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- If x and y are elements of an ordered integral domain D, prove the following inequalities. a. x22xy+y20 b. x2+y2xy c. x2+y2xy1. a) Let (x, d) be a metric space. Define a flow on (x, d). b) Let (x, {ϕt}) be a flow on a metric space X. When is xo in x a fixed point of the flow? c) When do you say that a fixed point xo in x is Poincare stable? d) When do you say that a fixed point xo is Lypanov stable?let {x, {phi t}} be a flow on a metric space x. when is x0 is in x a fixed point of the flow when do you say that a fixed point x0 in x is Pointcare stable? when do you say that a fixed point x0 is Lypunov stable?
- 2. Assume that (X, dX) and (Y, dY ) are complete spaces, and give X × Y themetric d defined byd((x1, y1),(x2, y2)) = dX(x1, x2) + dY (y1, y2)Show that (X × Y, d) is complete.1. a) Let (x, d) be a metric space. Define a flow on (x, d). b) Let (x, {phi_t}) be a flow on a metric space x. what is x0 in x a fixed point of the flow? c) When do you say that a fixed point x0 in x is Poincare stable? d) When do you say that a fixed point x0 is Lyapunov stable? Use Analysis to complete the following statements.2.) Let (S, d) be a metric space and suppose that ρ : S × S → R is defined byρ(x, y) = d(x, y)1 + d(x, y)for all points x, y ∈ S. Prove that (S, ρ) is a metric space, that it is bounded and thatρ(x, y) ≤ d(x, y) for all x, y ∈ S.
- 28) Bounded by the planes z=x, y=x, x+y=2, and z=0Let x denote the amount of time for which a book on 2-hour reserve at a college library is checked out by a student, and suppose that x has density func- tion f(x)5.5x for 0,x,2.If y equals sin squared x comma space t h e n space space y to the power of apostrophe equals space 2 sin x. cos x 0 2 sin x space minus space 2 space cos x space 4 sin x. cos x
- 5.) Suppose the joint density for X and Y is given by f(x, y) = 1/xfor 0 < y < x < 1. Find ρX,YThe graphs of f (x) = x2 and g (x) = cx3 , being c> 0, intersect at the points (0,0) and at (1 / c, 1 / c2 ). Determine c such that the bounded region between these graphs and on the interval (0, 1 / c) has area 2/3The following problem is similar in spirit to some which were studiedby Archimedes and others. Solve it using integral calculus: Let Ah be the closed regionin the coordinate plane defined by the vertical lines 1 = x and x = h (where h > 1), thex-axis, and the hyperbola y =((x^2) − 1)^1/2, and let Bh be the corresponding region definedby the vertical lines 0 = x and x = h (where h > 0), the x-axis, and the hyperbola’sasymptote y = x. Next, let Ph and Qh be the solids of revolution obtained by rotating Ahand Bh (respectively) about the x-axis. Compute the ratio|Vol (Ph)|/|Vol (Qh)|.[Hint: Draw a picture to make the problem more transparent.]