Complete the following sentences by choosing the word that best fits the context, based on information you infer from the use of the italicized word. Some word pairs will be antonyms, some will be synonyms, and some will simply be words often used in the same context. Ophelia exhorted her friends in their most _____ moments, compassionately urging them to move forward and to live life once again. A. pertinent, B. dolorous, C. gargantuan, D. grandiloquent, E. affable.

Complete the following sentences by choosing the word that best fits the context, based on information you infer from the use of the italicized word. Some word pairs will be antonyms, some will be synonyms, and some will simply be words often used in the same context. The king's royal whisperer was a sycophant who acted as a social ____ in the evenings, learned the latest gossip and rumors, and reported it to his liege every day at lunch. A. gadfly, B. hyperbole, C. epistle, D. lethargy, E. altercation.

Complete the following sentences by choosing the word that best fits the context, based on information you infer from the use of the italicized word. Some word pairs will be antonyms, some will be synonyms, and some will simply be words often used in the same context. The colonel marched his battered regiment northward to avoid a[n] _____ with the colonial militias comprised of vengeful relatives and formidable marksmen. A. lexicon, B. hue, C. sanction, D. acrimony, E. altercation.

List as many words as you can think of that contain the prefix circum- and the prefix intro-. Try to define each word literally.

Derive the differential equation(s) describing the given physical situation. A rocket is shot vertically upward from the surface of the earth. After all its fuel has been expended, the mass of the rocket is a constant m. Use Newton's second law of motion and the fact that the force of gravity varies inversely as the square of the distance to find the differential equation for distance y from the earth's center to the rocket at time t after burnout. State appropriate conditions at t=0 associated with this differential equation.

Let $\phi$ be a 3cnf-formula. An $\neq$-assignment to the variables of $\phi$ is one where each clause contains two literals with unequal truth values. In other words, an $\neq$-assignment satisfies $\phi$ without assigning three true literals in any clause. a. Show that the negation of any $\neq$-assignment to $\phi$ is also an $\neq$-assignment. b. Let $\neq S A T$ be the collection of 3cnf-formulas that have an $\neq$-assignment. Show that we obtain a polynomial time reduction from 3SAT to $\neq S A T$ by replacing each clause $c_{i}$ $\left(y_{1} \vee y_{2} \vee y_{3}\right)$ with the two clauses $\left(y_{1} \vee y_{2} \vee z_{i}\right)$ and $\left(\overline{z_{i}} \vee y_{3} \vee b\right)$, where $z_{i}$ is a new variable for each clause $c_{i}$, and b is a single additional new variable. c. Conclude that $\neq S A T$ is NP-complete.

Let P be any nontrivial property of the language of a Turing machine. Prove that the problem of determining whether a given Turing machine’s language has property P is undecidable. In more formal terms, let P be a language consisting of Turing machine descriptions where P fulfills two conditions. First, P is nontrivial-it contains some, but not all, TM descriptions. Second, P is a property of the TM’s language-whenever $L\left(M_{1}\right)=L\left(M_{2}\right)$, we have $\left\langle M_{1}\right\rangle \in P$ iff $\left\langle M_{2}\right\rangle \in P$. Here, $M_{1}$ and $M_{2}$ are any TMs. Prove that P is an undecidable language.

Graph each point in coordinate space. (2, 3, 0)

Complete the following sentences by choosing the word that best fits the context, based on information you infer from the use of the italicized word. Some word pairs will be antonyms, some will be synonyms, and some will simply be words often used in the same context. Historians discovered the lost epistles of the famous physicist, but only a few scientists could follow the obscure _____. A. sycophant, B. altercation, C. hue, D. lexicon, E. sanction.

Complete the sentence in a way that shows you understand the meaning of the italicized vocabulary word. The old medium attempted to evoke the spirit of...

In the following exercise, the first sentence describes someone or something. Infer information from the first sentence, and then choose the word from the Word Bank that best completes the second sentence. altercation, inscrutable, hyperbole, formidable, incisive, expound, dolorous, humility. The government's explanation of the strange lights in the desert was far too vague to satisfy the local residents, who vowed to contact their congressional representatives if some real answers were not provided. From this sentence, we can infer that the citizens want the government to _____ on the explanation of the strange lights.

a) Show that in the finite complement topology on $\mathbb{R}$, every subspace is compact. (b) If IR has the topology consisting of all sets A such that $\mathbb{R}-A$ is either countable or all of $\mathbb{R}$, is [0, 1] a compact subspace?

How did economic opportunities expand for African Americans, Mexican Americans, and women during World War I?

Let A={Dirk, Johan, Frans, Sarie}, B={Frans, Sarie, Tina, Klaas, Henrika}, C={Hans, Frans}. Find the numbers indicated. $n(A \cup C)$