Jacobi elliptic functionsIn mathematics, the Jacobi elliptic functions are a set of basic elliptic functions. They are found in the description of the motion of a pendulum (see also pendulum (mathematics)), as well as in the design of electronic elliptic filters. While trigonometric functions are defined with reference to a circle, the Jacobi elliptic functions are a generalization which refer to other conic sections, the ellipse in particular. The relation to trigonometric functions is contained in the notation, for example, by the matching notation for .
AsthenosphereThe asthenosphere () is the mechanically weak and ductile region of the upper mantle of Earth. It lies below the lithosphere, at a depth between ~ below the surface, and extends as deep as . However, the lower boundary of the asthenosphere is not well defined. The asthenosphere is almost solid, but a slight amount of melting (less than 0.1% of the rock) contributes to its mechanical weakness. More extensive decompression melting of the asthenosphere takes place where it wells upwards, and this is the most important source of magma on Earth.
Weierstrass elliptic functionIn mathematics, the Weierstrass elliptic functions are elliptic functions that take a particularly simple form. They are named for Karl Weierstrass. This class of functions are also referred to as ℘-functions and they are usually denoted by the symbol ℘, a uniquely fancy script p. They play an important role in the theory of elliptic functions. A ℘-function together with its derivative can be used to parameterize elliptic curves and they generate the field of elliptic functions with respect to a given period lattice.
Elliptic curveIn mathematics, an elliptic curve is a smooth, projective, algebraic curve of genus one, on which there is a specified point O. An elliptic curve is defined over a field K and describes points in K^2, the Cartesian product of K with itself. If the field's characteristic is different from 2 and 3, then the curve can be described as a plane algebraic curve which consists of solutions (x, y) for: for some coefficients a and b in K. The curve is required to be non-singular, which means that the curve has no cusps or self-intersections.
MagmaMagma () is the molten or semi-molten natural material from which all igneous rocks are formed. Magma (sometimes colloquially but incorrectly referred to as lava by laypeople) is found beneath the surface of the Earth, and evidence of magmatism has also been discovered on other terrestrial planets and some natural satellites. Besides molten rock, magma may also contain suspended crystals and gas bubbles. Magma is produced by melting of the mantle or the crust in various tectonic settings, which on Earth include subduction zones, continental rift zones, mid-ocean ridges and hotspots.
SubductionSubduction is a geological process in which the oceanic lithosphere and some continental lithosphere is recycled into the Earth's mantle at convergent boundaries. Where the oceanic lithosphere of a tectonic plate converges with the less dense lithosphere of a second plate, the heavier plate dives beneath the second plate and sinks into the mantle. A region where this process occurs is known as a subduction zone, and its surface expression is known as an arc-trench complex.
Magma chamberA magma chamber is a large pool of liquid rock beneath the surface of the Earth. The molten rock, or magma, in such a chamber is less dense than the surrounding country rock, which produces buoyant forces on the magma that tend to drive it upwards. If the magma finds a path to the surface, then the result will be a volcanic eruption; consequently, many volcanoes are situated over magma chambers. These chambers are hard to detect deep within the Earth, and therefore most of those known are close to the surface, commonly between 1 km and 10 km down.
BasaltBasalt (UKˈbæsɔːlt,_-əlt; USbəˈsɔːlt,_ˈbeɪsɔːlt) is an aphanitic (fine-grained) extrusive igneous rock formed from the rapid cooling of low-viscosity lava rich in magnesium and iron (mafic lava) exposed at or very near the surface of a rocky planet or moon. More than 90% of all volcanic rock on Earth is basalt. Rapid-cooling, fine-grained basalt is chemically equivalent to slow-cooling, coarse-grained gabbro. The eruption of basalt lava is observed by geologists at about 20 volcanoes per year.
Cumulate rockCumulate rocks are igneous rocks formed by the accumulation of crystals from a magma either by settling or floating. Cumulate rocks are named according to their texture; cumulate texture is diagnostic of the conditions of formation of this group of igneous rocks. Cumulates can be deposited on top of other older cumulates of different composition and colour, typically giving the cumulate rock a layered or banded appearance. Cumulate rocks are the typical product of precipitation of solid crystals from a fractionating magma chamber.
Divergent boundaryIn plate tectonics, a divergent boundary or divergent plate boundary (also known as a constructive boundary or an extensional boundary) is a linear feature that exists between two tectonic plates that are moving away from each other. Divergent boundaries within continents initially produce rifts, which eventually become rift valleys. Most active divergent plate boundaries occur between oceanic plates and exist as mid-oceanic ridges.
VolcanismVolcanism, vulcanism or volcanicity is the phenomenon of eruption of molten rock (magma) onto the surface of the Earth or a solid-surface planet or moon, where lava, pyroclastics, and volcanic gases erupt through a break in the surface called a vent. It includes all phenomena resulting from and causing magma within the crust or mantle of the body, to rise through the crust and form volcanic rocks on the surface. Magmas that reach the surface and solidify form extrusive landforms.
Partial meltingPartial melting is the phenomenon that occurs when a rock is subjected to temperatures high enough to cause certain minerals to melt, but not all of them. Partial melting is an important part of the formation of all igneous rocks and some metamorphic rocks (e.g., migmatites), as evidenced by a multitude of geochemical, geophysical and petrological studies. The parameters that influence partial melting include the composition of the source rock, the pressure and temperature of the environment, and the availability of water or other fluids.
Lava domeIn volcanology, a lava dome is a circular, mound-shaped protrusion resulting from the slow extrusion of viscous lava from a volcano. Dome-building eruptions are common, particularly in convergent plate boundary settings. Around 6% of eruptions on Earth are lava dome forming. The geochemistry of lava domes can vary from basalt (e.g. Semeru, 1946) to rhyolite (e.g. Chaiten, 2010) although the majority are of intermediate composition (such as Santiaguito, dacite-andesite, present day) The characteristic dome shape is attributed to high viscosity that prevents the lava from flowing very far.
Elliptic-curve cryptographyElliptic-curve cryptography (ECC) is an approach to public-key cryptography based on the algebraic structure of elliptic curves over finite fields. ECC allows smaller keys compared to non-EC cryptography (based on plain Galois fields) to provide equivalent security. Elliptic curves are applicable for key agreement, digital signatures, pseudo-random generators and other tasks. Indirectly, they can be used for encryption by combining the key agreement with a symmetric encryption scheme.
Lemniscate elliptic functionsIn mathematics, the lemniscate elliptic functions are elliptic functions related to the arc length of the lemniscate of Bernoulli. They were first studied by Giulio Fagnano in 1718 and later by Leonhard Euler and Carl Friedrich Gauss, among others. The lemniscate sine and lemniscate cosine functions, usually written with the symbols sl and cl (sometimes the symbols sinlem and coslem or sin lemn and cos lemn are used instead), are analogous to the trigonometric functions sine and cosine.
Continental collisionIn geology, continental collision is a phenomenon of plate tectonics that occurs at convergent boundaries. Continental collision is a variation on the fundamental process of subduction, whereby the subduction zone is destroyed, mountains produced, and two continents sutured together. Continental collision is only known to occur on Earth. Continental collision is not an instantaneous event, but may take several tens of millions of years before the faulting and folding caused by collisions stops.
Elliptic integralIn integral calculus, an elliptic integral is one of a number of related functions defined as the value of certain integrals, which were first studied by Giulio Fagnano and Leonhard Euler (1750). Their name originates from their originally arising in connection with the problem of finding the arc length of an ellipse. Modern mathematics defines an "elliptic integral" as any function f which can be expressed in the form where R is a rational function of its two arguments, P is a polynomial of degree 3 or 4 with no repeated roots, and c is a constant.
Elliptic partial differential equationSecond-order linear partial differential equations (PDEs) are classified as either elliptic, hyperbolic, or parabolic. Any second-order linear PDE in two variables can be written in the form where A, B, C, D, E, F, and G are functions of x and y and where , and similarly for . A PDE written in this form is elliptic if with this naming convention inspired by the equation for a planar ellipse.
Elliptic operatorIn the theory of partial differential equations, elliptic operators are differential operators that generalize the Laplace operator. They are defined by the condition that the coefficients of the highest-order derivatives be positive, which implies the key property that the principal symbol is invertible, or equivalently that there are no real characteristic directions. Elliptic operators are typical of potential theory, and they appear frequently in electrostatics and continuum mechanics.
ShaleShale is a fine-grained, clastic sedimentary rock formed from mud that is a mix of flakes of clay minerals (hydrous aluminium phyllosilicates, e.g. kaolin, Al2Si2O5(OH)4) and tiny fragments (silt-sized particles) of other minerals, especially quartz and calcite. Shale is characterized by its tendency to split into thin layers (laminae) less than one centimeter in thickness. This property is called fissility. Shale is the most common sedimentary rock.
