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  1. B

    Joint probability density function problem

    Suppose that h is the probability density function of a continuous random variable. Let the joint probability density function of X, Y, and Z be f(x,y,z) = h(x)h(y)h(z) , x,y,zER Prove that P(X<Y<Z)=1/6 I don't know how to do this at all. This is suppose to be review since this is a...
  2. B

    Deduce taylor series

    Deduce that the Taylor series about 0 of 1/sqrt(1-4x) is the series summation (2n choose n) x^n. From this conclude that summation (2n choose n) x^n converges to 1/sqrt(1-4x) for x in (-1/4,1/4). Then show that summation (2n choose n) (-1/4)^n = 1/sqrt(1-4(-1/4)) = 1/sqrt(2) What I know...
  3. B

    Variational calculus - dual problem

    the primal problem was: min (x^T)Px i found g(r) and the partial derivative of g(r) w.r.t. x to be: x=-1/2(P^-1)(A^T)r i have found the dual problem to be: max -1/4(r^T)A(P^(-1))(A^T)r - (b^T)r subject to r>= 0 I am told to find x* and r* (which i think is just x and r): i have not...
  4. B

    Radius of convergence

    I am looking for radius of convergence of this power series: \sum^{\infty}_{n=1}a_{n}x^{n}, where a_{n} is given below. a_{n} = (n!)^2/(2n)! I am looking for the lim sup of |a_n| and i am having trouble simplifying it. I know the radius of convergence is suppose to be 4, so the lim sup...
  5. B

    Showing it is orthogonally diagonalizable

    Suppose that the real matrices A and B are orthogonally diagonalizable and AB=BA. Show that AB is orthogonally diagonalizable. I know that orthogonally diagonalizable means that you can find an orthogonal matrix Q and a Diagonal matrix D so Q^TAQ=D, A=QDQ^T. I am aware of the Real Spectral...
  6. B

    Finding the function, given the gradient.

    the gradient function is |x|^p-2 x and i need to find the function, which apparantly is 1/p |x|^p but i can't figure out how to show this. This is for a bigger problem where the function must be convex. and also p>1 I tried, finding the derivative of 1/p |x|^p , but i don't get the gradient...
  7. B

    Subdifferentiation at a point

    Define f:R->R by f(x) = { x^2 sin(1/x) x!=0, 0 x=0 Compute subdifferential f(0) I went through my notes on subdifferentiation and still do not have a clue how to do this process, is there a formula to directly do this? any help would be greatly appreciated.
  8. B

    Gateaux derivative

    suppose A:H->H is a symmetric operator for some Hilbert Space H, define \varphi: H->R by \varphi(u) = <Au,u>. Compute the Gateaux derivative of \varphi I know the definition of the Gateaux derivative, I just don't know how to use the information given to compute it. The definition of the...
  9. B

    Variational methods - minimization problem proof

    Consider the minimization problem Inf (u E D) F(u) where F(u) = 1/2 integ(0->T) |u (with circle on top)|^2 dt + 1/2 integ(0->T) |u|^2 dt + 1/2 integ(0->T) f(t)u(t) dt, f E L^2 [0,T], and H = {u:[0,T]->R, uEL^2[0,T], u(circle on top) E L^2 [0,T]} is a Hilbert space equipped with the norm...
  10. B

    Variational methods - prove f is convex in R->R

    Suppose f:R^N -> R is twice differentiable. Prove that f is convex if and only if its Hessian gradiant^2 f(x) is nonnegative. How do I go about proving this? and my professor said I only need to consider when N=1. so R->R. any help would be greatly appreciated. For proving it backwards...
  11. B

    Variational methods - properties of convex hull

    Show the following properties of convex hull: (a) Co(CoA) = Co(A) (b) Co(AUB) \supseteqCo(A) U Co(B) (c) If A\subseteqB then Co(AUB)=Co(B) (d) If A\subseteqB then Co(A)\subseteqCo(B) The definition of a convex hull is a set of points A is the minimum convex set containing A. (c) is quite...
  12. B

    Variational methods conjugate points

    (a will be alpha and b will be beta) Let y=y(x,a,b) be a general solution of Euler's equation, depending on two parameters a and b. Prove that if the ratio (subdifferential y/subdifferential a)/(subdifferential y/subdifferential b) is the same at the points, the points are conjugate. I...
  13. B

    Variational methods - conjugate of function

    Let F:H->R bar be a function and F*:H->R bar its conjugate. Fix aEH and show that the conjugate of the new function G(u)=F(u-a) is G*(u*)=F*(u*)+<a,u> Verify the case where F:R^2->R, F(x)=1/2(x)^2 and a = (2,-1) I don't really know how to show this. please help
  14. B

    Conjugate points of extermals of functional

    Show that the extermals of any functional of the form integ (a->b) F(x,y') dx have no conjugate points. Not sure how to start this question, any help would be appreciated
  15. B

    Linear Algebra: show it is diagonalizable and find eigenbasis

    Let T1 be the C-vector space with basis B = (1, cosx, sinx). Define J: T1->T1 by (Jf)(x) = integ(0->pi) f(x-t)dt. Show that J is diagonalizable and find an eigenbasis. J(1) = integ(0->pi) 1 dt J(1) = t | (0->pi) J(1) = pi J(cos(x)) = integ(0->pi) cos(x-t) dt J(cos(x)) = - sin (x-t) | (0->pi)...
  16. B

    Linear Algebra : prove geometric multiplicities are the same

    Let A and B be similar matrices. Prove that the geometric multiplicities of the eigenvalues of A and B are the same. Some help I have gotten so far but still don't know how to proceed from there: To prove that the geometric multiplicities of the eigenvalues of A and B are the same, we can...
  17. B

    D-admissible directions and extermal points

    Let X = C[a,b], J(y) = integ(a to b) sin^3(x) + y(x)^2 dx and D={yEX; integ(a to b) y(x)dx = 1} (a) what are the D-admissible directions for J? (b) Find all possible (local) external points for J on D? so far i have: (let e be epsilon) lim e->0 J(y+ev) - J(y) / e = lim e->0 integ (a to...
  18. B

    Finding adjoint of an operator

    Find the adjoint of the operator A:L^2[0,1] -> L^2[0,1] defined by (Af)(x) = integ from (0 to x) f(t)dt so from my notes it says: the operator A* is called the adjoint of A if: <Ax, y> = <x, A*y> for all x, vE H i am not sure how to do this, and need to know how to do it for a test
  19. B

    Hilber space and linear bounded operator

    Let H be a Hilbert space and A: H-> H be a Linear Bounded Operator. Show that A can be written as A=B+C where B and C are Linear Bounded Operators and B is self-adjoint and C is skew. This is suppose to be an easy question but i'm not sure where to start. I know that self-adjoint is (B*=B)...
  20. B

    Linear Algebra Lagrange polynomials & Basis [Broken] for (a): does that mean i must compute l0(t), l1(t) and l2(t), and i wasn't sure how to do this with the lagrange polynomial formula given, so i found one online and did it, i'm not sure if this is correct, but my l0(t) looks like...
  21. B

    Linear Algebra Subspace Basis Problem

    1. The set of all traceless (nxn)-matrices is a subspace sl(n) of (bold)K^(nxn). Find a basis for sl(n). What is the dimension of sl(n)? Not sure how to go about finding the basis. I know a basis is a list of vectors that is linearly independent and spans. and for the dimension of sl(n), is...
  22. B

    Statistics Unbiased Estimators

    1. Let X1,X2, ... ,Xn be independent identically distributed random variables with ex- pected value \mu and variance \sigma^2: Consider the class of linear estimators of the form \mu\widehat{} = a1X1 + a2X2 + ... + anXn (1) for the parameter \mu, where a1, a2, ... an are arbitrary constants. a)...
  23. B

    Real Analysis related to Least Upper Bound

    Give an example of a function f for which \exists s \epsilon R P(s) ^ Q(s) ^ U(s) P(s) is \forall x \epsilon R f(x) >= s Q(s) is \forall t \epsilon R ( P(t) => s >= t ) U(s) is \exists y\epsilon R s.t. \forall x\epsilon R (f(x) = s => x = y) So this was actually a two part question, and...
  24. B

    Vector calculus - line integral computation

    Compute the line integral \int_{C} F\cdot dr where F = -y i + x j. The directed path C in the xy-plane consists of two parts: i) a left semicircle from (0, -1) to (0, 1) with center at the origin, and ii) a straight line segment from (0,1) to (2,1). i) r(t) = cos t i + sin t j [pi/2 <=t<=...
  25. B

    Vector calculus - line integral

    Suppose that F is an inverse square force field; this is, F(r) = cr/ |r|^{3} for some constant c, where r = xi + yj + zmbfk. Find the work done by F in moving an object from a point P1 along a path to a point P2 in terms of the distances d1 and d2 from these points to the origin. Not exactly...
  26. B

    Linear algebra - solve linear system with complex constants

    Solve the following linear system: ix + (1+i)y = i (1-i)x + y - iz = 1 iy + z = 1 I am getting nowhere with this. is there a trick to do these? I keep getting more and more variations of i. like i^2-1, and (1-i^2)-1 ix + (1+i)y=i (1-i)x + y-iz=1 y + z = 1 ix + (1+i)y = i i(1-i)x - (i^2-1)z =...
  27. B

    Linear algebra - find all solutions with complex numbers

    (a)Find all t \epsilon C such that t^{2} + 3t + (3-i) = 0. Express your solution(s) in teh form x+iy where x,y \epsilon R. (b) Prove that | 1+iz | = | 1-iz | if and only if z is real. Okay so I tried to use the quadratic formula to find the roots to find the solutions, but I am stuck...
  28. B

    Double integral of mass of circular cone

    Find the mass of a right circular cone of base radius r and height h given that the density varies directly with the distance from the vertex does this mean that density function = K sqrt (x^2 + y^2 + z^2) ? is it a triple integral problem?
  29. B

    Multiple integrals - volume of part of a unit sphere

    Find the volume of that portion of a unit sphere for which 0<theta<alpha, where theta is one of the spherical coordinates So i know the equation is z^2 = x^2 + y^2 , but what is the meaning of 0<theta<alpha? where do i start? I know i must convert to polar coordinates. z^2 = r^2
  30. B

    Double integral of volume bounded by plane and paraboloid

    Evaluate the volume of the solid bounded by the plane z=x and the paraboloid z = x^2 + y^2 I have tried to graph this, and they don't bound anything? have i graphed it wrong. and is there a way to do these problems where you don't need to draw the graph.