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\documentstyle[12pt]{article}
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\pagestyle{empty}
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\begin{document}
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\begin{eqnarray*}
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U^{cond} = \displaystyle\sum_{i=1}^{N} u_{i}^{cond} \\
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U^{mech} = \displaystyle\sum_{i=1}^{N} u_{i}^{mech} \\
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U = \displaystyle\sum_{i=1}^{N} (u_{i}^{cond} + u_{i}^{mech}) \\
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\theta_{avg} = (\frac{1}{N}\displaystyle\sum_{i=1}^{N} \frac{1}{\theta_{i}})^{-1} \\
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\end{eqnarray*}
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\end{document}
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\documentstyle[12pt]{article}
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\pagestyle{empty}
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\begin{document}
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$$
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u_{i} = u^{mech}_{i} + u^{cond}_{i} = C_{V} \theta_{i}
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$$
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\end{document}
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\documentclass[12pt]{article}
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\pagestyle{empty}
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\begin{document}
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$$
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F^C = A \omega_{ij} \qquad \qquad r_{ij} < r_c
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$$
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\end{document}
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\documentclass[12pt]{article}
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\pagestyle{empty}
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\begin{document}
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$$
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\omega_{ij} = 1 - \frac{r_{ij}}{r_{c}}
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$$
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\end{document}
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