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A secant line is a straight line that intersects a curve at two distinct points. In this specific problem, the secant line is the straight line that passes through the two points on the parabola $y = x^2 - 4$ located at the specified x-values $x = -1$ and $x = 2$.

Conditions: The curve is a quadratic function, specifically $y = x^2 - 4$.; The line must intersect the curve at exactly two distinct points.; The x-values of the intersection points are given as $x = -1$ and $x = 2$.

Prevalence determines the size of the diseased and healthy subgroups in the population. It acts as the prior probability that influences the final posterior probability of disease given a positive test.

Conditions: The model uses a fictional population.; Prevalence is set to $1\%$.

Once the new remainder is 0, the division is exact, meaning the current divisor perfectly divides the previous dividend. The algorithm's termination rule states that the greatest common divisor of the original pair is the last nonzero remainder, which is the divisor of this final exact division.

Conditions: The Euclidean algorithm has been applied to two positive integers.; A division step has produced a remainder of 0.; The inputs are 10 and 45.

Edge B-E is skipped because both of its endpoints (B and E) are already in the visited set (MST). Prim's algorithm only considers edges that connect a visited vertex to an unvisited vertex to avoid creating cycles.

Conditions: Both endpoints of the edge are already marked as visited.; The algorithm strictly follows the rule of adding only crossing edges.

The locality condition means that each data point influences the interpolated surface only up to a certain distance. Equivalently, the value at a given point will be similar to the values at nearby points, and distant points do not continue to influence one another.

Conditions: Presented as a basic condition fulfilled by almost all interpolation methods.; Applies to local interpolation methods.

A block cipher is a type of cipher that groups bits into blocks of plaintext before applying the encryption, adding complexity by operating on these blocks rather than individual characters.

Conditions: The cipher operates on grouped bits or blocks of plaintext.; The encryption is applied to the blocks.

For the arc length formula $\int_{a}^{b} \sqrt{1 + (f'(x))^2} dx$ to be valid, the function $f(x)$ must be continuous on the closed interval $[a, b]$, and its derivative $f'(x)$ must also be continuous on $[a, b]$. These conditions ensure that the curve is smooth enough for the integral to accurately represent its length.

Conditions: The curve is defined by a function $y = f(x)$.; The interval of integration is $[a, b]$.; $f(x)$ is continuous on $[a, b]$.; $f'(x)$ is continuous on $[a, b]$.

Geometrically, the determinant of $a 2\times 2$ matrix represents the factor by which the linear transformation scales areas. Specifically, it is the signed area of the parallelogram formed by the matrix's column vectors.

Conditions: The matrix is $2\times 2$.; The transformation is linear.; Use ordinary Euclidean area in standard orthonormal coordinates.

The formula means that the state of a cell at the current time step $t$ is computed by applying a transition function $f$ to the states of its neighboring cells at the previous time step $t-1$.

Conditions: $t$ represents the current discrete time step or generation index.; $t-1$ represents the previous discrete time step.; $f$ is the transition rule mapping previous neighborhood states to the next cell state.

Construct a matrix $P$ whose columns are the alternative basis vectors expressed in standard coordinates. Multiplying $P$ by a coordinate vector in the alternative basis yields the standard coordinates of that vector.

Conditions: The alternative basis vectors are known in standard coordinates.; The input vector is given as coordinates in the alternative basis.

A geodesic is treated as a curve in space-time and is written componentwise as functions of a parameter $\lambda$. Every space-time component of the curve can be expressed as $x^\mu = x^\mu(\lambda)$, which converts the geometric object into coordinate functions suitable for differentiation.

Conditions: $\lambda$ is the chosen curve parameter.; Each space-time component $x^\mu$ is expressed as a function of $\lambda$.

The proof demonstrates this by showing that d is a linear combination of a and b. Since any common divisor divides both a and b, it must also divide any integer linear combination of them, including d.

Conditions: d is the least positive linear combination of a and b.; c is a common divisor of a and b.