(1) Thanks to this slide Circuit, State Diagram, State Table. achieves state if the remainder obtained by dividing the sum of input values up to that time by is . (2) The states can be held in a 2-bit register. Their corresponding representation in 2-bit is: State 0: 00. State 1: 01. State 2: 10. [...]
(1) See Unilateral Z-transform on Wikipedia. (2) Under superpower of Transfer Function of a Circuit: (3) As hinted by Wikipedia: Q.E.D (4) (5) Thanks to Wikipedia (corresponding official source: Z Transform of Difference Equations and Direct-Form I, also Shifting Theorem of Z Transform): Let be [...]
(1) AEFDG (2) BCAEFDG (3) Real-world problem: Task scheduling. If there are many dependent tasks to do then topological sort will point out the order of tasks to be completed (reference). (4) function DFS: Replace “print u” in function DFS2 for “s.push(u)”. (5) The above code block visited each node of the graph once (thanks [...]
Problem link (1) We prove and (note that and , I doubt if zero can be considered as an even number though). Base case: . if a red card was taken. if a white card was taken. So and are true for . Induction step: Let . if a red card was taken. if a [...]
Lagrange multipliers (given in 2018.1.4) Pseudoinverse (given in 2018.1.5). Positive-semidefinite (given in 2018.1.5). Geometric sum (given in 2018.2.3). Limit properties (Paul’s Online Notes).
Problem link (1) Base case: Show that holds true. We have: . Induction step: Show that holds true leads to the fact that holds true. So the statement follows. (2) This problem is interestingly tricky. First we find the critical points of : If then (due to the fact [...]
Problem link (1) (2) Under the superpower of this Youtube Lecture, this WikiBooks Article, this Table of Laplace Transforms, and Khan Academy Lectures: As the Heaviside step function has multiple definitions (see my unanswered question on Math StackExchange), so [...]
Problem link (1) Use Gaussian Elimination to address (i). All the rest can be easily inferred. (2) Proof by contradiction: Assume and the solution of the simultaneous linear equation does not exist. If then . However, and , so is guaranteed to have linearly dependent columns. Therefore, we can always represent as a linear combination [...]