Curvature of Riemannian manifoldsIn mathematics, specifically differential geometry, the infinitesimal geometry of Riemannian manifolds with dimension greater than 2 is too complicated to be described by a single number at a given point. Riemann introduced an abstract and rigorous way to define curvature for these manifolds, now known as the Riemann curvature tensor. Similar notions have found applications everywhere in differential geometry of surfaces and other objects. The curvature of a pseudo-Riemannian manifold can be expressed in the same way with only slight modifications.
ProjectileA projectile is an object that is propelled by the application of an external force and then moves freely under the influence of gravity and air resistance. Although any objects in motion through space are projectiles, they are commonly found in warfare and sports (for example, a thrown baseball, kicked football, fired bullet, shot arrow, stone released from catapult). In ballistics mathematical equations of motion are used to analyze projectile trajectories through launch, flight, and impact.
Riemann curvature tensorIn the mathematical field of differential geometry, the Riemann curvature tensor or Riemann–Christoffel tensor (after Bernhard Riemann and Elwin Bruno Christoffel) is the most common way used to express the curvature of Riemannian manifolds. It assigns a tensor to each point of a Riemannian manifold (i.e., it is a tensor field). It is a local invariant of Riemannian metrics which measures the failure of the second covariant derivatives to commute. A Riemannian manifold has zero curvature if and only if it is flat, i.
Stochastic gradient descentStochastic gradient descent (often abbreviated SGD) is an iterative method for optimizing an objective function with suitable smoothness properties (e.g. differentiable or subdifferentiable). It can be regarded as a stochastic approximation of gradient descent optimization, since it replaces the actual gradient (calculated from the entire data set) by an estimate thereof (calculated from a randomly selected subset of the data).
Tax incidenceIn economics, tax incidence or tax burden is the effect of a particular tax on the distribution of economic welfare. Economists distinguish between the entities who ultimately bear the tax burden and those on whom the tax is initially imposed. The tax burden measures the true economic effect of the tax, measured by the difference between real incomes or utilities before and after imposing the tax, and taking into account how the tax causes prices to change.
EarthEarth is the third planet from the Sun and the only astronomical object known to harbor life. This is enabled by Earth being a water world, the only one in the Solar System sustaining liquid surface water. Almost all of Earth's water is contained in its global ocean, covering 70.8% of Earth's surface. The remaining 29.2% of Earth's surface is land, most of which is located in the form of continental landmasses within one hemisphere, Earth's land hemisphere.
Earth massAn Earth mass (denoted as or , where ⊕ is the standard astronomical symbol for Earth), is a unit of mass equal to the mass of the planet Earth. The current best estimate for the mass of Earth is , with a relative uncertainty of 10−4. It is equivalent to an average density of 5,515kg/m3. Using the nearest metric prefix, the Earth mass is approximately six ronnagrams, or 6.0 Rg. The Earth mass is a standard unit of mass in astronomy that is used to indicate the masses of other planets, including rocky terrestrial planets and exoplanets.
Radius of curvatureIn differential geometry, the radius of curvature (Rc), R, is the reciprocal of the curvature. For a curve, it equals the radius of the circular arc which best approximates the curve at that point. For surfaces, the radius of curvature is the radius of a circle that best fits a normal section or combinations thereof. In the case of a space curve, the radius of curvature is the length of the curvature vector. In the case of a plane curve, then R is the absolute value of where s is the arc length from a fixed point on the curve, φ is the tangential angle and κ is the curvature.
Gradient descentIn mathematics, gradient descent (also often called steepest descent) is a iterative optimization algorithm for finding a local minimum of a differentiable function. The idea is to take repeated steps in the opposite direction of the gradient (or approximate gradient) of the function at the current point, because this is the direction of steepest descent. Conversely, stepping in the direction of the gradient will lead to a local maximum of that function; the procedure is then known as gradient ascent.
Shell (projectile)A shell, in a military context, is a projectile whose payload contains an explosive, incendiary, or other chemical filling. Originally it was called a bombshell, but "shell" has come to be unambiguous in a military context. Modern usage sometimes includes large solid kinetic projectiles, which are more properly termed shot. Solid shot may contain a pyrotechnic compound if a tracer or spotting charge is used.
TurbineA turbine ('tɜːrbaɪn or 'tɜːrbɪn) (from the Greek τύρβη, tyrbē, or Latin turbo, meaning vortex) is a rotary mechanical device that extracts energy from a fluid flow and converts it into useful work. The work produced can be used for generating electrical power when combined with a generator. A turbine is a turbomachine with at least one moving part called a rotor assembly, which is a shaft or drum with blades attached. Moving fluid acts on the blades so that they move and impart rotational energy to the rotor.
Launch escape systemA launch escape system (LES) or launch abort system (LAS) is a crew-safety system connected to a space capsule. It is used in the event of a critical emergency to quickly separate the capsule from its launch vehicle in case of an emergency requiring the abort of the launch, such as an impending explosion. The LES is typically controlled by a combination of automatic rocket failure detection, and a manual activation for the crew commander's use. The LES may be used while the launch vehicle is still on the launch pad, or during its ascent.
