國立中正大學八十一學年度應用數學研究所 碩士班研究生招生考試試題
基礎數學
I.(20%) Test for convergence or divergence of the following infinite series (a)
∞
X
n=1
cos(πn)
n (b)
∞
X
n=1
sin(πn) n (c)
∞
X
n=2
1
n(log n)p (p > 1) (d)
∞
X
n,m=1
1 n2+ m2
II.(15%) Compute the following integrals and differentiation (a)
Z a12 0
dt a2−√
t (a < 1) (b)
Z 2π
0 sin x2n+1dx (n ≥ 0 integer) (c)∂f
∂z where f (x, y, z) = φ(xe−z, ye−2z) · e−3z, φ(u, v) = euv
III.(15%) Find the maximum and minimum of f (x) = 3x − 2y + z subject to the condition x2+ 3y2+ 6z2 = 1.
IV.(10%) Let A, B be compact subsets of Rn. f : A → B is 1-1, onto and continuous.
Show that f−1 is continuous.
V.(5%) Given 2 × 2 matrix A = a b c d
!
with det A < 0. Show that A is diagonal- izable.
VI.(5%) Let A be an n×n matrix with the property Ak = 0 for some k > 0, integer.
Show that both I − A, I + A are not invertible.
1
VII.(10%) Let A =
1 a1 · · · an
−a1 1 ... . .. 0
0
−an 1
(a) Find A−1 (b) Find all eigenvalues of A.
VIII.(10%) (a) Define an “inner product” space.
(b) State and prove the Cauchy-Schwarz inegaulity for an inner product space.
IX.(10%) Prore or disprove following statement:
Let V be any vector space T : V → V is a linear map. If T is 1-1, then T is onto!
統計學
(20%)1. Let X1, X2, · · · , Xn be a random sample of size n from the distribution with p.d.f. f (x; θ) = θxθ−1, 0 < x < 1, 0 < θ < ∞.
(5%)(a) Find the method of moments estimator of θ.
(5%)(b) Find the maximam likelihood estimator of θ.
(10%)(c) Let n = 1, find the most powerful test with significant level α = .05 for testing H0 : θ = 1 versus H1 : θ = 2.
(20%)2. Let X1, X2, · · · , Xn be a random sample of size n from N(µ, σ2), where both µ and σ2 are unknown. For testing H0 : σ2 = σ02 verses H1 : σ2 6= σ02, show that the likelihood ratio test is equivalent to the χ2 (Chi-squared) test for variances.
(25%)3. Consider the simple linear regression model:
Yi = βi+ β1xi+ ǫi i = 1, 2, · · · , n.
where var(ǫi) = σ2 and cov(ǫi, ǫj) = 0 i 6= j.
(10%)(a) Derive the least squares estimates of β0 and β1 (denoted by ˆβ0 and ˆβ1
respectively).
(10%)(b) Show that E( ˆβ1) = β1 and find V ar( ˆβ1) (5%)(c) Show that cov(Y , ˆβ1) = 0
(25%)4. Let X1, X2, · · · , Xnbe a random sample of size n from U(0, θ), the uniform distribution over (0, θ).
(5%)(a) Show that X(n) is a sufficient statistic for θ, where X(n)= max(X1, · · · , Xn).
(10%)(b) Construct a 100(1 − α)% confidence interval for θ based on the sufficient statistic X(n).
(10%)(c) Find the best unbiased estimator of θ.
(10%)5. Let X be a random variable with continuous distribution function F , and let F−1 be the inverse of F . Show that the random variable Y = F (X) is distributed as U(0, 1).
計算統計
1 Part I: 數值計算方法
Reminder: The answer will not be accepted without proper explanation.
1. Let P (x) = 9.5x20 + 8.1x16 + 7.2x12 + 6.5x8. What is the least number of multiplications required for evaluating P (x)? (10%)
2. What is the polynomial P (x) with the least degree which satisfies P (0) = 1, P′(0) = 0, P (1) = 4 and P′(1) = 9? (10%)
3. Let f (x) = (x − 1)3, x ∈ R. Suppose that the initial x(0) = 0 and x(n), n ≥ 1, is defined by the Newton’s method. Will the sequence {x(n)} converges to 1? If so, what is the order of convergence? (15%)
4. Let the system Ax = b be nonsingular where A ∈ Rn×n; x, b ∈ Rn. In particular, we may actually solve the perturbed system Ay = b + ∆b with k∆bk small under some vector norm. Let cond (A) be the condition number of A under some matrix norm. Show that cond(A)1 k∆bkkbk ≤ ky−xkkxk ≤ cond(A)k∆bkkbk . (15%)
2 Part II: 計算機系統概念
1. 試用您熟悉的一種程式語言 (譬如 C, Fortran, or Pascal, etc.) 把計算機系 統是如何地 來計算出ex寫成一個副程式。(10%)
2. 就您所熟悉或使用過的兩種計算機系統 (譬如 IBM PC and SUN Work Station, etc.), 簡述他們的特性以及比較他們之間的異同 (可以從軟、 硬體和相關方面來回答這個問題)。
(15%)
3. 您知道計算機系統中有那些硬體部份可以用來儲存資料呢? 如何的歸類? 並依您的歸類
方式略述他們的特性和差異性。 進一步我們要透過計算機系統來儲存和找尋資料的時候, 則系統是如何地來幫助我們呢? (可以就您所熟悉的 File and Data Structures 說明 之)。(15%)
4. 一個 Computer Word (譬如說有 4 bytes) 可能存放著一個指令 (Instruction), 也有可 能被解釋成放的是一組資料 (Data), 計算機系統是如何地來區別呢? 並請您略述一下他 們各有那些歸類方式? 例如有那些 Instruction Formats 以及那些不同型態的 Data?
(可以就您所熟悉的概念略述之)。(10%)
線性代數
1. For vectors x = [x1, . . . , xn] and y = [y1, . . . , yn] in the vector space Rn, the length and the inner product are given by the following:
kxk2 = x21+ . . . + x2n, hx, yi =
n
X
j=1
xjyj. Suppose that v1, . . . , vm, m ≤ n, is an orthonomal set of Rn, i.e.
hvi, vji =
(1 if i = j, 0 if i 6= j.
Prove that for any vector g in Rn,
m
X
j=1
hg, vji2 ≥ kgk2. (15%)
2. Let W and V be vector subspaces of Rn. Prove that
dim W + dim V = dim (W + V ) + dim(W ∩ V ). (15%) Here dim X denotes the dimension of X.
3. Find real constants c0, c1 and c2 so that the following integral has minimal value.
Z 1
0 (ex− c0− c1x − c2x2)2dx. (20%) 4. For any n × n matrix A, we define eA= P∞
n=0 An
n!. (a) Prove that eA+B = eAeB if AB = BA. (10%) (b) Find eA if A =
"
2 3 0 2
#
. (5%)
(c) Find eB if B =
"
0 1 1 0
#
. (5%)
(d) Find the general solution to dudt = Au if A =
1 1 1 0 1 1 0 0 1
. (10%)
5. Let A =
0 1 0
0 0 1
6 −11 6
. Find max
kxk=1kAxk and min
kxk=1kAxk. (20%)
高等微積分
(20%)#1. Let f (s) = P∞
n=1n−s. Show that f is continuous on [2, ∞).
(20%)#2. Let f (x) = 3x2+ x + 100, ∀x ∈ R′. Show that f is not uniformly continuous on R1.
(15%)#3. S ⊆ Rn. Suppose for each x in S there exists an open set N(x) such that N(x) ∩ S is countable. Show that S is countable.
(15%)#4. Let f be an one to one and real-valued continuous function on [0, 1].
Show that f is strictly monotonic on [0, 1].
(15%)#5. Let f be a positive continuous real-valued function on [0, 1]. Suppose M = max
0≤x≤1f (x). Show that
n→∞lim(
Z 1
0 fn(x)dx)n1 = M.
(15%)#6. f and the derivative f′are continuous on [0, ∞). Suppose thatR0∞|f′(x)|dx <
∞. Show that the limit of f(x) exists as x tends to ∞.
微 分方程
1. Solve the following Differential Equations (50%) a. y′ = x + 4y − 2
4x − y + 1 b. y′ = y
yey − 2x c. y′ = 3y
x + y
d. y′ = x
x2y + y + y3 (hint: let u = x2+ 1) e. x2y′′+ xy′+ y = 0
2. Solve the following system: Y′ = AY + B where Y = y1
y2
!
A = 1 0
6 −1
!
B = 1 t
!
(15%)
3. By the method of infinite series, find two linealy independent solutions for y2′′xy′+ 2y = 0 (15%)
4. Let y = f (x) satisfy y′′ = xy, y(0) = 0 y′(0) = 1.
(a) Show that f (x) is strictly positive in (0, ∞).
(b) What is lim
x→∞f (x)? (10%)
5. Prove the uniqueness of the solution for the differential equation y′ = sin y, y(o) = 1. (10%)
數值分析
Reminder: The answer without the proper explanation will not be accepted.
1. Suppose a simple zero α of a C2 function f : IR → IR is to be approximated by applying the Newton’s method under the tolerance ǫ. We may have two possible stopping criteria:
(A) |f(xn)| ≤ ǫ, or (B) |xn+1− xn| ≤ ǫ,
where {xn} is the sequence of Newton’s iterates in the program. Which criterion is better? Why?
(15%) 2. Given a data table as follows:
x -3 -2 -1 0 1 2
p(x) -62 -15 0 1 6 33 ,
where p(x) is a polynomial with deg(p) ≤ 5. What is the expression of p(x)?
(10%)
3. Let I(f ) = R01f (x)dx where f ∈ C[0, 1]. A quadrature of I(f) is defined by In(f ) = Pni=1aif (xi) for some nodes xi ∈ [0, 1] and coefficients ai. Also let P3 = {p(x) : p(x) is a polynomial on [0, 1] with deg(p) ≤ 3}. Show that the quadrature In(f ) derived from the Simpson’s rule is exact for all p in P3. Hint:
I(p) =In(p). (20%)
4. Given an initial value problem (IVP)
dy/dx = f (x, y), x ∈ [0, 1], y(0) = y0 ∈ R,
where f is Lipschitz continuous in y. Derive a weakly stable numerical method for solving (IVP). (15%)
5. For any matrix A ∈Rn×n, it is known that A = Q · R where Q is orthonormal and R is upper triangular in Rn×n. Suppose
A =
1 1 1
2 −1 −1
2 −4 5
. What are Q and R? (15%)
6. Given a linear system A · x = b where A =
−4 1 0
1 −4 1
0 1 −4
and b = [1, 1, 1, ]T. Please derive an iterative method for solving the system whose iterates convege for any choice of initial guess in R3. (15%)
7. Let
B =
5 −1 0 0
−1 3 2 0
0 2 3 1
0 0 1 1
.
Show that all the eigenvalues of B must lie in the interval [0, 6]. (6%)
