In the

mathematical Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...

field of

differential equation In mathematics, a differential equation is an equation that relates one or more unknown functions and their derivatives. In applications, the functions generally represent physical quantities, the derivatives represent their rates of change, an ...

s, the ultrahyperbolic equation is a

partial differential equation In mathematics, a partial differential equation (PDE) is an equation which imposes relations between the various partial derivatives of a Multivariable calculus, multivariable function. The function is often thought of as an "unknown" to be sol ...

(PDE) for an unknown

scalar function In mathematics and physics, a scalar field is a function associating a single number to every point in a space – possibly physical space. The scalar may either be a pure mathematical number (dimensionless) or a scalar physical quantity (wi ...

of variables of the form

\frac + \cdots + \frac - \frac - \cdots - \frac = 0.

More generally, if is any

quadratic form In mathematics, a quadratic form is a polynomial with terms all of degree two ("form" is another name for a homogeneous polynomial). For example, :4x^2 + 2xy - 3y^2 is a quadratic form in the variables and . The coefficients usually belong to a ...

in variables with

signature A signature (; from la, signare, "to sign") is a handwritten (and often stylized) depiction of someone's name, nickname, or even a simple "X" or other mark that a person writes on documents as a proof of identity and intent. The writer of a ...

, then any PDE whose

principal part In mathematics, the principal part has several independent meanings, but usually refers to the negative-power portion of the Laurent series of a function. Laurent series definition The principal part at z=a of a function : f(z) = \sum_^\infty a_k ...

a_u_

is said to be ultrahyperbolic. Any such equation can be put in the form above by means of a change of variables.See Courant and Hilbert. The ultrahyperbolic equation has been studied from a number of viewpoints. On the one hand, it resembles the classical

wave equation The (two-way) wave equation is a second-order linear partial differential equation for the description of waves or standing wave fields — as they occur in classical physics — such as mechanical waves (e.g. water waves, sound waves and s ...

. This has led to a number of developments concerning its characteristics, one of which is due to

Fritz John Fritz John (14 June 1910 – 10 February 1994) was a German-born mathematician specialising in partial differential equations and ill-posed problems. His early work was on the Radon transform and he is remembered for John's equation. He was a 1 ...

: the

John equation John's equation is an ultrahyperbolic partial differential equation satisfied by the X-ray transform of a function. It is named after Fritz John. Given a function f\colon\mathbb^n \rightarrow \mathbb with compact support the ''X-ray transform'' i ...

. In 2008, Walter Craig and Steven Weinstein proved that under a nonlocal constraint, the initial value problem is well-posed for initial data given on a codimension-one

hypersurface In geometry, a hypersurface is a generalization of the concepts of hyperplane, plane curve, and surface. A hypersurface is a manifold or an algebraic variety of dimension , which is embedded in an ambient space of dimension , generally a Euclidean ...

. And later, in 2022, a research team at the

University of Michigan , mottoeng = "Arts, Knowledge, Truth" , former_names = Catholepistemiad, or University of Michigania (1817–1821) , budget = $10.3 billion (2021) , endowment = $17 billion (2021)As o ...

extended the conditions for solving ultrahyperbolic wave equations to complex-time (kime), demonstrated space-kime dynamics, and showed data science applications using tensor-based linear modeling of

functional magnetic resonance imaging Functional magnetic resonance imaging or functional MRI (fMRI) measures brain activity by detecting changes associated with blood flow. This technique relies on the fact that cerebral blood flow and neuronal activation are coupled. When an area o ...

data. The equation has also been studied from the point of view of

symmetric space In mathematics, a symmetric space is a Riemannian manifold (or more generally, a pseudo-Riemannian manifold) whose group of symmetries contains an inversion symmetry about every point. This can be studied with the tools of Riemannian geometry, l ...

s, and

elliptic differential operator In 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 i ...

s. In particular, the ultrahyperbolic equation satisfies an analog of the

mean value theorem In mathematics, the mean value theorem (or Lagrange theorem) states, roughly, that for a given planar arc between two endpoints, there is at least one point at which the tangent to the arc is parallel to the secant through its endpoints. It i ...

for

harmonic functions In mathematics, mathematical physics and the theory of stochastic processes, a harmonic function is a twice continuously differentiable function f: U \to \mathbb R, where is an open subset of that satisfies Laplace's equation, that is, : \f ...

Notes

References

* * * * * Differential operators {{mathanalysis-stub