The boundary of this set is the Cauchy horizon.

If the Cauchy surface were noncompact, then the image has a boundary.

If the Cauchy surface were compact, i. e. space is compact, the null geodesic generators of the boundary can intersect everywhere because they can intersect on the other side of space.

For example, there is the Cauchy boundary condition, or the mixed boundary condition which is a combination of the Dirichlet and Neumann conditions.

* Cauchy boundary condition

If the boundary has the form of a curve or surface that gives a value to the normal derivative and the problem itself then it is a Cauchy boundary condition.

* Cauchy boundary condition

More commonly, however, the Hilbert transform refers to a convolution with the Cauchy kernel, for functions defined on the real line R ( the boundary of the upper half-plane ).

But attempts to incorporate quantum effects into general relativity using semiclassical gravity seem to make it plausible that vacuum fluctuations would drive the energy density on the boundary of the time machine ( the Cauchy horizon of the region where closed timelike curves become possible ) to infinity, destroying the time machine at the instant it was created or at least preventing anyone outside it from entering it.

In physics, a Cauchy horizon is a light-like boundary of the domain of validity of a Cauchy problem ( a particular boundary value problem of the theory of partial differential equations ).

For example, there is the Cauchy boundary condition, or the mixed boundary condition which is a combination of the Neumann and Dirichlet conditions.

* Cauchy boundary condition

In mathematics, a Cauchy boundary condition () imposed on an ordinary differential equation or a partial differential equation specifies both the values a solution of a differential equation is to take on the boundary of the domain and the normal derivative at the boundary.

Cauchy boundary conditions can be understood from the theory of second order, ordinary differential equations, where to have a particular solution one has to specify the value of the function and the value of the derivative at a given initial or boundary point, i. e.,

It is sometimes said that Cauchy boundary conditions are a weighted average of imposing Dirichlet boundary conditions and Neumann boundary conditions.

That is, the Cauchy – Riemann equations are the conditions for a function to be conformal.

This only applies to initial conditions which are outside of the chronology-violating region of spacetime, which is bounded by a Cauchy horizon.

Even if self-consistent extensions can be found for arbitrary initial conditions outside the Cauchy Horizon, the finding that there can be multiple distinct self-consistent extensions for the same initial condition — indeed, Echeverria et al.

When it comes to asking for all solutions, it may be the case that conditions from mathematical analysis should be applied ; for example, in the case of the Cauchy equation mentioned above, the solutions that are continuous functions are the ' reasonable ' ones, while other solutions that are not likely to have practical application can be constructed ( by using a Hamel basis for the real numbers as vector space over the rational numbers ).

In mathematics, in the study of differential equations, the Picard – Lindelöf theorem, Picard's existence theorem or Cauchy – Lipschitz theorem is an important theorem on existence and uniqueness of solutions to first-order equations with given initial conditions.

The above properties mean, more precisely, that Laplace's equation for u and Laplace's equation for v are the integrability conditions for solving the Cauchy – Riemann equations.

where σ < sub > ij </ sub > are the Cauchy stresses, r is the distance from the crack tip, θ is the angle with respect to the plane of the crack, and f < sub > ij </ sub > are functions that are independent of the crack geometry and loading conditions.

This should not be confused with statistical objects such as the weighted mean, the weighted geometric mean or the weighted harmonic mean, since no such formulas are used upon imposing Cauchy boundary conditions.

Since the parameter is usually time, Cauchy conditions can also be called initial value conditions or initial value data or simply Cauchy data.

Notice that although Cauchy boundary conditions imply having both Dirichlet and Neumann boundary conditions, this is not the same at all as having Robin or impedance boundary condition.

Suppose that the temperature is held at zero on the curved portion of the boundary, while the straight portion of the boundary is insulated, i. e., we define the Cauchy boundary conditions as

A Cauchy problem in mathematics asks for the solution of a partial differential equation that satisfies certain conditions which are given on a hypersurface in the domain.

The Cauchy – Kowalevski theorem says that Cauchy problems have unique solutions under certain conditions, the most important of which being that the Cauchy data and the coefficients of the partial differential equation be real analytic functions.

The utility of Cauchy sequences lies in the fact that in a complete metric space ( one where all such sequences are known to converge to a limit ), the criterion for convergence depends only on the terms of the sequence itself.

The customary acceptance of the fact that any real number x has a decimal expansion is an implicit acknowledgment that a particular Cauchy sequence of rational numbers ( whose terms are the successive truncations of the decimal expansion of x ) has the real limit x.

When U and V are two independent normally distributed random variables with expected value 0 and variance 1, then the ratio U / V has the standard Cauchy distribution.

If are independent and identically distributed random variables, each with a standard Cauchy distribution, then the sample mean has the same standard Cauchy distribution ( the sample median, which is not affected by extreme values, can be used as a measure of central tendency ).

But in the case of the Cauchy distribution, both the positive and negative terms of ( 2 ) are infinite.

Because the mean and variance of the Cauchy distribution are not defined, attempts to estimate these parameters will not be successful.

For example, if n samples are taken from a Cauchy distribution, one may calculate the sample mean as:

The Cauchy – Riemann equations on a pair of real-valued functions of two real variables u ( x, y ) and v ( x, y ) are the two equations:

The reason why Euler and some other authors relate the Cauchy – Riemann equations with analyticity is that a major theorem in complex analysis says that holomorphic functions are analytic and viceversa.

which are the Cauchy – Riemann equations ( 2 ) at the point z < sub > 0 </ sub >.

In other words, if u and v are real-differentiable functions of two real variables, obviously u + iv is a ( complex-valued ) real-differentiable function, but u + iv is complex-differentiable if and only if the Cauchy – Riemann equations hold.

There are Cauchy – Riemann equations, appropriately generalized, in the theory of several complex variables.

Viewed as conjugate harmonic functions, the Cauchy – Riemann equations are a simple example of a Bäcklund transform.

Nevertheless, equivalence classes of Cauchy sequences are defined as above, and the set of equivalence classes is easily shown to be a field that has the rational numbers as a subfield.

This definition corresponds to the classical definition using Cauchy sequences, except with a constructive twist: for a classical Cauchy sequence, it is required that, for any given distance, there exists ( in a classical sense ) a member in the sequence after which all members are closer together than that distance.

However, such theories in general do not have a well-defined Cauchy problem ( for reasons related to the issues of causality discussed above ), and are probably inconsistent quantum mechanically.

The Cauchy – Schwarz inequality proves that this definition is sensible, by showing that the right hand side lies in the interval, and justifies the notion that ( real ) Hilbert spaces are simply generalizations of the Euclidean space.

Various generalizations of the Cauchy – Schwarz inequality exist in the context of operator theory, e. g. for operator-convex functions, and operator algebras, where the domain and / or range of φ are replaced by a C *- algebra or W *- algebra.

Without using countable choice, it is not possible to constructively prove the fundamental theorem of algebra for complex numbers based on the Dedekind real numbers ( which are not constructively equivalent to the Cauchy real numbers without countable choice < ref > For the minimum necessary to prove their equivalence, see Bridges, Schuster, and Richman ; 1998 ; < cite > A weak countable choice principle </ cite >; available from .</ ref >).

The Cauchy – Kowalevski theorem states that the Cauchy problem for any partial differential equation whose coefficients are analytic in the unknown function and its derivatives, has a locally unique analytic solution.