Draw a Haworth projection for each of the following compounds: (a) $\beta$-D-Galactopyranose (b) $\alpha$-d-Mannopyranose (c) $\alpha$-d-Allopyranose (d) $\beta$-D-Mannopyranose (e) $\beta$-D-Glucopyranose (f) $\alpha$-D-Glucopyranose

Let T be a linear operator on $V=\mathbb{R}^{n}$ whose matrix A is a real symmetric matrix. (a) Prove that V is the orthogonal sum $V=(\operatorname{ker} T) \oplus(\mathrm{im} T)$ (b) Prove that T is an orthogonal projection onto im T if and only if, in addition to being symmetric, $A^{2}=A .$

If $\sin \alpha=\frac{1}{3}, \frac{\pi}{2}<\alpha<\pi,$ and $\cos \beta=-\frac{1}{3}, \pi<\beta<\frac{3 \pi}{2}$ find the exact value of: $\cos (\alpha+\beta)$

One projection of the world population by the United Nations (a low-projection scenario) is given in the table below. The data can be modeled by $y=-0.36 x^2+38.52 x+5822.86$ million people, where x is the number of years after 1990. In what year after 1990 does this model predict the world population will first reach 6,702,000,000?

Suppose you held a diversified portfolio consisting of a $7,500 investment in each of 20 different common stocks. The portfolio’s beta is 1.25. Now suppose you decided to sell one of the stocks in your portfolio with a beta of 1.0 for$7,500 and use the proceeds to buy another stock with a beta of 0.80. What would your portfolio’s new beta be?

Decide if the given statement is true or false, and give a brief justification for your answer. If true, you can quote a relevant definition or theorem. If false, provide an example, illustration, or brief explanation of why the statement is false. If A is an invertible

$$n \times n$$

matrix, then the projection matrix is

$$P = I _ { n }.$$

Find (a) the orthogonal projection of b onto Col A and (b) a least-squares solution of Ax=b.

$$A=\left[ \begin{array}{rr}{1} & {2} \\ {-1} & {4} \\ {1} & {2}\end{array}\right], \mathbf{b}=\left[ \begin{array}{r}{3} \\ {-1} \\ {5}\end{array}\right]$$

Find the exact value of the given functions. $\\$ Given$\sin \alpha=-$\frac{7}{25}$, \alpha$in Quadrant IV, and$\cos \beta=$\frac{8}{17},$\beta$in Quadrant IV, find$$c.$\tan (\alpha+\beta)$

Find the exact value of the given functions. $\\$ Given$\sin \alpha=$\frac{24}{25}$, \alpha$in Quadrant II, and$\cos \beta=-$\frac{4}{5}$, \beta$in Quadrant III, find$$c.$\tan (\alpha+\beta)$

The thin rectangular plate is attached to the rectangular frame by pins. The frame rotates with constant angular velocity $\omega_0$. Show that

$$\frac{d^2 \beta}{d t^2}=-\omega_0^2 \sin \beta \cos \beta .$$

The Mercator map projection was proposed by the Flemish geographer Gerardus Mercator (1512-1594). The stretching factor of the Mercator map as a function of the latitude $\theta$ is given by the function $G(\theta)=\int_{0}^{\theta} \sec x d x$. Graph G, for $0 \leq \theta<\pi / 2$.

Find all $\alpha$ and $\beta >0$ so that the matrix

$$A=\left[\begin{array}{ccc}{4} & {\alpha} & {1} \\ {2 \beta} & {5} & {4} \\ {\beta} & {2} & {\alpha}\end{array}\right]$$

is strictly diagonally dominant.

Determine whether the statement is true or false. If it is true, explain why. If it is false, explain why or give an example that disproves the statement. The projection of the curve

$$\mathbf{r}(t)=\langle\cos 2 t, t, \sin 2 t\rangle$$

onto the xz-plane is a circle.

Express the $\beta$ decays $n \rightarrow p+e^{-}+\bar{\nu}$ and $p \rightarrow n+e^{+}+\nu$ in terms of $\beta$ decays of quarks. Check to see that the conservation laws for a charge, lepton number, and baryon number are satisfied by the quark $\beta$ decays.