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Shubham C.
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Java Programming
TutorMe
Question:

Consider the condition for a database containing the names as well as the marks of a student. How will you publish the rank list of the class?

Shubham C.

Let us assume that the names and the marks are stored in a double dimensional array student_name[] and student_marks[] respectively. Then let us consider that the number of students in num_student Then the code for the sorting block will be: for(int i=0;i<num_student-1;i++) { for(int j=i+1;j<num_student;j++) { if(student_marks[i]<student_marks[j]) { double temp=student_marks[i]; student_marks[i]=student_marks[j]; student_marks[j]=temp; String t=student_name[i]; student_name[i]=student_name[j]; student_name[j]=t; } } } Then, displaying the rank list would be: for(int i=0;i<num_student;i++) System.out.println(student_name[i]+"\t\t"+student_marks[i]);

Calculus
TutorMe
Question:

The delta function, which is defined as $$\delta(x)=0; x\neq 0\\ undefined,\, x=0$$. Inspite of being undefined at x=0, due to an infinite discontinuity, we know that $$\int_{-\infty}^{\infty}\delta(x)dx=1$$. Is this an anomaly and if not, why is it true?

Shubham C.

Let us assume a pure phase of frequency $$K_0$$, then f(x)=$$e^{\iota K_0x}\\$$ Then, fourier transform of f(x)=F($$\omega)=\int_{-\infty}^{\infty}\exp(\iota K_0x)\exp(-\iota Kx)dx\\=\int_{-\infty}^{\infty}\exp(\iota (K_0-K)x)dx$$. Now, as this function is a pure phase, one can interpret its Fourier transform as possessing a single strand of frequency equal to $$K_0$$. Thus, in other words, we define this function as $$\delta(K_0-K)=\frac{1}{2\pi}\int_{-\infty}^{\infty}\exp(\iota (K_0-K)x)dx$$ Then coming to the inverse Fourier Transform, $$f(x)=\int_{-\infty}^{\infty}F(x)\exp(\iota Kx)dK$$. Thus, $$\exp(\iota K_0x)=2\pi(\frac{1}{2\pi})\int_{-\infty}^{\infty}\delta(K-K_0)\exp(\iota Kx)dK$$. Hence for any $$x\epsilon\mathbb{R}$$ it satisfies the same condition, therefore let us cosider the situation at x=0. 1=$$\int_{\infty}^{\infty}\delta(K-K_0)dK$$.If $$K_0=0, then\\\boxed{\int_{-\infty}^{\infty}\delta(x)dx=1}(Proved)$$ (Changing the variable K to x).

Physics
TutorMe
Question:

Let us assume that a team A consisting of me and 10 other players is to play against RMA. I play as the defender in my team. Hence, it is natural that I have to face off Ronaldo as he comes to score. In the game, once he grabbed the ball, he kept running and neither me nor my teammates could keep up with him and he scored against us. Later, my teammates alleged that I could easily have stopped him as he was running quite slow according to them. However, I couldn't possibly have caught him as he was running quite fast, according to me. Why does this disparity exist between our observations?? Assume that everyone was running along the same straight line, along the same direction, according to an observer on the stands.

Shubham C.

Let us assume that the game takes place in an inertial frame, which can't be the Earth because it has rotational acceleration. Then, let us assume that velocity of Ronaldo is u and velocity of some random player chasing after him be v. Then, let displacement in Ronaldo's frame be x and in the other player's frame be x' and similarly for the times be t and t', where the starting position and time is same for both the players. Then, Velocity of Ronaldo with respect to the player=f(v)= $$\frac{dx'}{dt'}\\=\gamma_u\frac{(dx-udt)}{\gamma_u(dt-\frac{dx}{c^2}u)}\\=\frac{\frac{dx}{dt}-1}{1-\frac{dx}{dt}\frac{u}{c^2}}\\=\frac{v-u}{1-\frac{vu}{c^2}}$$. Thus, differentiating this function with respect to v, we get f'(v)=$$-\frac{1-\frac{u^2}{c^2}}{(1-\frac{uv}{c^2})^2}$$ Solving for this, we get relative speed is always a decreasing function with respect to v. Hence. my teammates must have been running with a speed much greater than mine and they hence, perceived the speed of Ronaldo as less than what I observed. The fact that "I" could have caught up with Ronaldo, and not them implies that I was in a position which was closer to Ronaldo in the frame of the observer. But I was running at a lesser speed as compared to my teammates according to the observers' frame. Thus, all of us are correct and this illustrates why one should always consider reference frames in a question.

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