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 »

The following data apply to a 125-kW, 2300-V, three-phase, four pole, 60-Hz squirrel cage induction motor: Stator-resistance between phase terminals = 2.23 Ohm No-load test at rated frequency and voltage: Line current = 7.7 A Three-phase power = 2870 W Blocked-rotor test at 15 Hz: Line voltage = 268 V Line current = 50.3 A Three-phase power = 18.2 kW a. Calculate the sum of rotational and core losses. b. Calculate the equivalent-circuit parameters in ohms- Assume that X 1 = X 2. You can use the cantilever approximation. c. Compute the stator current, input power and power factor, output power and efficiency when this motor is operating at rated voltage and frequency at a slip of 2.95 percent.

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