TITLE: What is Method of Characteristics?
QUESTION [6 upvotes]: I am a final year student of BS Mechanical Engineering and method of characteristics is not a part of our curriculum. In-fact I heard of it first time after finally picking my FYP. My final year project is to design analyze and manufacture a CD Nozzle which would achieve Mach 3. I was told by my supervisor that I would have to build the profile of the nozzle using method of characteristics because he wants the profile to be generalized so that every time the inlet condition changes, the profile changes automatically. My supervisor who himself is only a doctorate has never studied MOC himself so he can not help me. I want to know what it is and how could I start practicing it. Moreover what are the pre-requisites to start learning it. Any help would highly be appreciated.

REPLY [0 votes]: I just found that second volume of Gas Dynamics by Zucrow wholly deals with the use of Method of Characteristics to solve various kinds of flow problems. I hope it will solve my problem.


TITLE: Describe the Kernel of the map from this polynomial ring
QUESTION [2 upvotes]: Let $\mathbb R[x]$ denote the ring of all polynomials with real coefficients. The mapping $f(x)\rightarrow f(1)$ is a ring homomorphism from $\mathbb R[x]$ onto $\mathbb R$.

Question:
  Describe the kernel of this ring homomorphism.

Recall the definition of kernel of a $\left ( ring \right )$ homomorphism:
$\ker\left ( \varphi \right )=\left \{ f\left ( x \right ) \in\mathbb  R\left [ x \right ] \mid \left [ f\left ( x \right ) \right ]\varphi=0 \right \}$
Now, we define:
Define
$\varphi:R\left [ x \right ]\rightarrow R$
$f\left ( x \right ) \mapsto f\left ( 1 \right )$
I was almost certain the kernel is the trivial kernel but my solution sheet dictates otherwise.
Hint is appreciated.
Thanks in advance

REPLY [2 votes]: Hint: This is primarily a counterexample that shows the kernel to be non-trivial, but it also serves as a strong hint. Let $f(x) = x^2 - 3x + 2$. Then $\varphi(f(x)) = f(1) = 1 - 3 + 2 = 0 \implies f(x) \in \ker \varphi$.

Solution:
\begin{align*}
\ker \varphi & = \{\, f(x) \in \mathbb R[x] \mid \varphi(f(x)) = 0 \in \mathbb R \,\}\\
& = \{\, f(x) \in \mathbb R[x] \mid f(1) = 0 \,\}\\
& = \{\, f(x) \in \mathbb R[x] \mid (x - 1)\ \text{is a factor of}\ f(x) \,\}\\
& = \{\, (x -1)g(x) \mid g(x) \in \mathbb R[x] \,\}\\
& = \langle x - 1 \rangle
\end{align*}
That is, $\ker \varphi$ is the ideal generated by $x - 1$.


TITLE: Formulating a logical constraints - integer linear programming
QUESTION [1 upvotes]: If I have two binary(0-1) decision variables: $P_1$ and $P_2$, and my constraint is that:
$P_2$ may be chosen if and only if $P_1$ is not chosen, is this equivalent to the formulation that:
$$P_1 + P_2=1$$

REPLY [2 votes]: You can derive linear constraints via conjunctive normal form as follows.
$$
P_2 \iff \lnot P_1 \\
(P_2 \implies \lnot P_1) \land (\lnot P_1 \implies P_2) \\
(\lnot P_2 \lor \lnot P_1) \land (P_1 \lor P_2) \\
(1 - P_2) + (1 - P_1) \ge 1 \land P_1 + P_2 \ge 1 \\
1 \ge P_2 + P_1 \land P_1 + P_2 \ge 1 \\
P_1 + P_2 = 1
$$