NIST TN 1297: Equations

Eq A1

$$Y = f(X_1, ~X_2, ~\ldots, ~X_N) ~ $$

Eq A2

$$y = f(x_1, ~x_2, ~\ldots, ~x_N) ~ $$

Eq A3

\begin{eqnarray*} u_c^2(y) &=& \sum_{i=1}^N ~ \left( {{\partial f}\over{\partial x_i}} \right)^2 ~ u^2(x_i)\\ &~& + 2 \sum_{i=1}^{N-1} ~ \sum_{j=i+1}^N ~ {{\partial f}\over{\partial x_i}} ~ {{\partial f}\over{\partial x_j}} ~ u(x_i,x_j) ~ \end{eqnarray*}

Eq A4

$$ x_i = \bar{X}_i = {1\over n} ~ \sum_{k=1}^n ~ X_{i,k} ~$$

Eq A5

\begin{eqnarray*} u(x_i) &=& s(\bar{X}_i)\\ &=& \left( {1\over{n(n-1)}} ~ \sum_{k=1}^n ~ (X_{i,k} - \bar{X}_i)^2 \right)^{1/2} ~ \end{eqnarray*}

Eq A6

$$ x_i = (a_+ + a_-)/2 ~ $$

Eq A7

$$u(x_i) = a/\sqrt{3} ~ $$

Eq B1

$$\nu_{\rm eff} = {\textstyle{u_c^4(y)}\over{\sum_{i=1}^N ~ {\textstyle{c_i^4 \, u^4(x_i)}\over{\textstyle\nu_i}}}} ~ $$

Eq B2

$$\nu_{\rm eff} \leq \sum_{i=1}^N ~ \nu_i. $$

Eq D1.1

$$u_c^2 = \sum_{i=1}^N ~ [c_i \, u(x_i)]^2 \equiv \sum_{i=1}^N ~ u_i^2(y) ~ $$

Eq D3.1

\begin{eqnarray*} x_1 &=& g_1(w_1, ~w_2. ~..., ~w_K)\\ x_2 &=& g_2(z_1, ~z_2. ~..., ~z_L)\\ &~& {\rm etc.}\end{eqnarray*}

Eq D3.2

$$y = x + C_1 + C_2 + \ldots + C_M ~$$

Eq D5.1

$$1 - \alpha = \int_{-\infty}^{\textstyle{t_{1-\alpha}}} f(t,\nu) {\rm d}t $$