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When measuring a speed, the most common way to calculate it is by recording how far something went and the time it took to go that far. In the case of light, this is very difficult. One could conceivably shine a light over a vast distance and have someone else record when they see the light, but this would be difficult even at large distances. The person recording when they see it will need to have terrific reflexes to accurately measure a correct time as the time will be very short. A better method involves the use of a quickly rotating mirror and a beam of light. By aiming a beam of light o the rotating mirror, then reflecting it o a second stationary mirror back into the rotating mirror, calculations can be made on the speed of light. After first hitting the rotating mirror, the mirror will rotate very slightly in the time it takes the beam of light to return and will reflect back to a different position from where it came from. By measuring the displacement of the round trip, a measurement of the speed of light can be made.

The differential wave equation can be used to describe electromagnetic waves in a vacuum. In the one dimensional case, this takes the form $\frac{\partial^2\phi}{\partial x^2}-\frac{1}{c^2}\frac{\partial^2\phi}{\partial t^2} = 0$. A general function $f(x,t) = x \pm ct$ will propagate with speed c. To represent the properties of electromagnetic waves, however, the function $\phi(x,t) = \phi _0 sin(kx-\omega t)$ must be used. This gives the Electric and Magnetic field equations to be $E (z,t) = \hat{x} E _0 sin(kz-\omega t)$ and $B (z,t) = \hat{y} B _0 sin(kz-\omega t)$. Using this solution as well as Maxwell's equations the relation $\frac{E_0}{B_0} = c$ can be derived. In addition, the average rate of energy transfer can be found to be $\bar{S} = \frac{E_0 ^2}{2 c \mu _0} \hat{z}$ using the poynting vector of the fields.

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Chronic Myeloid Leukemia is a type of cancer that starts in certain blood-forming cells of the bone marrow. Interferon-\(\alpha\) was once the standard front-line treatment producing remission rates of only 28.3 percent in 1991. After the highly effective drugs first became available in 2001, survival rates have increased immensely. According to the American Cancer Society, one large study of CML patients treated with a drug called imatinib found that about 90 percent of them were still alive 5 years after starting treatment. While imatinib has changed the way oncologists treat CML, remission is common after extended gaps in treatment. In this paper, we will explore the long-term dynamics of CML under treatment through the use of use of theoretical and mathematical components. We closely base our methods upon the approach of Urszula Ledzewicz and Helen Moore. We will introduce our unique model and explain component selections, while we move towards understanding the optimal interactions of imatinib and interferon-\(\alpha\) against dormant and proliferating CML cells. Our future work involving optimal control dynamics will be briefly introduced and further solutions are currently ongoing.

We present a geometric proof of the addition formulas for the hyperbolic sine and cosine functions, using elementary properties of linear transformations.

Trabalho de Física Experimental II

Trabalho de Introdução à Sociologia - Engenharia da Computação - CEFET/MG