(ii)と(iii)がどうしてこうなるのか分かりません お願いします

b. [1+] Let t(n) be the number of total partitions of n, as defined in Exam- ple 5.2.5. Let g(n) have the same meaning as in Exercise 5.26. Deduce from (a) that g(n) = 2"t(n) for n >1. c. [2+] Give a simple combinatorial proof of (b). 5,37. a. [2+] Let 1=D po(x), pi(x), be a sequence of polynomials (with coeffi- cients in some field K of characteristic O0), with deg pn=n for all nE N. Show that the following four conditions are equivalent: ) Pn(x + y) =DE>o (") Pe(x)pnーk(y), for all n eN. (i) There exists a power series f(u)=aju+azu'+ E K [[u]] such that と P(x)- un expxf(u). (5.110) n! n>0 仮定 NOTE: The hypothesis that deg pPn=nimplies that aj ¥ 0. () E20 Pa(x) = (E>0 Pn(1)). (iv) There exists a linear operator Q on the vector space K[x] of all poly- nomials in x, with the following properties: ●Ox is a nonzero constant ●Qis a shift-invariant operator, i.e., for all aeK,Qcommutes with the shift operator E4 defined by E® p(x)=D p(x +a). ● We have Qpn(x) =D npn-1(x) for all n e P. NOTE: A sequence po, Pi, .. . of polynomials satisfying the above con- ditions is said to be of binomial tvpe. The operator Q is called a delta

分かる方教えて頂きたいです。

2 CONVERSATION A Read the sentences and agree or disagree. Use expressions from the box and give your reasonS. Yeah, you're right. Really? I don't think so. Sorry, but l disagree. I think so, too. Example: Your friend: I think history is a boring subject. You: Really?l don't think so.l like to read about heroes. 1. Your friend: I think English is a difficult language. You: 2. Your friend: Bill Gates is a great hero. You: 3. Your friend: I think our city is a really boring place. You: 4. Your friend: Everything is very expensive in our country. You: Our teacher gives us too much homework! 5. Your friend: You: Number the sentences in order to make a conversation. B Nelson Mandela. I'm writing a paper about a hero. Really? Why is he your choice? Well, he was a great leader who united his country. That's interesting! Who are you writing about?

解答がなくて困っています。 わかる方教えて頂きたいです

| VOCABULARY A Circle the best word to complete each sentence. 1. Educators / Travelers teach people. 2. Doctors / Authors write books for a living. 3. Politicians / Explorers lead research trips. 4. Speakers / Scientists give presentations to people. 5. Musicians/ Ambassadors work for the government. 6. Founders / Activists run their own companies. B Which word is different? Cross out the word that doesn't fit. Example: scientist speaker researcher 1. teacher educator activist 2. director traveler leader politician ambassador 3. doctor explorer author 4. writer C Complete the sentences. Use words from A and B. There may be more than one correct answer. need passports. 1. talk to large groups of people. 2. try to change the world. are leaders of companies. 3. 4. discover new things. 5. Which jobs require a person to be brave? Explain your answer.

この問題は自分でf(x)を求めて積分するという解釈で合っていますか？教えて欲しいです、よろしくお願いします

(1 point) You are given the four points in the plane A - (8,-2), B- (10,6), C = (12,-5), and D = (14,6). The graph of the 14 function f() consists of the three line segments AB, BC and CD. Find the integral | f(z) da by interpreting the integral in terms of sums and/or differences of areas of elementary figures. 14 f(z) da = 8

速度の問題

Ifa ball is thrown into the air with a velocity of 40 ft/s, its height in feet after t seconds is given by y= 40t - 162. (a) Find the average velocity for the time period beginning with t =D 2: (1).5 second (2).1 second (3) .05 second (4) .01 second (b) Find the instantaneous velocity when t 3 2. ft/s ft/s ft/s ft/s ft/s

67の答えがCなのですがおかしくないですか？恐らくCEOの事を書いているのだと思いますがCEOと社長presidentは別の役職で同じではないと思ったのですが

65-67 refer to the following conversation. W: Richard, we were deeply impressed with your presentation this morning. You concentrated on the benefits the customers 65. What did the man do this morning? OEIC (A) He had a talk with an executivetsTENING will get from our new products. That was awesome. The sales manager wants you to give a presentation on the same topic to the board of directors next week. (B) He gave a talk. (C) He made a presentation to the board of directors. (D) He put together handouts. 66. What does the woman suggest? I'm glad you liked it. I'l try my best to please the board of directors. Maybe l could use some technology to supplement my presentation. Don't you think using a video allows the audience to understandit (A) Preparing more informative materials (B) Using a video (C) Getting advice from the sales manager (D) Choosing a new topic M: ,Com, /。 better? W: That's a good idea. You should prepare more extensive handouts as well. I will be free this afternoon, so l can help you put them together. M: I'd appreciate it. Let's make it our top priority to ensure that our executives are satisfied. Even the CEO will be there. 67. What does the man say about next week's presentation? (A) It will take place in the afternoon. (B) It will concentrate on the benefits of video presentations. (C) The president will see it. (D) The sales manager will help them prepare for it. 65B 66A 6

多様体の接空間に関する基底定理の証明です。g(q)=∫〜と定義した関数を微積分学の基本定理を用いながら変形してg(q)=g(0)+∑gᵢuⁱと導出するのですが、これがうまくいきません。 自分は、g(q)の式をまず両辺tで微分して、次に両辺uⁱで積分して、最後に両辺tで積分... 続きを読む

12. Theorem.If{ = (x', , x") is a coordinate system in M at p, then its coordinate vectors d, lp, …… 0,l, forma basis for the tangent space T,(M); and D= E(x) 。 i=1 for all ve T(M). Proof. By the preceding remarks we can work solely on the coordinate neighborhood of G. Since u(c) = Othere is no loss of generality in assuming ど(p) = 0eR". Shrinking W if necessary gives E(W) = {qe R":|q| < } for some 8. Ifg is a smooth function on E(W) then for each 1 <isndefine og (tq) dt du g(9) = for all qe {(W). It follows using the fundamental theorem of calculus that g= g(0) + E&,u' on (W). Thus if fe &(M), setting g = f。' yields f= f(P) + Ex on U. Applying d/ax' gives f(p) = (f /0x)(P). Thus applying the tangent vector e to the formula gives (f) = 0+ E(x'(p) + E Ap)u(x) = E(Px). ず ax Since this holds for all f e &(M), the tangent vectors v and Z Ux') d,l, are equal. It remains to show that the coordinate vectors are linearly independent. But if ) a, o.l, = 0, then application to x' yields dxi 0=24 (P) = 2q d」= 4. In particular the (vector space) dimension of T,(M) is the same as the dimension of M.

波線を引いた式の導出がよく分かりません。u~²を展開して整理するとたしかに1-e²/2は出てくるのですが、eφsinφなどはどうしたら出てくるのでしょうか？

To see this, recall from Appendix C that the Newtonian orbit equation (d'u/dp?)+ u= M/L? has à = helion at p = 0. The hypotheses imply r> L»M SO this is close to the relativistic solution. To obtain a more refined approximation, we use ü in (M/L")(1 + e cos φ) as solution with peri- the relativistic correction term thus 代人 (d'u/dp?) + u = (M/L?) + 3Mz?. A routine computation gives 3m3 (1+e cos p)+ L e? e? 1+ 2 M cos 2p + ep Sin p 6 u= L? as the solution with perihelion at φ = 0. To find the next perihelion we need 3M°ele p+ L* du Me sin L? sin 2p + sinφ+p cos φ 3 dp

（　）に入る答えは②なのですが、他の選択肢がなぜダメなのか分かりません…教えてください！！

) me to takea day off. ② allowed 566 He ( 図図図 D admitted 基本 ③ forgave さと ④ let

1から9までを埋めなければならないのですが、質問の意味もよくわからず…。どなたか質問の意味と解き方を教えて頂けると助かります…！

Vocabulary skills: Collocations Read the text. Then find verbs that collocate with the words and phrases below. Write the infinitive of each verb. KEY VOCABULARY SKILLS STUDY SHEET 7, PAGE 164 The Duke of Edinburgh’'s Award The Duke of Edinburgh's Award (known as the DofE) began in the U.K. in 1956. It gives young people from all backgrounds an opportunity to experience challenge and adventure, acquire new skills, and make new friends. Each year, participants between the ages of 14 and 24 complete many hours of voluntary community work. There are three levels: Bronze, Silver, and Gold. Each one requires increasing commitment, so you can choose which level is best for you. There are four sections at Bronze and Silver level, and five at Gold. These sections combine practical, cultural, and adventurous activities. Achieving a Duke of Edinburgh's Award helps young people to develop a sense of responsibility, and promotes values of persistence, commitment, and enterprise. It has a lasting impact on young people's behavior, skils, and life chances. In 1988, the DofE was expanded to become international. These days, over 635,000 participants around the world are achieving a DofE at any one time. Since the DofE began, Over six million young people in over 123 countries have taken part, and it continues to grow in popularity. 1 an appointment / small talk / progress success / your goals / independence dark / older / crops 2 4 a course / a form / an exercise someone a chance / a talk / feedback 5 awareness / growth / competition a qualification / knowledge / a reputation for something a problem / dificulty / something at first hand a friendship / a taste for something / an idea 6 7 8 9 Predicting Look at the title of the information sheet you’re going to read. Discus with a partner what. think you'll read about. Gnbal Youth Service Day