Which of the following sets is open in (R, Tµ)? О а. R { (-1)"3n³ :n E N} 5n+n³ b. O b. [2, 4) U (1, 5) (-1)" 2n² R – ({ n+n? :n E N} U {-2}) Ос. O d. [2, 4) e. R - (Q – Z) O e.

Q: Compute Div(F) and Curl(F): F--(4y-1) i+yj+(-3y) k

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Q: 1- iz If z = x +iy and w =- z-i show that w = 1= zis purely real

A: Use modulus property

Q: Let {u, v} be an orthonormal set in R" and let y = v13 u + 6v. Then, y|| is

A: Here, given that u,v are orthonormal. That is: u·u=1, v·v=1, u·v=v·u=0

Q: a. Show that W[5, sin2 t, cos(2t)] = 0 for all t by directly evaluating the Wrosnkian. b. Establish…

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Q: Letr(t) = (t2, 1- t,4t).Calculatethederivativeofr(t) • a(t)att = 2, assumingthata(2) = (1,3,3)…

A: Given that, rt=t2, 1-t, 4t a2=1, 3, 3a'2=-1, 4, 1 Need to evaluate derivative of rt.at

Q: (b) En=1 m

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Q: 2) Use Green's Theorem to compute f F df where F= (3x +4y)i + 2xyj and C is the triangular path from…

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Q: Q2= A) for E= 2xĭ-3Y3+K %3D prove That divc curl È)=0 %3D

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Q: 4. Prove that for n > 0: E, 2* = 2"+1 – 1. sk3D0

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Q: Show that under the transformation: k= the equally spaced z-values, z : 1o, To +h, xo +2h, .…

A: Just substitute the given values of x and proof will follow.

Q: Ex: O calkulake th gui for the disanse formula in a barred cordihute system defined by R = Ax, whene…

A: Here, one matrix function, A=1-223 is given Let the X=a1b1a2b2 So, X¯=AX=a1b1a2b2.1-223…

Q: Prove by mathematical induction: X n (U-, Y) = U=1(X nY), for V n E Z* – {1} and sets X,Y,,Y2,. Yp-…

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Q: For x = 1, 2, 3 and y = 1, 2, let the joint pmf of X and Y be defined by f X,Y (x, y) = (x+y)/21…

A: The joint probability mass function of X and Y is  f(x,y)=(x+y)/21   , x=1,2,3 and y=1,2

Q: Let, T:R' → R³;T(x,y,z)=(2x+ y, y- z,2y+4z) 2. Find the rank and nullity of T.

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Q: Let X₁, X₂,..., X be a r.s. from N(,0) and Y₁, Y₂...., be another independent r.s. from m…

A: Solution

Q: Let Z, = 2 cis 30° Z2 = 3elli %3D Z = In (1 + i) If Z = Z,Z, +Z3 Find IZ.

A: First we convert all expression into a+ib form then make necessary calculation.

Q: 2. Consider the space L ([1,2)) of complex square integrable functions on the interval [1, 2).…

A: Consider the  equation

Q: 4. Show that for any bounded linear operator T on H, the operators T₁=1/√(T+T*) and T₁=1 (1 ==…

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Q: Exercise 4 Show that for cach k EN it holds that kn-k < n! S n", Vn E N, n2 k. Deduce from this that…

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Q: Which elements are invertible in Z4[x,y]?

A: W need to find elements which are invertible inZ4x,y.

Q: dz Let z-ay" – 4r, * = uv and y-v - 3u. Compute Ən (u,r)-(1,1) -

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Q: Let po) - X In (x) , using the following approx im for the secoud denvative fcxo) = FOXO -h)-ZR…

A: Introduction: Apart from differentiating a function analytically, the derivative of a function can…

Q: prove that F(r) = 4z-3z is chaotic on the interval [-1,1]. (Hint: consider g(6) = 30 on S'.)

A: The given function: F(x)=4x3-3x The given interval is [-1,1] To prove that F(x) is chaotic: Chaotic…

Q: 2) Use Green's Theorem to compute f F. df where F=(3x +4y) i + 2xyj and C is the triangular path…

A: Use green's theorem

Q: 2) Use Green's Theorem to compute f F df where F=(3x + 4y)i + 2xyj and C is the triangular path from…

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Q: Let (An) and (Bn) be (Fn)-martingales on (2, F, P) and let Cn = A - Bn. Show that E [(Am - An) |Fn]…

A: Solution:

Q: The Wronskian of y,(t)=e 2, y,()=e" is-e St. Select one: True False

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Q: Let w = -4xy – 5yz – 4xz, x = st, y = et, z = t Compute (3,4) = as (3,4)%3D at

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Q: 3. Use Green's Theorem to evaluate G³ï+xj)•dr around the perimeter of the rectangle 0<x<2,0<y<3.…

A: Given ∫C(y2i+xj).dr We know that dr = dxi+dyj Thus ∫C(y2i+xj).dr = ∫C(y2i+xj).(dxi+dyj)  =…

Q: 4. Evaluate the Kullback-Leibler divergence between two Gaussians p(x) = N(xu, E) and %3D q(x) =…

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Q: Prove that F(z) = 81 8z + 1 is chaotic on (-1,1).

A: To prove: Fx=8x4-8x2+1 is chaotic on -1,1. First plot graph of given function with -1,1 as follows:

Q: @ Let A -a Find Cenditiens on a,b.c and d that AX =o has infailely many Solution. what must there…

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Q: Q1:- Let T =- where W-N(0, 1) and V~xShow that T2-F(1,r).

A: We know that  The ratio of chi-square divided by its degrees of freedom is called the…

Q: Assume that u = U (x,y) iu differentiible and x =r ca Ô and y=rsin o eapress 2. entinely in termv 0…

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Q: If f(z,y, z) = I COs(2y – 2). compute and af and Te Əz =I Cos(2y -2). compute O = cos(2y2) I sin(2y…

A: If fx,y,z =xcos2y-z....icompute dfdx and dfdz

Q: Consider Uu=4uzzy u(x,0) = u(x), u₁(x,0) = u₁(x), on the domain -∞<x<∞, 0≤t<∞o. Suppose that both…

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Q: Which of the following is a nonlinear second order nonhomogenous PDE a. xux-Uxx=tuxt O b.…

A: To find - Which of the following is a nonlinear second order non-homogenous PDE x ux - uxx = t uxt…

Q: For x, y, z, w > 0, prove that (x/z)+(y/z)+(z/w)+(w/x)≥4

A: It should by x/y, instead of x/z

Q: Suppose {V1, V2}is an orthonormal set. Let w = 301 – 252 and z = 201 + 02. %3D Find w· Z, the dot…

A: v1,v2 is an orthonormal set.w=3v1-2v2 and z=2v1+v2

Q: prove for n=l,2,3. e that +2+3+..+h = ncn+l) 2. %3D %3D

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Q: Suppose the goind densiy Runction Xanef y i ginen by frxy) = 44 (ス-8) ○ZXC0,○ ス %3D otheraise…

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Q: Determine Z- Trans furm in cludig vegiva of con vergence in n 7o メ(At n<o

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Q: 1. Obtain the weak formulation for the following BVP: in Ω, -V (kVu) = f u =g on ô0.

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Q: 19: Find two linear operator T, and T,on V(R) such that T,T, = 0 but T,7, - 0.

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Q: Suppose that T :C C is R-linear. Show that T is C-linear if and if T(iz) = iTz for all z.

A: First we describe the definition of C-linear function:

Q: 2) Suppose S,T E L(V) are such that ST = TS. Prove that null S is invariant under T.

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Q: 4) Let f(r, y, z) = In(r + 2y + 3z). Find Vf = . %3D

A: Given: f(x, y, z)=ln (x+2y+3z)

Q: 2 - Find the inter val on which f are Con Cave down and infleobion Points a -y = x2 5X+10 6- ソ= (¥+…

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Q: Let z = F(x , y) = xy2 + - , where x = u2 + v2 - 4 and y = uv . az Use the chain rule to find at (u…

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Q: 2. Use Wronskian to show that fi = x, f2 = ln x, and f3 C (R). x In x are linearly independent in

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