多様体を構成するために、位相空間に完全アトラスを導入するところで質問です。 完全アトラスを導入するメリットとして、この文章の下線部を「異なる座標系を用いたのに同じ計算ができてしまうという問題が解消される」解釈したのですが、そこがよくわかりません。座標系を変えて計算する... Read More

1 Two n-dimensional coordinate systems & and ŋ in S overlap smoothly provided the functions on¯¹ and ŋo §¯¹ are both smooth. Explicitly, if : U → R" and ŋ: R", then ŋ 1 is defined on the open set ε (ur) → ° (UV) V and carries it to n(u)—while its inverse function § 4-1 runs in the opposite direction (see Figure 1). These functions are then required to be smooth in the usual Euclidean sense defined above. This condition is con- sidered to hold trivially if u and do not meet. Č (UV) R" Ĕ(U) n(UV) R" S n(v) Figure 1. 1. Definition. An atlas A of dimension n on a space S is a collection of n-dimensional coordinate systems in S such that (A1) each point of S is contained in the domain of some coordinate system in, and (A2) any two coordinate systems in ✅ overlap smoothly. An atlas on S makes it possible to do calculus consistently on all of S. But different atlases may produce the same calculus, a technical difficulty eliminated as follows. Call an atlas Con S complete if C contains each co- ordinate system in S that overlaps smoothly with every coordinate system in C. 2. Lemma. Each atlas ✅ on S is contained in a unique complete atlas. Proof. If has dimension n, let A' be the set of all n-dimensional coordinate systems in S that overlap smoothly with every one contained in A. (a) A' is an atlas (of the same dimension as ✅).

この問題の途中式を教えてください。答えに載ってなかったので。 答えはH(X)から順に2.0bit、1.5bit、0bitです

roduced in any form without permission 問題 2.2 確率変数 X, Y, Z の確率分布が下表のように与えられている とする。このとき, これらの確率分布に対するエントロピー H(X), H(Y), H(Z) をそれぞれ求めよ. a b C d 1/4 1/4 1/4 1/4 1/4 0 0 0 Px(X) Py (Y) 1/2 1/4 Pz(Z) 1 0

10,68の答えがどうしてこのようになるか教えてください。 分野は重積分のストークスの定理です

By Green's theorem in space (divergence theorem). Prove that that (V x A) - n ds for any closed surface S. S Prove that 10.66. dS ff n ds = 0. where n is the outward drawn normal to any closed surface S. (Hint: Let A = Oc, SS S where c is an arbitrary vector constant.) Express the divergence theorem in this special case. Use the arbitrary property of c. 10.67. If n is the unit outward drawn normal to any closed surface S bounding the region V, prove that fff div n dv = S V Stokes's theorem 40.68. Verify Stokes's theorem for A = 2yi + 3xj - z²k, where S is the upper half surface of the sphere x² + y² + ² = 9 and C is its boundary. Ans. Common value = 9T 10.65. , y = 0,

後1週間後に受験を控えているのですが志望校の過去問の答えが公表されてなくて困ってます。赤本も出てないです。なのでできれば解答解説、せめて解答だけでも教えて下さい。お願いします。

[I]次の問いに答えよ。 (1) a, bを実数の定数とする。 3次方程式 + (a-3)- (a 1 3)α+b=0の実数解がz= 1 だけであるとき, aの値の範囲とbの値を求めよ。 (2) tan a = 13, tan β = 4, tany=2のとき, tan(α+B+)を求めよ。 (3) 2-1< 31275< 2" を満たす自然数nをすべて求めよ。ただし, log1o2= 0.301, log1o3 = 0.477 とする。 (4) 5個の整数 a, 1, 3, 9, 10からなるデータの分散が12のとき, aの値とこのデータの平均 値を求めよ。 (5) さいころをn回続けて投げるとき, k回目に出る目の数をXxとし, Y,を Y,= Xi+ X2++X, とする。Y,が7で割り切れる確率を pn とする。 このとき, Phを求めよ。 [II] f(c) = -とおくとき, 次の問いに答えよ。 (1) f(a) = zの2つの解を α, β (a<β)とする。 このとき, a, βの値を求めよ。 (2) f(f(a))の値を求めよ。 (3) 関数 f(F(x)) を求めよ。 (4) 方程式 f(f(a)) = " の解を求めよ。

問題としてはこのURLのやつでexercise2.2.9の問題です。 2.2.9. Define T : ℓ^2(Zn ) → ℓ^2(Zn ) by (T(z))(n) =z(n + 1) − z(n). Find all eigenvalues of T.... Read More

16:22マ l 全 の Exerc: 164/520 matrices, convolution operators, and Fourier r operators. 2.2.9. Define T:l'(Zn) - → e°(ZN) by ニ Find all eigenvalues of T. 2.2.10. Let T(m):e'(Z4) → '(Z) be the Fourier multipliei (mz)' where m = (1,0, i, -2) defined by T (m)(2) = i. Find be l(Z4) such that T(m) is the convolutior Tb (defined by Th(Z) = b*z). ii. Find the matrix that represents T(m) with resp standard basis. 2.2.11. i. Suppose Ti, T2:l(ZN) → e(ZN) are tra invariant linear transformations. Prove that th sition T, o T, is translation invariant. ii. Suppose A and B are circulant NxN matric directly (i.e., just using the definition of a matrix, not using Theorem 2.19) that AB is Show that this result and Theorem 2.19 imp Hint: Write out the (m + 1,n+1) entry of the definition of matrix multiplication; compare hint to Exercise 2.2.12 (i). iii. Suppose b,, bz e l'(Zn). Prove that the cor Tb, o Tb, of the convolution operators Tb, and convolution operator T, with b = 2 bz * b.. E Exercise 2.2.6. iv. Suppose m,, mz € l"(Z). Prove that the cor T(m2) ° T(m) and T(m) is the Fourier multiplier operator T) m(n) = m2(n)m」(n) for all n. v. Suppose Ti, T2:l"(Zw) → e'(Zn) are linear tra tions. Prove that if Ti is represented bya matri respect to the Fourier basis F (i.e., [T; (z)]F =A Tz is represented by a matrix Az with respect t the composition T20T, is represented by the ma with respect to F. Deduce part i again. Remark:ByTheerem 2.19, we have just proved of the Fourier multiplier operat Aresearchgate.net - 非公開

問題としてはこのURLのやつでP.144 exercise2.2.1の問題です。 https://www.researchgate.net/profile/Seyed-Yahya-Moradi/post/Can-anyone-suggest-a-wavele... Read More

留学先の統計学の問題です。 4、5、2問あります。 後、数時間で提出となり、焦っております。 どなたかお力添えをお願いします😭😭😭 時差があり、深夜回答はたいへんありがたいです。 朝までにお願いします。

4. The temperature of the Earth at different sites can be measured in two ways. One, by taking readings using thermometers on the ground (x), which is extremely tedious and time consuming. Second, bylasers positioned on satellites revolving about the Earth (v). which ie a less accurate method and may be biased. The readings for both are given below: Ground Therm., x Satellite Laser, y Site 1 4.6 4.7 2 17.3 19.5 3 12.2 12.5 4 3.6 4.2 5 6.2 6.0 6 14.8 15.4 7 11.4 14.9 8 14.9 17.8 emignaia 9 9.3 9.7 10 10.4 10.5 11 7.2 7.4 You would like to test the claim that the Laser method gives a significantly higher reading than the ground therm. method. You may assume that the difference between pairs of scores is approximately normal. A) Would testing for a difference in means or a paired difference test be better to use here? Why? B) Perform the test you concluded in part A). 200 seetV C) Would you have any reservations about yourinference? Why?

fが不連続の時、xを導出すると言う問題です。 2番以外の問題が分かりません。 宜しければご教授お願いします。

Find values of x, if any, at which is not continuous. /、 沙ゅ : グー8 5 、 † : - オイバ 。 +ス メ ナw ~ 一 / る 2%+ | -] | /特」 2すめてイだfr S ナリ * 3 |

x=tany を微分すると、 dx=1/cos^2y dyになりますよね？？ このdyとはどういう意味ですか？？( .. )