Level 1 — RecognitionAdvanced Topics (Elite Level)

Advanced Topics (Elite Level)

⏱ 20 minutes40 marksprintable — key stays hidden on paper

Time limit: 20 minutes Total marks: 40 Instructions: Answer all questions. For True/False and matching items, a brief justification is required to earn full marks. Use $...$ notation for mathematics.

Section A — Multiple Choice (2 marks each)

Q1. The Cauchy–Riemann equations for $f(z)=u(x,y)+iv(x,y)$ are:

(a) $u_x=v_y,\ u_y=v_x$
(b) $u_x=v_y,\ u_y=-v_x$
(c) $u_x=-v_y,\ u_y=v_x$
(d) $u_x=u_y,\ v_x=v_y$

Q2. By Cauchy's integral theorem, $\oint_C f(z)\,dz=0$ provided:

(a) $f$ is bounded on $C$
(b) $f$ is analytic on and inside the closed contour $C$
(c) $C$ is a straight line
(d) $f$ has exactly one pole inside $C$

Q3. The residue of $f(z)=\dfrac{1}{z^2+1}$ at $z=i$ is:

(a) $\tfrac{1}{2i}$
(b) $-\tfrac{1}{2i}$
(c) $i$
(d) $\tfrac{1}{2}$

Q4. In the Einstein summation convention, $a_i b^i$ denotes:

(a) a single product term
(b) the sum $\sum_i a_i b^i$ over the repeated index
(c) a rank-2 tensor
(d) an undefined expression

Q5. The Euler–Lagrange equation for $J[y]=\int_a^b F(x,y,y')\,dx$ is:

(a) $\dfrac{\partial F}{\partial y}=0$
(b) $\dfrac{\partial F}{\partial y}-\dfrac{d}{dx}\dfrac{\partial F}{\partial y'}=0$
(c) $\dfrac{d}{dx}\dfrac{\partial F}{\partial y}=0$
(d) $\dfrac{\partial F}{\partial y'}=0$

Q6. The metric tensor $g_{ij}$ is used to:

(a) differentiate scalars
(b) lower an index, converting a contravariant component to covariant
(c) count dimensions
(d) diagonalize any matrix

Q7. A function $f$ is convex on a convex set if for all $x,y$ and $\lambda\in[0,1]$ :

(a) $f(\lambda x+(1-\lambda)y)\ge \lambda f(x)+(1-\lambda)f(y)$
(b) $f(\lambda x+(1-\lambda)y)\le \lambda f(x)+(1-\lambda)f(y)$
(c) $f''(x)<0$ everywhere
(d) $f$ is linear

Q8. The stationary curve minimizing descent time in the brachistochrone problem is:

(a) a straight line
(b) a parabola
(c) a cycloid
(d) a circular arc

Q9. For a Markov chain with transition matrix $P$ , the steady-state (stationary) distribution $\pi$ satisfies:

(a) $\pi P=\pi$
(b) $P\pi=0$
(c) $\pi=P$
(d) $\pi^2=\pi$

Q10. The KKT conditions for a constrained minimization include, for inequality constraint $g(x)\le0$ with multiplier $\mu$ :

(a) $\mu<0$
(b) complementary slackness $\mu\, g(x)=0$ and $\mu\ge0$
(c) $g(x)=0$ always
(d) $\nabla f=0$ only

Section B — True/False WITH Justification (2 marks each: 1 verdict + 1 reason)

Q11. "Every function differentiable at a point in the complex sense is analytic there." — True or False? Justify.

Q12. "Uniform continuity of $f$ on a set implies pointwise continuity." — True or False? Justify.

Q13. "A function sequence $f_n\to f$ pointwise necessarily converges uniformly." — True or False? Justify.

Q14. "The Cauchy–Tukey FFT reduces the DFT of $N$ points (with $N$ a power of 2) from $O(N^2)$ to $O(N\log N)$ operations." — True or False? Justify.

Q15. "A pole of order 2 has a principal part in its Laurent expansion containing exactly one negative-power term." — True or False? Justify.

Section C — Matching (5 marks total, 1 each)

Q16. Match each concept (A–E) to its correct description (1–5).

Concept		Description
A. Residue theorem		1. Symmetric object measuring lengths/angles in a manifold
B. Christoffel symbols		2. $\oint_C f\,dz = 2\pi i\sum \text{Res}$ inside $C$
C. Metric tensor		3. Iterative first-order minimization $x_{k+1}=x_k-\eta\nabla f$
D. Gradient descent		4. Encode how basis vectors change; connection coefficients
E. Lebesgue measure		5. Assigns "size" to subsets of $\mathbb{R}^n$ , generalizing length

Answer keyMark scheme & solutions

Section A

Q1. (b). CR equations require $u_x=v_y$ and $u_y=-v_x$ ; these come from equating the two directional limits of $f'(z)$ . (2)

Q2. (b). Cauchy's theorem needs analyticity on and inside a closed contour in a simply-connected region. (2)

Q3. (a). Simple pole: $\text{Res}_{z=i}\frac{1}{(z-i)(z+i)}=\frac{1}{2i}$ . (2)

Q4. (b). A repeated upper/lower index pair implies summation. (2)

Q5. (b). Standard Euler–Lagrange stationarity condition. (2)

Q6. (b). $v_i=g_{ij}v^j$ lowers an index. (2)

Q7. (b). Convexity means the graph lies below chords: Jensen inequality with $\le$ . (2)

Q8. (c). The brachistochrone extremal is a cycloid. (2)

Q9. (a). Left eigenvector with eigenvalue 1: $\pi P=\pi$ (row vector convention). (2)

Q10. (b). KKT: stationarity, primal/dual feasibility ( $\mu\ge0$ ), and complementary slackness $\mu g=0$ . (2)

Section B

Q11. False. Analyticity requires complex differentiability in an open neighborhood, not merely at a single point. E.g. $f(z)=|z|^2$ is differentiable only at $z=0$ but nowhere analytic. (1 verdict + 1 reason)

Q12. True. Uniform continuity uses a $\delta$ independent of the point, which in particular works at each fixed point; hence it implies (indeed is stronger than) pointwise continuity. (2)

Q13. False. Pointwise ≠ uniform. Counterexample: $f_n(x)=x^n$ on $[0,1]$ converges pointwise to a discontinuous limit, but not uniformly. (2)

Q14. True. Cooley–Tukey recursively splits the DFT (even/odd indices), giving the recurrence $T(N)=2T(N/2)+O(N)$ , i.e. $O(N\log N)$ . (2)

Q15. False. An order-2 pole has principal part $\frac{a_{-2}}{(z-z_0)^2}+\frac{a_{-1}}{(z-z_0)}$ with $a_{-2}\ne0$ — generally two negative-power terms. (2)

Section C

Q16. A–2, B–4, C–1, D–3, E–5. (1 mark each correct pairing.) (5)

[
  {"claim":"Residue of 1/(z^2+1) at z=i is 1/(2i)","code":"z=symbols('z'); r=residue(1/(z**2+1), z, I); result = simplify(r - 1/(2*I))==0"},
  {"claim":"Metric lowers index consistency: Res at z=-i is -1/(2i) confirming symmetry","code":"z=symbols('z'); r=residue(1/(z**2+1), z, -I); result = simplify(r + 1/(2*I))==0"},
  {"claim":"Convexity: for f(x)=x^2 midpoint value <= average of endpoints","code":"x,y=symbols('x y'); lhs=((x+y)/2)**2; rhs=(x**2+y**2)/2; result = simplify(rhs-lhs) == ((x-y)**2)/4 and (((x-y)**2)/4).subs({x:1,y:0})>=0"},
  {"claim":"x^n at x=1 stays 1 while limit for x<1 is 0 (nonuniform)","code":"n=symbols('n', positive=True); result = (limit(Rational(1,2)**n, n, oo)==0) and (1**n==1)"}
]