue is known, the corresponding eigenvectors can be found by finding nonzero solutions of the eigenvalue equation, that becomes a system of linear equations with known coefficients. For example, once it is known that 6 is an eigenvalue of the matrix
we can find its eigenvectors by solving the equation , that is
This matrix equation is equivalent to two linear equations
that is | 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 of the form , for any nonzero real number .
Simple iterative methods
The converse approach, of first seeking the eigenvectors and then determining each eigenvalue from its eigenvector, turns out to be far more tractable for computers. The easiest algorithm here consists of picking an arbitrary starting vector and then repeatedly multiplying it with the matrix (optionally normalizing the vector to keep its elements of reasonable size); this makes the vector converge towards an eigenvector. A variation is to instead multiply the vector by this causes it to converge to an eigenvector of the eigenvalue closest to
If is (a good approximation of) an eigenvector of , then the corresponding eigenvalue can be computed as
where denotes the conjugate transpose of .
Modern methods
Efficient, accurate methods to compute eigenvalues and eigenvectors of arbitrary matrices were not known until the QR algorithm was designed in 1961. Combining the Householder transformation with the LU decomposition results in an algorithm with better convergence than the QR algorithm. For large Hermitian sparse matrices, the Lanczos algorithm is one example of an efficient iterative method to compute eigenvalues and eigenvectors, among several other possibilities.
Most numeric methods that compute the eigenvalues of a matrix also determine a set of corresponding eigenvectors as a by-product of the computation, although sometimes implementors choose to discard the eigenvector information as soon as it is no longer needed.
Applications
Geometric transformations
Eigenvectors and eigenvalues can be useful for understanding linear transformations of geometric shapes.
The following table presents some example transformations in the plane along with their 2×2 matrices, eigenvalues, and eigenvectors. | 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 special cases, a rotation changes the direction of every nonzero vector in the plane.
A linear transformation that takes a square to a rectangle of the same area (a squeeze mapping) has reciprocal eigenvalues.
Principal component analysis
The eigendecomposition of a symmetric positive semidefinite (PSD) matrix yields an orthogonal basis of eigenvectors, each of which has a nonnegative eigenvalue. The orthogonal decomposition of a PSD matrix is used in multivariate analysis, where the sample covariance matrices are PSD. This orthogonal decomposition is called principal component analysis (PCA) in statistics. PCA studies linear relations among variables. PCA is performed on the covariance matrix or the correlation matrix (in which each variable is scaled to have its sample variance equal to one). For the covariance or correlation matrix, the eigenvectors correspond to principal components and the eigenvalues to the variance explained by the principal components. Principal component analysis of the correlation matrix provides an orthogonal basis for the space of the observed data: In this basis, the largest eigenvalues correspond to the principal components that are associated with most of the covariability among a number of observed data.
Principal component analysis is used as a means of dimensionality reduction in the study of large data sets, such as those encountered in bioinformatics. In Q methodology, the eigenvalues of the correlation matrix determine the Q-methodologist's judgment of practical significance (which differs from the statistical significance of hypothesis testing; cf. criteria for determining the number of factors). More generally, principal component analysis can be used as a method of factor analysis in structural equation modeling. | 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), where is a diagonal matrix with equal to the degree of vertex , and in , the th diagonal entry is . The th principal eigenvector of a graph is defined as either the eigenvector corresponding to the th largest or th smallest eigenvalue of the Laplacian. The first principal eigenvector of the graph is also referred to merely as the principal eigenvector.
The principal eigenvector is used to measure the centrality of its vertices. An example is Google's PageRank algorithm. The principal eigenvector of a modified adjacency matrix of the World Wide Web graph gives the page ranks as its components. This vector corresponds to the stationary distribution of the Markov chain represented by the row-normalized adjacency matrix; however, the adjacency matrix must first be modified to ensure a stationary distribution exists. The second smallest eigenvector can be used to partition the graph into clusters, via spectral clustering. Other methods are also available for clustering.
Markov chains
A Markov chain is represented by a matrix whose entries are the transition probabilities between states of a system. In particular the entries are non-negative, and every row of the matrix sums to one, being the sum of probabilities of transitions from one state to some other state of the system. The Perron–Frobenius theorem gives sufficient conditions for a Markov chain to have a unique dominant eigenvalue, which governs the convergence of the system to a steady state.
Vibration analysis
Eigenvalue problems occur naturally in the vibration analysis of mechanical structures with many degrees of freedom. The eigenvalues are the natural frequencies (or eigenfrequencies) of vibration, and the eigenvectors are the shapes of these vibrational modes. In particular, undamped vibration is governed by
or
That is, acceleration is proportional to position (i.e., we expect to be sinusoidal in time). | 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, which are the eigenvectors of alone. Furthermore, damped vibration, governed by
leads to a so-called quadratic eigenvalue problem,
This can be reduced to a generalized eigenvalue problem by algebraic manipulation at the cost of solving a larger system.
The orthogonality properties of the eigenvectors allows decoupling of the differential equations so that the system can be represented as linear summation of the eigenvectors. The eigenvalue problem of complex structures is often solved using finite element analysis, but neatly generalize the solution to scalar-valued vibration problems.
Tensor of moment of inertia
In mechanics, the eigenvectors of the moment of inertia tensor define the principal axes of a rigid body. The tensor of moment of inertia is a key quantity required to determine the rotation of a rigid body around its center of mass.
Stress tensor
In solid mechanics, the stress tensor is symmetric and so can be decomposed into a diagonal tensor with the eigenvalues on the diagonal and eigenvectors as a basis. Because it is diagonal, in this orientation, the stress tensor has no shear components; the components it does have are the principal components.
Schrödinger equation
An example of an eigenvalue equation where the transformation is represented in terms of a differential operator is the time-independent Schrödinger equation in quantum mechanics:
where , the Hamiltonian, is a second-order differential operator and , the wavefunction, is one of its eigenfunctions corresponding to the eigenvalue , interpreted as its energy.
However, in the case where one is interested only in the bound state solutions of the Schrödinger equation, one looks for within the space of square integrable functions. Since this space is a Hilbert space with a well-defined scalar product, one can introduce a basis set in which and can be represented as a one-dimensional array (i.e., a vector) and a matrix respectively. This allows one to represent the Schrödinger equation in a matrix form. | 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 operator, the infinite-dimensional analog of Hermitian matrices. As in the matrix case, in the equation above is understood to be the vector obtained by application of the transformation to .
Wave transport
Light, acoustic waves, and microwaves are randomly scattered numerous times when traversing a static disordered system. Even though multiple scattering repeatedly randomizes the waves, ultimately coherent wave transport through the system is a deterministic process which can be described by a field transmission matrix . The eigenvectors of the transmission operator form a set of disorder-specific input wavefronts which enable waves to couple into the disordered system's eigenchannels: the independent pathways waves can travel through the system. The eigenvalues, , of correspond to the intensity transmittance associated with each eigenchannel. One of the remarkable properties of the transmission operator of diffusive systems is their bimodal eigenvalue distribution with and . Furthermore, one of the striking properties of open eigenchannels, beyond the perfect transmittance, is the statistically robust spatial profile of the eigenchannels.
Molecular orbitals
In quantum mechanics, and in particular in atomic and molecular physics, within the Hartree–Fock theory, the atomic and molecular orbitals can be defined by the eigenvectors of the Fock operator. The corresponding eigenvalues are interpreted as ionization potentials via Koopmans' theorem. In this case, the term eigenvector is used in a somewhat more general meaning, since the Fock operator is explicitly dependent on the orbitals and their eigenvalues. Thus, if one wants to underline this aspect, one speaks of nonlinear eigenvalue problems. Such equations are usually solved by an iteration procedure, called in this case self-consistent field method. In quantum chemistry, one often represents the Hartree–Fock equation in a non-orthogonal basis set. This particular representation is a generalized eigenvalue problem called Roothaan equations.
Geology and glaciology | 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 compared graphically or as a stereographic projection. Graphically, many geologists use a Tri-Plot (Sneed and Folk) diagram,. A stereographic projection projects 3-dimensional spaces onto a two-dimensional plane. A type of stereographic projection is Wulff Net, which is commonly used in crystallography to create stereograms.
The output for the orientation tensor is in the three orthogonal (perpendicular) axes of space. The three eigenvectors are ordered by their eigenvalues ;
then is the primary orientation/dip of clast, is the secondary and is the tertiary, in terms of strength. The clast orientation is defined as the direction of the eigenvector, on a compass rose of 360°. Dip is measured as the eigenvalue, the modulus of the tensor: this is valued from 0° (no dip) to 90° (vertical). The relative values of , , and are dictated by the nature of the sediment's fabric. If , the fabric is said to be isotropic. If , the fabric is said to be planar. If , the fabric is said to be linear.
Basic reproduction number
The basic reproduction number () is a fundamental number in the study of how infectious diseases spread. If one infectious person is put into a population of completely susceptible people, then is the average number of people that one typical infectious person will infect. The generation time of an infection is the time, , from one person becoming infected to the next person becoming infected. In a heterogeneous population, the next generation matrix defines how many people in the population will become infected after time has passed. The value is then the largest eigenvalue of the next generation matrix.
Eigenfaces | 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 example of principal component analysis. They are very useful for expressing any face image as a linear combination of some of them. In the facial recognition branch of biometrics, eigenfaces provide a means of applying data compression to faces for identification purposes. Research related to eigen vision systems determining hand gestures has also been made.
Similar to this concept, eigenvoices''' represent the general direction of variability in human pronunciations of a particular utterance, such as a word in a language. Based on a linear combination of such eigenvoices, a new voice pronunciation of the word can be constructed. These concepts have been found useful in automatic speech recognition systems for speaker adaptation. | 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 antecessor; but in recent studies, most regard it either as a form of Homo heidelbergensis sharing affinities with African forms, or an early morph of Neanderthal.
History
During excavation in preparation for a highway on March 13, 1994, in the Campo Grande area near Ceprano, Italy, a partial hominin calvaria was discovered. Although damaged by a bulldozer, it was recognized, documented, and described by archeologist Italo Biddittu, who happened to be present when the fossil was discovered. Mallegni et al. (2003) proposed the introduction of a new human species, dubbed Homo cepranensis, based on the fossil. As the specimen was believed to be around 700 ka, they believed that this specimen and Homo antecessor suggested a wave of dispersals into Europe 0.9-0.8 Ma through Iberia and the Middle East. The most recent belief is that it is associated with Homo heidelbergensis or rhodesiensis, or that it is ancestral to Neanderthals.
Dating
The fossil was first estimated to be between 690,000 and 900,000 years old determined on the basis of regional correlations and a series of absolute dates. Taking the circumstances of the recovery of the fossil into account, Ascenzi (2001) noted that "an age between 800 and 900 ka is at present our best chronological estimate" based on "the absence in the sediments containing the cranium of any leucitic remnants of the more recent volcanic activity known in the region . . . and the presence above the cranium itself of a clear stratigraphic unconformity that marks" After clarification of its geostratigraphic, biostratigraphic and archaeological relation to the well known and nearby Acheulean site of Fontana Ranuccio, dated to , Muttoni et al. (2009) suggested that Ceprano is most likely about 450,000 years old. Manzi et al. (2010) agree with this, citing an age of 430 and 385 ka. | 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 analysis and context of the find. They note that a lack of gnawing, weathering or abrasions induced by transport supports the theory that the skull was buried once by rising and falling water levels, which is evidenced by the pedofeatures of the clay it was found in situ. This would have dispersed the remaining skeleton and rapidly filled the cranium for fossilization.
Description
The reconstruction of the skull made in 2000 by Clarke and tweaked by M.A. de Lumley and Mallegni features repositioning of the parietal, removal of dental plaster, midsagittal plane was established, added two zygomatic frontal processus previously missing, added an occipital fragment, and rid of unnecessary plaster and glue reinforcements. DI Vincenzo et al. (2017) provided a virtual reconstruction wherein all plaster and glue was removed and the remains were repositioned to most closely fit their life position. They noticed misplacement and misalignment in the temporo-parietal region, left mastoid process, and occipital squama, and worked to correct some taphonomic distortion through retrodeformation and other methods. Most of this work is reflected in the vault rather than the face, and most of the peculiar aspects of the skull are now gone. For example, the single autapomorphy used to distinguish it as a new, valid species is a foreshortened vault, which when compared to the new reconstruction, appears typical of H. heidelbergensis. | 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 caused by an altercation with a large animal, where the skull was butted and fractured; this is more plausible than another, more popular explanation that the blow was inflicted by another human wielding a stick (thus being, hypothetically, murder). They hypothesized that the individual was a young adult man (gender stated without evidence) whose activities consisted of hunting for themself or the group, and was "bold and aggressive" based on the accumulation of injuries. The fracture healed, suggesting that it did not cause death and the congenital malformation on the skull was not restrictive or painful enough to limit the subject's physical abilities.
Classification
Ascenzi et al. (1996) argue that the similarity to Chinese H. erectus and assignment to Homo heidelbergensis based on provenance (as Mauer cannot be compared to Ceprano) cannot justify attribution to any other species. Ascenzi and Segre (1997) compared an early cranial reconstruction with the Gran Dolina fossils and concluded that it was "late Homo erectus", being one of the latest occurrences of the species and earliest Italian hominin. This allocation was supported by them with vault profile data and metrics. However, Ascenzi and Segre also consider specimens such as Montmaurin, Arago, Petralona and Vertesszolos as H. erectus or a similar taxon. They suggested that Tighenif No. 3 mandible is a good fit for the skull, and hinted that connection between it and North Africa may be evident.
Clarke (2000) suggested that inconsistencies with minimal fontal breadth may be individual or geographic variation, and not taxonomically informative. Ascenzi et al. (2000) followed the cranial reconstruction by Clarke (2000) and modifications by M.A. de Lumley to reinforce assignment to H. erectus based on the tori, cranial capacity, bone thickness, and occipital profile angle. | 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 contemporary species, as they find the specimen is not referrable to Homo erectus, H. ergaster, H. heidelbergensis, or H. rhodesiensis. In fact, they recommend creation of a new name to represent a transition from late African to early European fossils. They also suggest that Early Pleistocene dispersals toted a new morphology that was lost, possibly by other Acheulean-using hominins.
Mallegni et al. (2003) noticed a lack of Homo heidelbergensis frontal morphology was similar to the Daka specimen, and as such they were recovered as sister individuals in their cladistic analysis. They propose that the Bouri population was the source for later European populations, and the resulting species did not contribute to the genomes of later Middle Pleistocene (MP) hominins. They also suggested that similarity with Homo rhodesiensis fossils may be reflective of an ancestral-descendant phylogenetic relationship; and since the fossil was appearing so distinct they named Homo cepranensis with the calvarium as the holotype and only specimen.
Bruner et al. (2007) recognize that the characters of the specimen exhibits a mix of early African and later European features, enough to be potentially distinct or, alternatively, considered an ancestral of Homo heidelbergensis. However, they caution other workers that no direct comparisons can yet be made based on fossil record incompleteness. Mounier et al. (2011) have identified the fossil as "an appropriate ancestral stock of [H. heidelbergensis] . . . preceding the appearance of regional autapomorphic features." They suggested that the specimen could be "an appropriate 'counterpart'" to the current, inadequate holotype due to its preservation and morphology. They also suggest ancestry with Neanderthals. | 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 results, both teams draw similar conclusions that the Ceprano calvarium and the Arago hominins are closest in morphology.
Manzi (2016) suggested that the species Homo heidelbergensis is the best descriptor for the calvaria, and further proposed two modes of classification. One uses a single species under that name with Ceprano being having ancestral characters, but noticed that subspecific distinctions may be made. The second incorporates this, using the following: H. h. heidelbergensis (Ceprano, Mauer, Arago, ?Hexian, Melka Kunture 2–3), daliensis (Dali, Denisova, Jinniushan, Narmada), rhodesiensis (Broken Hill, Irhoud, Florisbad, Eliye Springs, Ngaloba, Omo Kibish II), and steinheimensis (Steinheim, Petralona, Reilingen, Swanscombe, Sima). Di Vincenzo et al. (2017) found with their new reconstruction that it is typical of H. heidelbergensis, specifically Broken Hill and Petralona. They suggest that it is ancestral to the neanderthalensis-sapiens-Denisovan clade.
Manzi (2021) elaborates that the specimen is a lost morphology that lived in a refugium in Italy (much like the Neanderthal from Altamura) and retained plesiomorphic traits for an extended duration. This suite of old traits gave rise to the MP hominin diversity observed, but was absorbed. Manzi, again, recommends H. h. heidelbergensis for the specimen. The description of the Harbin skull suggests that it is associated specifically with H. rhodesiensis. Roksandic et al. (2022) considered it for inclusion in their Homo bodoensis, but this term was agreed to be valueless and does not comply with ICZN naming conventions. They suggest that it may have contributed to Arago and Petralona, among other specimens. | 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 suggested that these populations dispersed during the late Early Pleistocene with Mode I technologies and their morphology was lost by other Acheulean-using hominins.
Paleoecology
The Ceprano calvarium was discovered in the Camp Grande area by what is now a highway. It was associated with bone and lithic Acheulean artifacts and faunal remains, such as the straight-tusked elephant (Palaeoloxodon antiquus), the narrow-nosed rhinoceros (Stephanorhinus hemitoechus), Hippopotamus sp., the giant deer Praemegaceros verticornis, the fallow deer Dama clactoniana, beavers, aurochs (Bos primigenius), and the European pond turtle (Emys orbicularis). Pleistocene fossils occur in strata that is around 50 meters in thickness. The Ceprano basin splits into two sections: one of 22 meters in fluvial-colluvial facies with gravels and sands intercalacted with clays, and one of 24 meters with distinct limno-marsh facies, clays and silts. Further, six groups are split. The furthest to the top holds Acheulean and is common in aurochs and Palaeoloxodon. The skull was discovered in clays of a gray-green color above a travertine and with scattered nodular calcium carbonate concretions, mixed with yellow sands, and diffused with Ferromanganese. The area would have probably been forested due to the clay corresponding to a fluvial period that is in relation to terminal tectonics. It lived during the MIS 11, a warm stage at Lirino Lake, which was a refugium for archaic morphologies. It was buried once in a perilacustrine environment by rising and lowering water, scattering the skeleton and filling the cranium. | 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 Caves. Discovered in 1938, it was among the first early hominins described, and became the type species for the genus Paranthropus. However, it has been argued by some that Paranthropus is an invalid grouping and synonymous with Australopithecus, so the species is also often classified as Australopithecus robustus.
Robust australopithecines—as opposed to gracile australopithecines—are characterised by heavily built skulls capable of producing high stresses and bite forces, as well as inflated cheek teeth (molars and premolars). Males had more heavily built skulls than females. P. robustus may have had a genetic susceptibility for pitting enamel hypoplasia on the teeth, and seems to have had a dental cavity rate similar to non-agricultural modern humans. The species is thought to have exhibited marked sexual dimorphism, with males substantially larger and more robust than females. Based on 3 specimens, males may have been tall and females . Based on 4 specimens, males averaged in weight and females . The brain volume of the specimen SK 1585 is estimated to have been 476 cc, and of DNH 155 about 450 cc (for comparison, the brain volume of contemporary Homo varied from 500 to 900 cc). P. robustus limb anatomy is similar to that of other australopithecines, which may indicate a less efficient walking ability than modern humans, and perhaps some degree of arboreality (movement in the trees). | 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 harem society like gorillas or a multi-male society like baboons. P. robustus society may have been patrilocal, with adult females more likely to leave the group than males, but males may have been more likely to be evicted as indicated by higher male mortality rates and assumed increased risk of predation to solitary individuals. P. robustus contended with sabertooth cats, leopards, and hyenas on the mixed, open-to-closed landscape, and P. robustus bones probably accumulated in caves due to big cat predation. It is typically found in what were mixed open and wooded environments, and may have gone extinct in the Mid-Pleistocene Transition characterised by the continual prolonging of dry cycles and subsequent retreat of such habitat.
Taxonomy
Research history
Discovery
The first remains, a partial skull including a part of the jawbone (TM 1517), were discovered in June 1938 at the Kromdraai cave site, South Africa, by local schoolboy Gert Terblanche. He gave the remains to South African conservationist Charles Sydney Barlow, who then relayed them to South African palaeontologist Robert Broom. Broom began investigating the site, and, a few weeks later, recovered a right distal humerus (the lower part of the upper arm bone), a proximal right ulna (upper part of a lower arm bone) and a distal phalanx bone of the big toe, all of which he assigned to TM 1517. He also identified a distal toe phalanx which he believed belonged to a baboon, but has since been associated with TM 1517. Broom noted the Kromdraai remains were especially robust compared to other hominins. In August 1938, Broom classified the robust Kromdraai remains into a new genus, as Paranthropus robustus. "Paranthropus" derives from the Ancient Greek , beside or alongside; and , man. | 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 synonymised with A. africanus) from the Sterkfontein Caves only west from Kromdraai. All these species dated to the Pleistocene and were found in the same general vicinity (now called the "Cradle of Humankind"). Broom considered them evidence of a greater diversity of hominins in the Pliocene from which they and modern humans descended, and consistent with several hominin taxa existing alongside human ancestors.
The Kromdraai taxon, classified as Paranthropus robustus, was later discovered at the nearby Swartkrans Cave in 1948. P. robustus was only definitively identified at Kromdraai and Swartkrans until around the turn of the century when the species was reported elsewhere in the Cradle of Humankind at Sterkfontein, Gondolin, Cooper's, and Drimolen Caves. The species has not been found outside this small area. | 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 were classified into the family Hominidae, and non-human great apes into "Pongidae"; in 1950, Broom suggested separating early hominins into the subfamilies Australopithecinae (Au. africanus and "Pl. transvaalensis"), "Paranthropinae" (Pa. robustus and "Pa. crassidens"), and "Archanthropinae" ("Au. prometheus"). This scheme was widely criticised for being too liberal in demarcating species. Further, the remains were not firmly dated, and it was debated if there were indeed multiple hominin lineages or if there was only a single one leading to humans. Most prominently, Broom and South African palaeontologist John Talbot Robinson continued arguing for the validity of Paranthropus.
Anthropologists Sherwood Washburn and Bruce D. Patterson were the first to recommend synonymising Paranthropus with Australopithecus in 1951, wanting to limit hominin genera to only that and Homo, and it has since been debated whether or not Paranthropus is a junior synonym of Australopithecus. In the spirit of tightening splitting criteria for hominin taxa, in 1954, Robinson suggested demoting "P. crassidens" to subspecies level as "P. r. crassidens", and also moved the Indonesian Meganthropus into the genus as "P. palaeojavanicus". Meganthropus has since been variously reclassified as a synonym of the Asian Homo erectus, "Pithecanthropus dubius", Pongo (orangutans), and so on, and in 2019 it was again argued to be a valid genus. | 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 remains were already abundantly found in the cave. In 1957, though, Italian biologist Alberto Simonetta moved it to the genus "Pithecanthropus", and Robinson (without a specific reason why) decided to synonymise it with H. erectus (African H. erectus are sometimes called H. ergaster today). In 1965, South African palaeoanthropologist Phillip V. Tobias questioned whether this classification is completely sound or not.
By the 21st century, "P. crassidens" had more or less fallen out of use in favour of P. robustus. American palaeoanthropologist Frederick E. Grine is the primary opponent of synonymisation of the two species.
Gigantopithecus
In 1939, Broom hypothesised that P. robustus was closely related to the similarly large-toothed ape Gigantopithecus from Asia (extinct apes were primarily known from Asia at the time) believing Gigantopithecus to have been a hominin. Primarily influenced by the mid-century opinions of Jewish German anthropologist Franz Weidenreich and German-Dutch palaeontologist Ralph von Koenigswald that Gigantopithecus was, respectively, the direct ancestor of the Asian H. erectus or closely related, much debate followed over whether Gigantopithecus was a hominin or a non-human ape. | 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 he referred to as "Homo" africanus) and speculated it had some cultural and hunting abilities, being a member of the human lineage, which "paranthropines" lacked. With the popularisation of cladistics by the late 1970s to 1980s, and better resolution on how Miocene apes relate to later apes, Gigantopithecus was entirely removed from Homininae, and is now placed in the subfamily Ponginae with orangutans.
P. boisei
In 1959, another and much more robust australopithecine was discovered in East Africa, P. boisei, and in 1975, the P. boisei skull KNM-ER 406 was demonstrated to have been contemporaneous with the H. ergaster/H. erectus skull KNM ER 3733 (which is considered a human ancestor). This is generally taken to show that Paranthropus was a sister taxon to Homo, both developing from some Australopithecus species, which at the time only included A. africanus.
In 1979, a year after describing A. afarensis from East Africa, anthropologists Donald Johanson and Tim D. White suggested that A. afarensis was instead the last common ancestor between Homo and Paranthropus, and A. africanus was the earliest member of the Paranthropus lineage or at least was ancestral to P. robustus, because A. africanus inhabited South Africa before P. robustus, and A. afarensis was at the time the oldest known hominin species at roughly 3.5 million years old. Now, the earliest-known South African australopithecine ("Little Foot") dates to 3.67 million years ago, contemporaneous with A. afarensis. The matter is still debated. | 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. boisei (they considered Paranthropus synonymous with Australopithecus), thus establishing the boisei lineage as beginning long before robustus had existed.
Classification
The genus Paranthropus (otherwise known as "robust australopithecines", in contrast to the "gracile australopithecines") now also includes the East African P. boisei and P. aethiopicus. It is still debated if this is a valid natural grouping (monophyletic) or an invalid grouping of similar-looking hominins (paraphyletic). Because skeletal elements are so limited in these species, their affinities with each other and with other australopithecines are difficult to gauge with accuracy. The jaws are the main argument for monophyly, but jaw anatomy is strongly influenced by diet and environment, and could have evolved independently in P. robustus and P. boisei. Proponents of monophyly consider P. aethiopicus to be ancestral to the other two species, or closely related to the ancestor. Proponents of paraphyly allocate these three species to the genus Australopithecus as A. boisei, A. aethiopicus, and A. robustus. In 2020, palaeoanthropologist Jesse M. Martin and colleagues' phylogenetic analyses reported the monophyly of Paranthropus, but also that P. robustus had branched off before P. aethiopicus (that P. aethiopicus was ancestral to only P. boisei). The exact classification of Australopithecus species with each other is quite contentious.
In 2023, fragmentary genetic material belonging to this species was reported from 2 million year-old teeth, being the oldest genetic evidence to be retrieved from a human.
Anatomy
Head
Skull | 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 especially at the tips of the cusps, whereas in humans it thickens at the base of the cusps.
P. robustus has a tall face with slight prognathism (the jaw jutted out somewhat). The skulls of males have a well-defined sagittal crest on the midline of the skullcap and inflated cheek bones, which likely supported massive temporal muscles important in biting. The cheeks project so far from the face that, when in top-view, the nose appears to sit at the bottom of a concavity (a dished face). This displaced the eye sockets forward somewhat, causing a weak brow ridge and receding forehead. The inflated cheeks also would have pushed the masseter muscle (important in biting down) forward and pushed the tooth rows back, which would have created a higher bite force on the premolars. The ramus of the jawbone, which connects the lower jaw to the upper jaw, is tall, which would have increased lever arm (and thereby, torque) of the masseter and medial pterygoid muscles (both important in biting down), further increasing bite force.
The well-defined sagittal crest and inflated cheeks are absent in the presumed-female skull DNH-7, so Keyser suggested that male P. robustus may have been more heavily built than females (P. robustus was sexually dimorphic). The Drimolen material, being more basal, is comparatively more gracile and consequently probably had a smaller bite force than the younger Swartkrans and Kromdraii P. robustus. The brows of the former also are rounded off rather than squared, and the sagittal crest of the presumed-male DNH 155 is more posteriorly (towards the back of the head) positioned. | 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 thought to aid in stabilisation while running, which could mean P. robustus was not an endurance runner.
Brain
Upon describing the species, Broom estimated the fragmentary braincase of TM 1517 as 600 cc, and he, along with South African anthropologist Gerrit Willem Hendrik Schepers, revised this to 575–680 cc in 1946. For comparison, the brain volume of contemporary Homo varied from 500 to 900 cc. A year later, British primatologist Wilfrid Le Gros Clark commented that, since only a part of the temporal bone on one side is known, brain volume cannot be accurately measured for this specimen. In 2001, Polish anthropologist Katarzyna Kaszycka said that Broom quite often artificially inflated brain size in early hominins, and the true value was probably much lower.
In 1972, American physical anthropologist Ralph Holloway measured the skullcap SK 1585, which is missing part of the frontal bone, and reported a volume of about 530 cc. He also noted that, compared to other australopithecines, Paranthropus seems to have had an expanded cerebellum like Homo, echoing what Tobias said while studying P. boisei skulls in 1967. In 2000, American neuroanthropologist Dean Falk and colleagues filled in frontal bone anatomy of SK 1585 using the P. boisei specimens KNM-ER 407, OH 5, and KNM-ER 732, and recalculated the brain volume to about 476 cc. They stated overall brain anatomy of P. robustus was more like that of non-human apes.
In 2020, the nearly complete skull DNH 155 was discovered and was measured to have had a brain volume of 450 cc. | 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 in earlier hominins, and considered this a derived characteristic due to increased brain capacity. It has since been demonstrated that, at least for P. boisei, the parietal branch could originate from either the anterior or posterior branches, sometimes both in a single specimen on opposite sides of the skull.
Regarding the dural venous sinuses, in 1983, Falk and anthropologist Glenn Conroy suggested that, unlike A. africanus or modern humans, all Paranthropus (and A. afarensis) had expanded occipital and marginal (around the foramen magnum) sinuses, completely supplanting the transverse and sigmoid sinuses. They suggested the setup would have increased blood flow to the internal vertebral venous plexuses or internal jugular vein, and was thus related to the reorganisation of the blood vessels supplying the head as an immediate response to bipedalism, which relaxed as bipedalism became more developed. In 1988, Falk and Tobias demonstrated that early hominins (at least A. africanus and P. boisei) could have both an occipital/marginal and transverse/sigmoid systems concurrently or on opposite halves of the skull. | 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 lumbar vertebra. The 12th thoracic vertebra is relatively elongated, and the articular surface (where it joins with another vertebra) is kidney-shaped. The T12 is more compressed in height than that of other australopithecines and modern apes. Modern humans who suffer from spinal disc herniation often have vertebrae that are more similar to those of chimpanzees than healthy humans. Early hominin vertebrae are similar to those of a pathological human, including the only other 12th thoracic vertebra known for P. robustus, the juvenile SK 853. Conversely, SK 3981 is more similar to those of healthy humans, which could be explained as: SK 3981 is abnormal, the vertebrae took on a more humanlike condition with maturity, or one of these specimens is assigned to the wrong species. The shape of the lumbar vertebrae is much more similar to that of Turkana Boy (H. ergaster/H. erectus) and humans than other australopithecines. The pedicles (which jut out diagonally from the vertebra) of the lower lumbar vertebra are much more robust than in other australopithecines and are within the range of humans, and the transverse processes (which jut out to the sides of the vertebra) indicate powerful iliolumbar ligaments. These could have bearing on the amount of time spent upright compared to other australopithecines. | 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 sacrum, and indicates lumbar lordosis (curvature of the lumbar vertebrae) and thus bipedalism. The anatomy of the sacrum and the first lumbar vertebra (at least the vertebral arch), preserved in DNH 43, are similar to those of other australopithecines. The pelvis seems to indicate a more-or-less humanlike hip joint consistent with bipedalism, though differences in overall pelvic anatomy may indicate P. robustus used different muscles to generate force and perhaps had a different mechanism to direct force up the spine. This is similar to the condition seen in A. africanus. This could potentially indicate the lower limbs had a wider range of motion than those of modern humans. | 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 humans rather than the greater flexion achieved by non-human apes, but the head of radius (the elbow) seems to have been quite capable of maintaining stability when the forearm was flexed like non-human apes. It is possible this reflects some arboreal activity (movement in the trees) as is controversially postulated in other australopithecines. SKX 3602 exhibits robust radial styloid processes near the hand which indicate strong brachioradialis muscles and extensor retinaculae. Like humans, the finger bones are uncurved and have weaker muscle attachment than non-human apes, though the proximal phalanges are smaller than in humans. The intermediate phalanges are stout and straight like humans, but have stouter bases and better developed flexor impressions. The distal phalanges seem to be essentially humanlike. These could indicate a decreased climbing capacity compared to non-human apes and P. boisei. The P. robustus hand is consistent with a humanlike precision grip which would have made possible the production or usage of tools requiring greater motor functions than non-human primate tools. | 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 trabecular bone (at the hip joint) structure, which could indicate reduced mobility of the hip joint compared to non-human apes, and the ability to produce forces consistent with humanlike bipedalism. The femoral head StW 311, which either belongs to P. robustus or early Homo, seems to have habitually been placed in highly flexed positions based on the wearing patterns, which would be consistent with frequent climbing activity. It is unclear if frequent squatting could be a valid alternative interpretation. The textural complexity of the kneecap SKX 1084, which reflects cartilage thickness and thus usage of the knee joint and bipedality, is midway between modern humans and chimps. The big toe bone of P. robustus is not dextrous, which indicates a humanlike foot posture and range of motion, but the more distal ankle joint would have inhibited the modern human toe-off gait cycle. P. robustus and H. habilis may have achieved about the same grade of bipedality. | 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 using the presumed female humerus of STS 7 and comparing it with the presumed male femur of STS 14. He also had to estimate the length of the humerus using the femur assuming a similar degree of sexual dimorphism between P. robustus and humans. Comparing the ratio to humans, he concluded that P. robustus was a heavily built species with a height of and a weight of . Consequently, Robinson had described its locomotory habits as, "a compromise between erectness and facility for quadrupedal climbing." In contrast, he estimated A. africanus (which he called "H." africanus) to have been tall and in weight, and to have also been completely bipedal. | Paranthropus robustus | Wikipedia | 224 | 2165269 | https://en.wikipedia.org/wiki/Paranthropus%20robustus | Biology and health sciences | Australopithecines | Biology |
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