A research team is modeling the spread of a highly contagious pathogen in a densely packed, fully connected community where the homogeneous mixing assumption holds. Because they are modeling the disease beyond its initial early stages, they cannot use the simple exponential approximation. Instead, they use the exact SI model equation: $$i(t) = \frac{i_0 \exp(\beta t)}{1 - i_0 + i_0 \exp(\beta t)}$$ At the start of the observation ($t = 0$), exactly $10\%$ of the population is infected ($i_0 = 0.1$). After a certain number of days $t$, the infection's exponential growth factor reaches $e^{\beta t} = 9$. What is the fraction of infected individuals $i(t)$ in the population at this time? a) $10.0\%$ b) $33.3\%$ c) $50.0\%$ d) $81.8\%$ e) None of the above. Original idea by: Matheus de Oliveira Saldanha
Postagens
A social media platform is modeled as a network where users adopt a viral challenge if the fraction of their friends already participating exceeds a threshold $\phi = 0.5$. User Ana has $6$ friends, and currently $4$ of them are participating in the challenge. Which statement best describes what happens? a) Ana will not participate because at least all $6$ friends must participate. b) Ana will participate because the fraction of participating friends is $\frac{4}{6} \approx 0.67$ which is greater than the threshold $\phi = 0.5$. c) Ana will participate only if exactly $3$ friends participate. d) Ana’s decision is independent of the threshold $\phi$. e) None of the above. Original idea by: Matheus de Oliveira Saldanha
Evaluate the following affirmations regarding planarity and planarity testing algorithms: I. A multi-graph is considered planar when it admits a plane drawing . II. To postulate Euler's formula ( $N-L+F=2$ ), one can start with a single node and observe the effect on $N$ , $L$ , and $F$ when adding a new node connected to an existing node, or when adding a link between two existing nodes . III. Kuratowski's Theorem states that a graph is planar if and only if it contains a subdivision (the process of replacing a link by a path) of $K_{3,3}$ or of $K_{5}$ , effectively making them mandatory subgraphs for planarity . IV. The pseudo-code for the DMP algorithm begins by starting with a single node and iteratively selects a fragment $B$ with the maximum number of faces $F(B)$ that contain all its vertices of attachment . V. When using the DMP algorithm, a fragment of a graph $G$ with respect to a subgraph $H$ can be an...
In the heart of a bustling kitchen, Chef Leo is famous for his "Variable Soufflé." The height of the soufflé, $H$ , depends on two main factors: the temperature of the oven, $x$ , and the amount of whisking time, $y$ . Leo has discovered a mathematical model for his perfect bake: $$H(x, y) = x^2y + 3xy^2 + 5x - 2y$$ One morning, while the oven is preheating, Leo needs to know exactly how the height of the soufflé changes if he increases the temperature ( $x$ ) while keeping the whisking time ( $y$ ) perfectly constant. He needs to calculate the partial derivative of $H$ with respect to $x$ at the point where $x = 2$ and $y = 3$ . Given the function $H(x, y) = x^2y + 3xy^2 + 5x - 2y$ , what is the value of the partial derivative $\frac{\partial H}{\partial x}$ at the point $(2, 3)$? a) 37 b) 44 c) 39 d) 41 e) None of the above. Original idea by: Matheus de Oliveira Saldanha
We know that Kosaraju-Sharir's algorithm can be used to find the Strongly Connected Components (SCCs) of a graph. It is composed of three main tasks: Get Decreasing Order of DFS Finishing Times Traverse the Reverse Graph Each Tree is a Strongly Connected Component Given that, and the following graph: Select the alternative that has the correctly DFS decreasing order of finishing times starting from node A , the correctly reverse graph and, finally, each SCC colored correctly. e) None of the above. Original idea by: Matheus de Oliveira Saldanha
Jerry the mouse is trying to reach a piece of cheese located somewhere in a maze. The maze can be represented as a graph , where: Each node is labeled with a letter and a number (for example: A1, D2, F3). Edges represent paths between locations in the maze . Jerry starts at node D2 and explores the maze using Depth-First Search (DFS) . When DFS needs to choose which neighbor to visit next, Jerry must follow one consistent ordering rule . However, he is not sure which rule will lead him to the cheese the fastest. Jerry is considering the following four DFS neighbor ordering strategies: Order neighbors by the letter only, ascending (A → Z) Order neighbors by the letter only, descending (Z → A) Order neighbors by the number only, ascending (1 → 9) Order neighbors by the number only, descending (9 → 1) Assume that when sorting neighbors, the ignored part of the label is not considered . The cheese is located at node J2 . Which of the following alterna...
