Euler's formula, named after Leonhard Euler, is a mathematical formula in complex analysis that establishes the fundamental relationship between the trigonometric functions and the complex exponential function. Euler's formula states that, for any real number x, one has where e is the base of the natural logarithm, i is the imaginary unit, and cos and sin are the trigonometric functions cosine and sine respectively. This complex exponential function is sometimes denoted cis x. The formula is still valid if x is a complex number, and is also called Euler's formula in this more general case.
In classical mechanics, a harmonic oscillator is a system that, when displaced from its equilibrium position, experiences a restoring force F proportional to the displacement x: where k is a positive constant.
In mathematics, the trigonometric functions are real functions which relate an angle of a right-angled triangle to ratios of two side lengths. They are widely used in all sciences that are related to geometry, such as navigation, solid mechanics, celestial mechanics, geodesy, and many others. They are among the simplest periodic functions, and as such are also widely used for studying periodic phenomena through Fourier analysis.
In electrodynamics, elliptical polarization is the polarization of electromagnetic radiation such that the tip of the electric field vector describes an ellipse in any fixed plane intersecting, and normal to, the direction of propagation. An elliptically polarized wave may be resolved into two linearly polarized waves in phase quadrature, with their polarization planes at right angles to each other. Since the electric field can rotate clockwise or counterclockwise as it propagates, elliptically polarized waves exhibit chirality.
In mathematics, theta functions are special functions of several complex variables. They show up in many topics, including Abelian varieties, moduli spaces, quadratic forms, and solitons. Theta functions are parametrized by points in a tube domain inside a complex Lagrangian Grassmannian, namely the Siegel upper half space.
In telecommunications, particularly in radio frequency engineering, signal strength refers to the transmitter power output as received by a reference antenna at a distance from the transmitting antenna. High-powered transmissions, such as those used in broadcasting, are expressed in dB-millivolts per metre (dBmV/m). For very low-power systems, such as mobile phones, signal strength is usually expressed in dB-microvolts per metre (dBμV/m) or in decibels above a reference level of one milliwatt (dBm). In broadcasting terminology, 1 mV/m is 1000 μV/m or 60 dBμ.
In geometry, a disdyakis triacontahedron, hexakis icosahedron, decakis dodecahedron, kisrhombic triacontahedron or d120 is a Catalan solid with 120 faces and the dual to the Archimedean truncated icosidodecahedron. As such it is face-uniform but with irregular face polygons. It slightly resembles an inflated rhombic triacontahedron: if one replaces each face of the rhombic triacontahedron with a single vertex and four triangles in a regular fashion, one ends up with a disdyakis triacontahedron. That is, the disdyakis triacontahedron is the Kleetope of the rhombic triacontahedron. It is also the barycentric subdivision of the regular dodecahedron and icosahedron. It has the most faces among the Archimedean and Catalan solids, with the snub dodecahedron, with 92 faces, in second place.
Projectile motion is a form of motion experienced by an object or particle that is projected in a gravitational field, such as from Earth's surface, and moves along a curved path under the action of gravity only. In the particular case of projectile motion on Earth, most calculations assume the effects of air resistance are passive.
Mohr–Coulomb theory is a mathematical model describing the response of brittle materials such as concrete, or rubble piles, to shear stress as well as normal stress. Most of the classical engineering materials follow this rule in at least a portion of their shear failure envelope. Generally the theory applies to materials for which the compressive strength far exceeds the tensile strength.
In geometry, a rhombohedron is a special case of a parallelepiped in which all six faces are congruent rhombi. It can be used to define the rhombohedral lattice system, a honeycomb with rhombohedral cells. A rhombohedron has two opposite apices at which all face angles are equal; a prolate rhombohedron has this common angle acute, and an oblate rhombohedron has an obtuse angle at these vertices. A cube is a special case of a rhombohedron with all sides square.
A photon sphere or photon circle arises in a neighbourhood of the event horizon of a black hole where gravity is so strong that emitted photons will not just bend around the black hole but also return to the point where they were emitted from and consequently display boomerang-like properties. As the source emitting photons falls into the gravitational field towards the event horizon the shape of the trajectory of each boomerang photon changes, tending to a more circular form. At a critical value of the radial distance from the singularity the trajectory of a boomerang photon will take the form of a non-stable circular orbit, thus forming a photon circle and hence in aggregation a photon sphere. The circular photon orbit is said to be the last photon orbit. The radius of the photon sphere, which is also the lower bound for any stable orbit, is, for a Schwarzschild black hole,
In continuum mechanics, a material is said to be under plane stress if the stress vector is zero across a particular plane. When that situation occurs over an entire element of a structure, as is often the case for thin plates, the stress analysis is considerably simplified, as the stress state can be represented by a tensor of dimension 2. A related notion, plane strain, is often applicable to very thick members.
This uniform polyhedron compound is a symmetric arrangement of 6 cubes, considered as square prisms. It can be constructed by superimposing six identical cubes, and then rotating them in pairs about the three axes that pass through the centres of two opposite cubic faces. Each cube is rotated by an equal angle θ.
The compound of eight octahedra with rotational freedom is a uniform polyhedron compound. It is composed of a symmetric arrangement of 8 octahedra, considered as triangular antiprisms. It can be constructed by superimposing eight identical octahedra, and then rotating them in pairs about the four axes that pass through the centres of two opposite octahedral faces. Each octahedron is rotated by an equal angle θ.
The compound of four octahedra with rotational freedom is a uniform polyhedron compound. It consists in a symmetric arrangement of 4 octahedra, considered as triangular antiprisms. It can be constructed by superimposing four identical octahedra, and then rotating each by an equal angle θ about a separate axis passing through the centres of two opposite octahedral faces, in such a way as to preserve pyritohedral symmetry.
This uniform polyhedron compound is a symmetric arrangement of 12 pentagonal antiprisms. It can be constructed by inscribing one pair of pentagonal antiprisms within an icosahedron, in each of the six possible ways, and then rotating each by an equal and opposite angle θ.
In Einstein's theory of general relativity, the interior Schwarzschild metric is an exact solution for the gravitational field in the interior of a non-rotating spherical body which consists of an incompressible fluid and has zero pressure at the surface. This is a static solution, meaning that it does not change over time. It was discovered by Karl Schwarzschild in 1916, who earlier had found the exterior Schwarzschild metric.
Bimaximal mixing refers to a proposed form of the lepton mixing matrix. It is characterized by the neutrino being a bimaximal mixture of and and being completely decoupled from the , i.e. a uniform mixture of and . The is consequently a uniform mixture of and . Other notable properties are the symmetries between the and flavours and and mass eigenstates and an absence of CP violation. The moduli squared of the matrix elements have to be:
