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Import `StandardScaler` from `sklearn.preprocessing`, instantiate it, and call `fit_transform` on the selected DataFrame columns. This transforms the data to have mean 0 and standard deviation 1.

Conditions: Input data is numerical.; Scikit-learn library is installed.

The notation $\nabla^2$ (nabla squared) is mathematically equivalent to $\Delta$ (delta) when representing the Laplacian operator. Both symbols denote the sum of the unmixed second partial derivatives of a scalar field.

Conditions: The operator acts on a scalar field $u$.; The context is vector calculus or PDEs.; $\nabla$ is the del/nabla operator.; $\Delta$ is the Laplace operator.

The geometric distribution describes the number of failures before the first success, provided the trials are independent and the success probability remains fixed.

Conditions: Trials are independent.; Success probability is fixed.; The convention counts the number of failures before the first success.

The limit definition of a derivative is the finite real limit of the difference quotient as the nonzero increment tends to zero. At an interior point where this limit exists, it represents the derivative and the tangent slope.

Conditions: A finite real derivative limit must exist at the point.; The point is interior and $Δx\ne 0$ while taking the limit.

The Pareto front is the curve that links all non-dominated solutions in the objective-value plot. Visually, it is the lower-left boundary of the attainable set in this minimization problem, representing the best possible trade-offs between the conflicting objectives.

Conditions: Objectives are plotted in the (OBJ_1, OBJ_2) plane.; The example uses minimization of both objectives.

The least common multiple (LCM) of two positive integers is equal to their product if and only if the two integers are coprime, meaning their greatest common divisor (GCD) is 1.

Conditions: The two integers are positive.; The two integers are coprime (gcd(a,b) = 1).

Alpha is defined in the video as the exponential smoothing constant. It multiplies the previous forecast error and determines the size of the adjustment added to the previous forecast.

Conditions: Treated as a given constant in the problem.; Held fixed across all periods in the example.

If two congruences x ≡ r (mod m) and x ≡ r (mod n) share the same remainder r, then $x = r$ is an immediate particular solution. The full combined solution set is then r plus multiples of the LCM of the two moduli: $x = r + k$ * LCM(m, n).

Conditions: The two congruences have the identical remainder r.; The moduli m and n are positive integers.

To verify $a \equiv b \pmod{n}$, divide $a$ by the modulus $n$ using long division and check if the remainder is $b$. If the remainder matches $b$, the congruence is valid.

Conditions: The method is demonstrated for positive dividends and positive moduli.; The speaker calls it a colloquial explanation.

To find the magnitude of a vector from its components, you calculate the square root of the sum of the squares of those components. This process effectively measures the Euclidean length of the vector in the coordinate plane.

Conditions: The vector is defined by its components in a 2D Cartesian coordinate system.; The coordinate system is orthonormal.

Mutual information $I(X;Y)$ represents the reduction in uncertainty about $X$ after observing $Y$. $H(X)$ is the initial uncertainty about $X$, and $H(X|Y)$ is the remaining uncertainty after $Y$ is known.

Conditions: $X$ and $Y$ are random variables.; $H(X)$ and $H(X|Y)$ use the same unit system.

You can skip the parallel axis theorem when calculating the area moment of inertia directly about a coordinate axis (such as the x or y-axis) used in the integral setup. This applies specifically when there is no perpendicular offset distance between the axis of integration and the axis of interest, rendering the shift term zero.

Conditions: The requested axis is one of the coordinate axes (e.g., the x-axis or y-axis).; The integral is set up directly about that same axis.; No separate shifted-axis correction is being applied in the setup.