Bessel functions, first defined by the mathematician Daniel Bernoulli and then generalized by Friedrich Bessel, are canonical solutions y(x) of Bessel's differential equation for an arbitrary complex number , which represents the order of the Bessel function. Although and produce the same differential equation, it is conventional to define different Bessel functions for these two values in such a way that the Bessel functions are mostly smooth functions of .
In mathematical analysis, the Dirac delta function, also known as the unit impulse, is a generalized function on the real numbers, whose value is zero everywhere except at zero, and whose integral over the entire real line is equal to one. Since there is no function having this property, modelling the delta "function" rigorously involves the use of limits or, as is common in mathematics, measure theory and the theory of distributions.
In geometry, Heron's formula gives the area of a triangle in terms of the three side lengths Letting be the semiperimeter of the triangle, the area is
In differential geometry, the Laplace–Beltrami operator is a generalization of the Laplace operator to functions defined on submanifolds in Euclidean space and, even more generally, on Riemannian and pseudo-Riemannian manifolds. It is named after Pierre-Simon Laplace and Eugenio Beltrami.
A yield surface is a five-dimensional surface in the six-dimensional space of stresses. The yield surface is usually convex and the state of stress of inside the yield surface is elastic. When the stress state lies on the surface the material is said to have reached its yield point and the material is said to have become plastic. Further deformation of the material causes the stress state to remain on the yield surface, even though the shape and size of the surface may change as the plastic deformation evolves. This is because stress states that lie outside the yield surface are non-permissible in rate-independent plasticity, though not in some models of viscoplasticity.
Expected shortfall (ES) is a risk measure—a concept used in the field of financial risk measurement to evaluate the market risk or credit risk of a portfolio. The "expected shortfall at q% level" is the expected return on the portfolio in the worst of cases. ES is an alternative to value at risk that is more sensitive to the shape of the tail of the loss distribution.
Hudson's equation, also known as Hudson formula, is an equation used by coastal engineers to calculate the minimum size of riprap (armourstone) required to provide satisfactory stability characteristics for rubble structures such as breakwaters under attack from storm wave conditions.
In probability theory and statistics, the skew normal distribution is a continuous probability distribution that generalises the normal distribution to allow for non-zero skewness.
In hyperbolic geometry, the "law of cosines" is a pair of theorems relating the sides and angles of triangles on a hyperbolic plane, analogous to the planar law of cosines from plane trigonometry, or the spherical law of cosines in spherical trigonometry. It can also be related to the relativistic velocity addition formula.
In fluid dynamics, a cnoidal wave is a nonlinear and exact periodic wave solution of the Korteweg–de Vries equation. These solutions are in terms of the Jacobi elliptic function cn, which is why they are coined cnoidal waves. They are used to describe surface gravity waves of fairly long wavelength, as compared to the water depth.
An LC circuit can be quantized using the same methods as for the quantum harmonic oscillator. An LC circuit is a variety of resonant circuit, and consists of an inductor, represented by the letter L, and a capacitor, represented by the letter C. When connected together, an electric current can alternate between them at the circuit's resonant frequency:
The uncertainty theory invented by Baoding Liu is a branch of mathematics based on normality, monotonicity, self-duality, countable subadditivity, and product measure axioms.
In fluid dynamics, the Iribarren number or Iribarren parameter – also known as the surf similarity parameter and breaker parameter – is a dimensionless parameter used to model several effects of (breaking) surface gravity waves on beaches and coastal structures. The parameter is named after the Spanish engineer Ramón Iribarren Cavanilles (1900–1967), who introduced it to describe the occurrence of wave breaking on sloping beaches. The parameter used to describe breaking wave types on beaches; or wave run-up on – and reflection by – beaches, breakwaters and dikes.
In applied mathematics, oblate spheroidal wave functions are involved in the solution of the Helmholtz equation in oblate spheroidal coordinates. When solving this equation, , by the method of separation of variables, , with:
The Izbash formula is a mathematical expression used to calculate the stability of armourstone in flowing water environments.
Guderley–Landau–Stanyukovich problem describes the time evolution of converging shock waves. The problem was discussed by G. Guderley in 1942 and independently by Lev Landau and K. P. Stanyukovich in 1944, where the later authors' analysis was published in 1955.
Ramón Iribarren CavanillesIng.D was a Spanish civil engineer and professor of ports at the School of Civil Engineering in Madrid. He was chairman of the Spanish delegation to the Permanent International Association of Navigation Congresses (PIANC) and was elected as an academic at the Spanish Royal Academy of Sciences, although he did not take up the latter position. He made notable contributions in the field of coastal engineering, including methods for the calculation of breakwater stability and research which led to the development of the Iribarren number.
Wave overtopping is the time-averaged amount of water that is discharged per structure length by waves over a structure such as a breakwater, revetment or dike which has a crest height above still water level.
Wave run-up is the height to which waves run up the slope of a revetment, bank or dike, regardless of whether the waves are breaking or not. Conversely, wave run-down is the height to which waves recede. These heights are always measured vertically. The wave run-up height, denoted by , , or , is a very important parameter in coastal engineering as, together with the design highest still water level, it determines the required crest height of a dike or revetment.
