University of Hyderabad (UoH) 2007 Ph.D (Maths/Applied Maths) 2008-Entrance - Question Paper

(a) H is always separable.
(b) If H has an orthogonal Schauder basis, then H is separable.
(c) If H is separable, then H is locally compact.
(d) If H has a countable Hamel basis, then H is ¯nite dimensional.


21. For every n two N, let fn : [0; 1] ! [0; 1] be a continuous function and let f : [0; 1] ! [0; 1]
be de¯ned as f(x) = lim sup
n!1
fn(x). Then
(a) f is continuous and measurable.
(b) f is continuous but need not be measurable.
(c) f is measurable but need not be continuous.
(d) f need not be either continuous or measurable.

22. Let f; g : C ! C be holomorphic and let A = fx two R : f(x) = g(x)g : The minimum
requirement for the equality f = g is
(a) A is uncountable. (b) A has a positive Lebesgue measure.
(c) A contains a nontrivial interval. (d) A = R.

23. The critical point of the system x0(t) = ¡y + x2, y0(t) = x is
(a) a stable center. (b) unstable.
(c) an asymptotically stable node. (d) an asymptotically stable spiral.

24. An example of a subset of N which intersects every set of form fa + nd : n two Ng,
a; d two N, is
(a) f2k : k two Ng. (b) fk2 : k two Ng.
(c) fk + k! : k two Ng. (d) fk + k2 : k two Ng.

25. The characteristic number of the integral formula Á(x)¡¸Z 2¼
0
sin(x) sin(t)Á(t) dt = 0
is
(a) ¼. (b)1¼
(c) 2¼.12¼

Part B
ans any Ten ques.
1. Let f be a map from R to R such that f(a+b) = f(a)f(b). If f 6= 0 and it is continuous
at 0 then show that there exists a nonzero c two R such that f(x) = cx for all x two R.

2. provide an entire function whose image omits only the value 2¼. Also ¯nd a MÄobius map
whose only ¯xed point is 2¼.


3. Let f(z) = z6¡5z5+2z4+1 and K = fz two C : jz ¡ 2ij · 1g : Show that min fjf(z)j : z two Kg
is attained at a few point on the boundary of K.

4. Let f : W ! R3 be a linear transformation provided by f (¸1v1 + ¸2v2) = (¸1; ¸2; 0) where
W is the space generated by the vectors v1 = (1; 1;¡1) and v2 = (1;¡1; 1). define
how you would extend f to R3 so that the determinant of f is 1. De¯ne such an
extended f.

5. Consider the Banach space `1 of all complex sequences f®ng such that
1 Xn=1
j®nj < 1
with the norm jj f®ng jj1 =
1 Xn=1
j®nj. Let f¸ng be a sequence of complex numbers such
that f¸n®ng two `1 for all f®ng two `1. De¯ne T : `1 ! `1 by T (f®ng) = f¸n®ng. If T
is a bounded linear operator on `1 then show that f¸ng is bounded. In this case what
will be the value of jjTjj?

6. Determine the smallest m such that the ¯eld with 5m elements has a primitive 12th
root of 1.

7. Let A = f® two R j a®2 + b® + c = 0 for a few integers a; b; cg. Then prove that A is a
countably in¯nite set.

8. Let RN be the set of all sequences of real numbers. 2 members (an) and (bn)
are stated to be asymptotic if lim sup
n!1
(jan ¡ bnj) = 0; they are stated to be proximal if
lim inf
n!1
(jan ¡ bnj) = 0. Prove that asymptoticity is an equivalence relation on RN where
as proximality is not. provide an example of a proximal pair that is not asymptotic.


9. De¯ne a topology T on R by declaring a subset U ½ R to be open if U = Á or 0 two U.
define all ¯nite subsets of R which are dense in (R; T ). provide a basis of (R; T ) every
of whose element is a ¯nite set.
10. Let f : R ! R be a di®erentiable function with a bounded derivative. De¯ne
fn(x) = f µx +
1
n¶. Show that fn converges uniformly on R to f.


11. Let fn(x) = xn for 0 · x · 1. obtain the pointwise limit f of the sequence ffng. Prove
that lim
n!1Z 1
0
fn(x) dx = Z 1
0
f(x) dx. Is the convergence uniform?

12. obtain the extremal of the functional J[y] = Z 1
0 µx + 2y +
y02
2 ¶ dx, y(0) = 0, y(1) = 0.
Also test for extrema.

13. Construct the Green's function for the boundary value issue y00 + y = 0 subject to
the boundary conditions y(0) + y0(¼) = 0, y0(0) ¡ y(¼) = 0.
14. obtain the complete integral of p2q2 + x2y2 = x2q2(x2 + y2).
13
15. Solve the integral formula Á(x) ¡ ¸Z 2¼
0 jx ¡ tj sin(x)Á(t) dt = x:

Earning:   Approval pending.