Advanced Math

A more general and also complicated love between Romeo and Juliet is the IVP⎩⎨⎧​R˙=f(t,R,J),J˙=g(t,R,J),R(0)=R0​,J(0)=J0​,​wherefandgare two real functions dependent ont,R, andJ. Similarly, find conditions onfandgso that the IVP SYS. (14) has a solution. For example, a solution exists for IVP SyS. (14) wheref(R,J)=R(1−J)andg(R,J)=J(R−1)(this is also known as the Lotka-Volterra equations in Biology to model the interaction of two species). Give five specific examples of such IVPs without finding the exact solutions.

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36. Stock Prices of a Dividend Cycle In a discussion of stock prices of a dividend cycle, Palmon and Yaaril consider the function f given by (1 + r)l=2 ln(1 + r) u = f(t,r, z) = (1 + r)1–z – where u is the instantaneous rate of ask-price appreciation, r is an annual opportunity rate of return, z is the fraction of a dividend cycle over which a share of stock is held by a midcycle seller, and t is the effective rate of capital gains tax. They claim that ди t(1 + r)l-z In?(1 + r) [(1 + r)l–– t] Verify this. 37. Money Demand In a discussion of inventory theory of money demand, Swansonconsiders the function bᎢ , iᏟ F(b, C, T, i) + C 2 ƏF bT i and determines that + Verify this partial derivative. ac C2 2 az =-

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Two fair dice are thrown. Let A be the event the the first shows an odd number, B be the event that the second shows an even number, and C be the event the either both are odd or both are even.

Two fair dice are thrown. Let A be the event the the first shows an odd number, B be the event that the second shows an even number, and C be the event the either both are odd or both are even. Show that A, B, C are pairwise independent, but not independent. Are A …

Two fair dice are thrown. Let A be the event the the first shows an odd number, B be the event that the second shows an even number, and C be the event the either both are odd or both are even. Read More »

1 -3 2 -6 If T is defined by T(x)=Ax, find a vector x whose image under T is b, and determine whether x is unique. Let A-| 0 1-2 | and b-1-26 5-16 10 Find a single vector x whose image under T is b

EXPERT ANSWER 6 . Let the pre-image of (-6,-26,-2)T be x = (a,b,c)T where a,b, c are arbitrary real numbers. Then T(x) =Ax = (-6,-26,-2)T or, (a-3b+2c, b-2c, 5a-16b+10c)T = (-6,-26,-2)T . Hence, a-3b+2c = -6 …(1), b-2c = -26…(2) and 5a-16b+10c= 10…(3). The augmented matrix of the above linear system of equations is M (say)= 1 -3 2 -6 …

1 -3 2 -6 If T is defined by T(x)=Ax, find a vector x whose image under T is b, and determine whether x is unique. Let A-| 0 1-2 | and b-1-26 5-16 10 Find a single vector x whose image under T is b Read More »