find the order of 3 modulo 29

The Carolina Cougars is a major league baseball expansion team beginning its third year of operation. The team had losing records in each of its first 2 years and finished near the bottom of its division. However, the team was young and generally competitive. The team’s general manager, Frank Lane, and manager, Biff Diamond, believe that with a few additional good players, the Cougars can become a contender for the division title and perhaps even for the pennant. They have prepared several proposals for free- agent acquisitions to present to the team’s owner, Bruce Wayne.

Under one proposal the team would sign several good available free agents, including two pitchers, a good fielding shortstop, and two power-hitting outfielders for $52 million in bonuses and annual salary. The second proposal is less ambitious, costing $20 million to sign a relief pitcher, a solid, good-hitting infielder, and one power-hitting out- fielder. The final proposal would be to stand pat with the current team and continue to develop.

General Manager Lane wants to lay out a possible season scenario for the owner so he can assess the long-run ramifications of each decision strategy. Because the only thing the owner understands is money, Frank wants this analysis to be quantitative, indicating the money to be made or lost from each strategy. To help develop this analysis, Frank has hired his kids, Penny and Nathan, both management science graduates from Tech.

Penny and Nathan analyzed league data for the previous five seasons for attendance trends, logo sales (i.e., clothing, souvenirs, hats, etc.), player sales and trades, and revenues. In addition, they interviewed several other owners, general managers, and league officials. They also analyzed the free agents that the team was considering signing.

Based on their analysis, Penny and Nathan feel that if the Cougars do not invest in any free agents, the team will have a 25% chance of contending for the division title and a 75% chance of being out of contention most of the sea- son. If the team is a contender, there is a .70 probability that attendance will increase as the season progresses and the team will have high attendance levels (between 1.5 million and 2.0 million) with profits of $170 million from ticket sales, concessions, advertising sales, TV and radio sales, and logo sales. They estimate a .25 probability that the team’s attendance will be mediocre (between 1.0 million and 1.5 million) with profits of $115 million and a .05 prob- ability that the team will suffer low attendance (less than 1.0 million) with profit of $90 million. If the team is not a contender, Penny and Nathan estimate that there is .05 probability of high attendance with profits of $95 mil- lion, a .20 probability of medium attendance with profits of $55 million, and a .75 probability of low attendance with profits of $30 million.

If the team marginally invests in free agents at a cost of $20 million, there is a 50–50 chance it will be a contender. If it is a contender, then later in the season it can either stand pat with its existing roster or buy or trade for players that could improve the team’s chances of winning the division. If the team stands pat, there is a .75 probability that attendance will be high and profits will be $195 million. There is a .20 probability that attendance will be mediocre with profits of $160 million and a .05 probability of low attendance and profits of $120 million. Alternatively, if the team decides to buy or trade for players, it will cost $8 million, and the probability of high attendance with profits of $200 million will be .80. The probability of mediocre attendance with $170 million in profits will be .15, and there will be a .05 probability of low attendance, with profits of $125 million.

If the team is not in contention, then it will either stand pat or sell some of its players, earning approximately $8 million in profit. If the team stands pat, there is a .12 probability of high attendance, with profits of $110 million; a .28 probability of mediocre attendance, with profits of $65 million; and a .60 probability of low attendance, with profits of $40 million. If the team sells players, the fans will likely lose interest at an even faster rate, and the probability of high attendance with profits of $100 million will drop to .08, the probability of mediocre attendance with profits of $60 million will be .22, and the probability of low attendance with profits of $35 million will be .70.

The most ambitious free-agent strategy will increase the team’s chances of being a contender to 65%. This strategy will also excite the fans most during the off-season and boost ticket sales and advertising and logo sales early in the year. If the team does contend for the division title, then later in the season it will have to decide whether to invest in more players. If the Cougars stand pat, the probability of high attendance with profits of $210 million will be .80, the probability of mediocre attendance with profits of $170 million will be .15, and the probability of low attendance with profits of $125 million will be .05. If the team buys players at a cost of $10 million, then the probability of having high attendance with profits of $220 million will increase to .83, the probability of mediocre attendance with profits of $175 million will be .12, and the probability of low attendance with profits of $130 million will be .05.

If the team is not in contention, it will either sell some players’ contracts later in the season for profits of around $12 million or stand pat. If it stays with its roster, the prob- ability of high attendance with profits of $110 million will be .15, the probability of mediocre attendance with profits of $70 million will be .30, and the probability of low attendance with profits of $50 million will be .55. If the team sells players late in the season, there will be a .10 probability of high attendance with profits of $105 million, a .30 probability of mediocre attendance with profits of $65 mil- lion, and a .60 probability of low attendance with profits of $45 million.

Assist Penny and Nathan in determining the best strategy to follow and its expected value.

Set up the addition and multiplication tables for Z3 and Z6.
Use these tables to verify that (Z3, +), (Z3 \ {0}, ·) and (Z6, +) are groups, but (Z6 \ {0}, ·) is not a group.

In which finite field does "25 divided by 5 is 14" hold?

Let G be a graph. prove G has a Eulerian trail if and only if G has at most one non-trivial component and at most 2 vertices have odd degree

factorize the integer 2896753 by using the Quadratic Sieve method

describe your feelings toward math and list at least one positive and one negative experience you have had with a math course or mathematics you use in your daily life?

10.

You are valuing Soda City Inc. It has $150 million of debt, $70 million of cash, and 200 million shares outstanding. You estimate its cost of capital is 8.0%. You forecast that it will generate revenues of $740 million and $760 million over the next two years, after which it will grow at a stable rate in perpetuity. Projected operating profit margin is 40%, tax rate is 20%, reinvestment rate is 60%, and terminal EV/FCFF exit multiple at the end of year 2 is 8. What is your estimate of its share price? Round to one decimal place.

solve the inital value problem ?′′ + 6?′ + 9? = 2?^(-t) ; ?(0) =4, ?’(0) = −6

Consider the equivalence relation on Z defined by the prescription that all positive numbers are equivalent, all negative numbers are equivalent, and 0 is only equivalent to itself. Let f ∶ Z → {a, b} be the function that maps all negative numbers to a and all non-negative numbers to b. Does there exist a function F ∶ X/∼→ {a, b} such that f = F ○ π? If so, describe it

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3. Show that the Galois group of (x² − 3)(x² + 3) over Q is isomorphic to Z₂ × Z₂.

4. Let p(x) be an irreducible polynomial of degree n over a finite field K. Show that its Galois group over K is cyclic of order n.

Discrete Math: Give examples of relations on the set of humans that are:

a) asymmetric and transitive

b) symmetric and antisymmetric

c) reflexive and irreflexive.

Let F be a field and R = Mn(F) the ring of n×n matrices with entires in F. Prove that R has no two sided ideals except (0) and (1).

Give an example of a function whose Taylor polynomial of degree 1 about x = 0 is closer to the values of the function for some values of x than its Taylor polynomial of degree 2 about that point.

Sometimes a constant equilibrium solution has the property that solutions lying on one side of the equilibrium solution tend to approach it, whereas solutions lying on the other side depart from it. In this case the equilibrium solution is said to be semistable. Consider the equation dy/dt = y^2(4 − y 2 ) = f(y), where y(0) = y0 and −∞ < y0 < ∞.

(i) Sketch the graph of f(y) versus y.

(ii) Determine the critical points. (iii) Classify each one as asymptotically stable, unstable, or semistable. (iv) Illustrate several solutions in the ty-plane that illustrate how the different solutions depend upon y0.