## Todai Entrance Exam: Subject 2019 – Problem 5

This is Kronecker delta. The equation can be interpreted as: If there exists a that then we return , otherwise we return . (1) According the given formula at the end of the problem:     Reference: StackExchange (2) Reference: https://class.ece.uw.edu/235dl/EE235/Project/lesson19/lesson19.html     (3) By Modulation Property of the Fourier Transform, we have:     [...]

## Todai Entrance Exam: Subject 2019 – Problem 3

(1) Apply the Algorithm of (2), we get shortest paths:                 (2)     (3) Inside block if, add: . (4) There are 2 loops in the algorithm. The loop traversing the set is inside the loop traversing the set , so the time complexity is . (5) [...]

## Todai Entrance Exam: Subject 2019 – Problem 1

(1) Thanks to this lecture note by MIT, it seems like is to indicate that the path from the point to the point of the amplifier has current. And the point of ? Well, I’m not sure: a slide from University of Kansas.     (2)     Reference: Khan Academy and the slide from [...]

## Todai Entrance Exam: Subject 2018 – Problem 2

(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. [...]

## Todai Entrance Exam: Subject 2018 – Problem 5

(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 [...]

## Todai Entrance Exam: Subject 2018 – Problem 3

(1) GDFEA (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 [...]

## Todai Entrance Exam: Math 2018 – Problem 3

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 [...]

## Todai Entrance Exam: Math – Memo

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).

## Todai Entrance Exam: Math 2018 – Problem 2

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 [...]

## Todai Entrance Exam: Subject 2018 – Problem 1

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 [...]