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The final particular solution is $X_p(t) = 2te^{-t}$. This is obtained by substituting the solved coefficient $A=2$ back into the resonant trial solution $x=Ate^{-t}$.

Conditions: The ODE is $\ddot{x}+3\dot{x}+2x=2e^{-t}$.; The coefficient $A$ has been determined to be $2$.; The trial solution form is $x=Ate^{-t}$.

The Euclidean algorithm is a method for finding the greatest common divisor (gcd) of two natural numbers. It is set up by repeatedly applying the division algorithm.

Conditions: The inputs $a$ and $b$ are natural numbers.; The division algorithm is used at each step.

The example demonstrates that the chosen open cover $C$ of the finite space $X$ has a finite subcover $C_1$. This illustrates the core property used in the definition of compactness: every open cover admits a finite subcover.

Conditions: $X$ is the space under consideration.; $C$ is an open cover of $X$.; $C_1$ is a finite subcollection of $C$ covering $X$.

The object returned by the kmeans function contains a cluster assignment for each data point and the coordinate location of the center of each cluster.

A linear isometry is a map $F: V \to W$ between two inner product spaces that is both a vector space isomorphism and preserves inner products. This means it maintains the geometric structure defined by the inner product.

Conditions: $V$ and $W$ are inner product spaces.; $F$ is a vector space isomorphism.; $F$ preserves inner products for all $a, b \in V$.

The Greedy Algorithm fails to find the optimal solution for the Traveling Salesman Problem because it makes locally optimal choices at each step—always moving to the nearest unvisited city—without considering the global structure of the route. This often results in a sub-optimal route that crosses over itself.

Conditions: The algorithm starts at any city.; It repeatedly moves to the closest city that has not yet been visited.; It returns to the starting city after all others are visited.

Swapping the events changes both the numerator and the denominator in the conditional probability formula. $P(A\mid B)$ uses $P(A \cap B)$ as the numerator and $P(B)$ as the denominator, whereas $P(B\mid A)$ uses $P(A \cap B)$ as the numerator but $P(A)$ as the denominator.

Conditions: Events A and B have different marginal probabilities ($P(A) \neq P(B)$).; Both events have positive probability.

Capacity constraints must be linked to binary variables to ensure logical consistency between machine activation and production. Simple capacity limits ($X_i \le C_i$) allow production even if the machine is not activated ($Y_i=0$), which would incur variable costs without paying the fixed setup cost.

Conditions: Machines have fixed setup costs.; Production is bounded by machine capacity.; Binary variables indicate machine activation.; The model must prevent production on inactive machines.

In likelihood inference, the observed data $x_1, \dots, x_n$ are held fixed, and the parameter $\theta$ is varied to compare different candidate values.

Conditions: The data have already been observed.; One is comparing different candidate values of $\theta$.

An Arrow-Debreu security is a theoretical financial instrument that pays exactly 1 unit of currency in one specific state of the world and 0 in all other states. The state price vector is the collection of the current prices of all such Arrow-Debreu securities for the finite set of states in the market model.

Conditions: The market model has a finite set of states.; Payoffs are normalized to 1 in one state and 0 elsewhere.; The model is a one-period setting.

The base-rate fallacy occurs because learners often ignore the low prevalence of the disease (the base rate) and focus only on the high accuracy of the test. This leads to the incorrect assumption that a positive test result is highly likely to be a true positive, when in fact the large number of healthy people generates many more false positives than true positives.

Conditions: Prevalence is low ($1\%$).; Test accuracy (sensitivity/specificity) is high ($95\%$).; The observer focuses on test accuracy rather than population composition.

By comparing the mathematical properties of the traveling-wave solution with the physical wave equation, we can derive the wave speed. The traveling wave $f(x,t)=f(x-vt,0)$ satisfies $\frac{\partial^2 f}{\partial x^2} = \frac{1}{v^2} \frac{\partial^2 f}{\partial t^2}$.

Conditions: The wave can be written in the traveling-wave form $f(x-vt,0)$.; The physical system is a string with tension $F_T$ and linear density $\mu$.