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The key insight is that instead of treating each sample independently, dependent sampling uses the information from the previous sample to determine the next one. This creates a chain of information that flows from sample to sample, allowing the sampler to intelligently explore high-probability regions.

Conditions: Each new sample is drawn from a probability distribution conditioned on the previous sample.; The formula is $X_{n+1} \sim P(\cdot|X_n)$.; Applies to complex probability distributions with multiple peaks and weird shapes.

A jump discontinuity is identified when the left-hand limit and the right-hand limit at a point exist but are not equal. Because the one-sided limits differ, the two-sided limit does not exist, preventing the function from being continuous at that point.

Conditions: The function is defined on both sides of the point.; The left-hand limit and right-hand limit both exist as finite numbers.; The left-hand limit does not equal the right-hand limit.

The example proves the sequence is non-increasing. The mathematical derivation shows $a_{n+1}-a_n \le 0$, which implies $a_{n+1} \le a_n$.

Conditions: The sequence is defined by $a_{n+1}=\frac{1}{3}(a_n+1)$ with $a_1=1$.; The lower bound $a_n \ge \frac{1}{2}$ has been proven.

In the displayed Euclidean algorithm, $a$ and $b$ are the two initial natural numbers whose gcd is being found. $q_i$ represents the quotient at the $i$-th division step.

Conditions: The symbols are from the general statement of the Euclidean algorithm on the left board.

To prove that the set $W = \{(x_1, x_2, x_3) \in \mathbb{R}^3 \mid x_3 = x_1 + x_2\}$ is a subspace, you must verify three conditions: nonemptiness (containing the zero vector), closure under vector addition, and closure under scalar multiplication.

Conditions: The set is defined by the linear equation $x_3 = x_1 + x_2$ in $\mathbb{R}^3$.; The field of scalars is $\mathbb{R}$.

The span is considered the smallest containing subspace because every other subspace containing those vectors must also contain their span. Since the span consists of all linear combinations, any subspace closed under addition and scalar multiplication that includes the original vectors must include all their combinations.

Conditions: The collection of vectors is fixed.; We are comparing subspaces that contain this collection.

The analytic hierarchy process (AHP) is used to determine the weights of various indicators. The paper claims this method has a scientific basis and reduces the influence of subjective factors.

Conditions: Applied to determining weights of various indicators.; Presented as part of the comprehensive evaluation framework.

We set the two expected payoffs equal to enforce the indifference principle. In a mixed strategy Nash equilibrium, a player randomizes over their pure strategies only if they are completely indifferent between them.

Conditions: The game is in a mixed-strategy Nash equilibrium.; The player is randomizing over multiple pure strategies.; The expected payoffs for the player's pure strategies are calculated based on the opponent's mixed strategy.

Reversing the operands reverses the sign of the resulting vector. Specifically, $\mathbf{a} \times \mathbf{b} = -\mathbf{b} \times \mathbf{a}$.

Conditions: You are computing the cross product of two vectors.; You swap the order of the vectors.

The law of large numbers states that for independent repeated trials with a common success probability $p$, the observed success proportion approaches $p$ as the number of trials grows. This is a long-run tendency specified by the law, meaning that while individual short-term outcomes may vary, the cumulative average stabilizes around the theoretical value over many repetitions.

Conditions: The trials are independent.; Each trial shares a common success probability $p$.; The number of trials grows large.

The three essential ingredients are the decision variable, the real-valued objective, and the constraints. The decision variable represents what can vary, the objective defines the cost to minimize or reward to maximize, and the constraints specify the feasible choices.

Conditions: The problem involves making a choice to optimize a metric.; The objective is real-valued.; Constraints are defined as equalities or inequalities.

Yes, the dot product distributes over vector addition. This means $\mathbf{a} \cdot (\mathbf{b} + \mathbf{c}) = \mathbf{a} \cdot \mathbf{b} + \mathbf{a} \cdot \mathbf{c}$.

Conditions: The vectors are in a real inner product space.; The operation is the standard dot product.