Hole

A hole refers to a compact subset of the complex plane with a point excluded. In the context of the variance structures occuring in geostatistics, a variogram can exhibit a pattern known as the hole-effect where spatial cyclicity of the generating process contributes to a characteristic wave-form in the variogram; or in physics applications where the exchange (correlation) hole is related to the way electrons influence each others orbits.

To define it mathematically, let's consider the following:

Let $D$ be a compact subset of the complex plane $\mathbb{C}$, and let $z_0$ be a point that is contained in $D$. The hole, denoted as $H(D, z_0)$, is defined as the set obtained by removing the point $z_0$ from $D$. Mathematically, we can express it as:

$$H(D, z_0) = D \setminus {z_0}$$

where $\setminus$ denotes set subtraction.

To provide a more detailed mathematical description, we can express $D$ and $z_0$ using their respective mathematical representations:

1. Compact subset $D$ in the complex plane $\mathbb{C}$:

$$D = \{z \in \mathbb{C} : f(z) \leq 0\}$$

Here, $f(z)$ is a continuous function defined on $\mathbb{C}$, and the set $D$ represents the region where $f(z)$ is less than or equal to zero.

2. Point $z_0$ contained in $D$:

$$z_0 = a + bi$$

Here, $a$ and $b$ are real numbers representing the real and imaginary parts of $z_0$, respectively.

Combining these representations, we can express the hole $H(D, z_0)$ in terms of the function $f(z)$ and the point $z_0$:

$$H(D, z_0) = \{z \in \mathbb{C} \setminus \{z_0\}: f(z) \leq 0\}$$

This defines the hole in Fredholm theory as a compact subset of the complex plane with a point removed, represented using mathematical formulas and set notation.