# Study "Mathematics / Statistics" Essays 276-279

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#### Algebra Like Many Other Languages Term Paper…

… Keplar's laws relate to the orbit of an object moving around another in space as elliptical, with the stationary object located at one of the focal points of the ellipse. Simply put, the Earth travels around the sun in an ellipse, and the sun is a focal point in that ellipse; likewise for a satellite traveling around the earth.

Using Algebra, Keplar understood:

Ra=a (1+e) and Rp=a (1-e)

Where a = semi-major axis of the ellipse e= eccentricity of the ellipse

So that the elliptical shape of the orbit is the result of the inverse square force of gravity.

For instance, the eccentricity for a circle is zero. Earth's eccentricity is only 0.0167, while Pluto, the planet with the largest eccentricity, is .25.

Visually, this is what Keplar sought to explain:

Mathematically, however, linear algebra makes the picture an easily understood, computable…… [read more]

#### Stochastic Modeling Is a Mathematical Term Paper…

… The optimal policy from such a model is a single first-stage policy and a collection of recourse decisions (a decision rule) defining which second-stage action should be taken in response to each random outcome [5].

Solution approaches to stochastic programming models are driven by the type of probability distributions governing the random parameters. A common approach to handling uncertainty is to define a small number of scenarios to represent the future. In such cases, it is possible to compute a solution to the stochastic programming problem by solving a deterministic equivalent linear program. These problems are typically very large-scale problems, and so, much research effort in the stochastic programming community has been devoted to developing algorithms that exploit the problem structure, in particular in the hope of decomposing large problems into smaller more tractable components. When the probability distributions of random parameters are continuous, or there are many random parameters, one is faced with the problem of constructing appropriate scenarios to approximate the uncertainty. One approach to this problem constructs two different deterministic equivalent problems, the optimal solutions of which provide upper and lower bounds on the optimal value z* of the original problem.

Stochastic programming has been applied to a wide variety of areas in engineering. Some of the engineering problems such as electrical generation capacity planning, machine Scheduling, timber management, traffic management, automobile inventory management, and lake level management are complex indeterminate problems that require stochastic solutions.

References

[1] A. Prekopa. Stochastic Programming. Kluwer Academic Publishers, Netherlands, 1995.

[2] S.R. Tayur, R.R. Thomas, and N.R. Natraj. An algebraic geometry algorithm for scheduling in the presence of setups and correlated demands. Mathematical Programming, 69(3):369-401, 1995.

[3] C.C. Caroe and R.Schultz. Dual decomposition in stochastic integer programming. Operations Research Letters, 24:37-45, 1999.

[4] R. Hemmecke and R. Schultz. Decomposition of test sets in stochastic integer programming. Mathematical Programming, 94:323-341, 2003.

[5] H.D. Sherali and B.M.P. Fraticelli. A modification of Benders' decomposition algorithm for discrete subproblems: An approach for stochastic programs with integer…… [read more]

#### Personality Impressions Term Paper…

… Burkard, Alan W. And Knox, Sarah (2004). Effect of therapist color blindness on empathy and attributions in cross-cultural counseling. Journal of Counseling Psychology, 51:4, 387-397.

Whether research investigations are designed on the basis of an experimental, descriptive, historical, or case… [read more]

#### Frequency Distribution of Rejected Circuit Term Paper…

… In any statistical analysis the standard deviation mathematical quotient is representative of the degree to which a difference exists between a score distance and the mathematical mean of the group. As the Excel Program only permits cells up to a 30 count the more conventional hand calculation method was used. For the current set of data the calculated standard deviation is expressed by the following formula:

((((2-((()2

N = 38

(X = 684

((X)2 = 467,856

((2 = 12,528

Standard Deviation = ±2.416

A standard deviation of ±2.416 for the ungrouped date suggests that the rejection of the circuit boards within each week fluctuates above and below its mean of rejection by approximately ±2.416 points.

With a mean rejection rate of 18 pieces per week rejection the probability of a 15 piece rejection rate is highly probable with a standard deviation of ±2.416 (15 + 2.415 = 17.415). Caution must be exercised at this point, as "predictive" statistical measures must…… [read more]