Extra exercises, 01325Mathematics 4 · 2013-02-08 · Problem 106 (Extension of Problem 222, Exam...

Extra exercises, 01325 Mathematics 4

Problem 1: Let V denote the subset of C[−π, π] consisting of all finitelinear combinations of functions

1, cosx, cos 2x, . . . , cosnx, . . . , sin x, sin 2x, . . . sin nx, . . . .

(i) Show that V is a subspace of C[−π, π].

(ii) Is V closed in C[−π, π]? Hint: from Fourier analysis (e.g., example6.17 in the MAT 2 book from 2009) it is known that for x ∈ [−π, π],

∣∣∣∣∣x2 −

(π2

3+ 4

N∑

n=1

(−1)n

n2cosnx

)∣∣∣∣∣ ≤ 4∞∑

n=N+1

1

n2.

You may use this result without proof.

Problem 4: Consider the following functions defined on R (see p.20 for thedefinition of χ[a,b]):

f1(x) = x,

f2(x) = x2 − 1,

f3(x) = (x2 − 1)χ[−1,2](x),

f4(x) = (x2 − 1)χ[−1,1](x),

f5(x) = e−|x|,

(i) Make a rough sketch of the graph of each of the functions.

(ii) Which functions have compact support? Determine the support forthese functions.

(iii) Which functions belong to C0(R)?

(iv) Which functions belong to Cc(R)?

(v) Which functions belong to L1(R)?

1

Problem 6:

(i) Show that we can define an equivalence relation on L1(R) by

f ∼ g ⇔∫ ∞

−∞

|f(x)− g(x)| dx = 0.

Now let ∼ denote an equivalence relation on any set V.

(ii) Show that for any x1, x2 ∈ V we have

Ex1= Ex2

or Ex1∩ Ex2

= ∅.

(iii) Argue that there exists a collection {Exi}i∈I of elements in V such that

V =⋃

i∈I

Exiand Exi

∩ Exj= ∅ if i 6= j.

In words, the result in (iii) shows that an equivalence relation splits the setV into a disjoint union of equivalence classes. In the case of L1(R) we obtaina normed space if we identify all elements within a given equivalence class.Thus, strictly speaking L1(R) is not a space of functions, but a space ofequivalence classes of functions.

V

Ex1Ex2

Ex3. . . . . .

Problem 7 Let p ∈ [1,∞[ and consider the mapping

T : Lp(−2, 2) → Lp(−2, 2), (Tf)(x) := xf(x).

(i) Show that T indeed maps Lp(−2, 2) into Lp(−2, 2).

(ii) Show that T is linear and bounded.

(iii) Calculate the norm of the operator T .

Hint: let f = χ[ǫ,2] for some ǫ > 0 and show that

ǫ ||f ||p ≤ ||Tf ||p ≤ 2 ||f ||p

2

Problem 8 Consider the function

f(x) =1

|x|1/3 χ]0,∞[(x), x ∈ R.

(i) Make a draft of the function f.

(ii) Check whether there exists any p > 0 for which f ∈ Lp(R).

Problem 9 Consider the vector space C2. For x ∈ C2, we write x =(x1, x2), x1, x2 ∈ C.

(i) Does the expression

||(x1, x2)|| := |x1|

define a norm on C2?

(ii) Does the expression

||(x1, x2)|| := |x1| − |x2|


(iii) Does the expression

||(x1, x2)|| := max (|x1|, |x2|)


Problem 101 (extension of Problem 221 from the Exam 2011) Con-sider the function

f(x) := exχ[0,1](x).

(i) Calculate the convolution (f ∗ f)(x), x ∈ R.

(ii) Calculate the Fourier transform

f̂ ∗ f(γ), γ ∈ R.

3

Problem 102 Let r : [a, b] →]1,∞[ be a continuous function and considerthe weighted L2-space

L2r(a, b) = {f : [a, b] → C

∣∣∫ b

a

|f(x)|2r(x) dx <∞}.

We know (you are not expected to show this) that L2r(a, b) is a Hilbert

space with respect to the inner product

〈f, g〉r =∫ b

a

f(x)g(x)r(x) dx.

(i) Show that L2r(a, b) is a subspace of L2(a, b).

(ii) Show that if {ek(x)}∞k=1 is an orthonormal system in L2r(a, b), then

{ek(x)√r(x)}∞k=1 is an orthonormal system in L2(a, b).

Problem 103 Consider the Legendre differential equation, which, for a givenand fixed parameter λ ∈ R, is

(1− x2)d2u

dx2− 2x

du

dx+ λu = 0. (1)

Assume that (1) has a power series solution,

u(x) =∞∑

k=0

ckxk.

(i) Show that for k ≥ 2,

ck+2(k + 2)(k + 1)− ck(k(k + 1)− λ).

(ii) Show that if (1) has a solution u 6= 0 which is a polynomial, thennecessarily the parameter λ has the form

λ = ℓ(ℓ+ 1)

for some ℓ ∈ {0, 1, 2, . . .}.

4

Problem 104 Let

V := {f ∈ L2(R)∣∣ f is infinitely often differentiable, f ′ ∈ L2(R), [x 7→ xf(x)] ∈ L2(R)}.

Consider the mappings defined by

D : V → V, (Df)(x) := f ′(x),

and

M : V → V, (Mf)(x) := xf(x).

The domain V is chosen such that the operatorsD andM are well defined.

(i) Show that D and M are linear mappings

(ii) Calculate the operator

DM −MD.

Problem 105 We will consider the modulation operator Eb for complexvalues of the parameter b. That is, for b ∈ C we let Eb act on the functionf : R → C by

(Ebf)(x) := e2πibxf(x), x ∈ R.

(i) Show that the function

f(x) :=1√

1 + x2

belongs to L2(R).

(ii) For which values of the parameter b ∈ C does the operator Eb define abounded operator from L2(R) into L2(R)?

Hint: Find the values of b ∈ C for which Ebf ∈ L2(R) for the functionf in (i).

5

Problem 106 (Extension of Problem 222, Exam 2011) Consider acomplex Hilbert space H and a bounded linear operator

T : H → H.

We say that a number λ ∈ C is an eigenvalue for T if there exists an eigen-vector, i.e., a vector v 6= 0 such that

Tv = λv.

(i) Assume that λ ∈ C is an eigenvalue for T, and let v denote a corre-sponding eigenvector with ||v|| = 1. Calculate the numbers

〈Tv,v〉 and 〈v, Tv〉.

(ii) Assume that T is self-adjoint. Show that all the eigenvalues are realnumbers.

Now assume that H has an orthonormal basis {ek}∞k=1 and that for eachk ∈ N there exists a number λk ∈ R such that

Tek = λkek.

(iii) Show that for any f ∈ H,

T f =

∞∑

k=1

λk 〈f, ek〉 ek.

Problem 107 Assume that a 2π-periodic function f : R → C is given by

f(x) =∞∑

n=−∞

dneinx, (2)

where the series∑∞

n=−∞ dn is absolutely convergent. The goal of this exerciseis to show that then the coefficients dn are necessarily the well known Fouriercoefficients on complex form.

(i) Show that the series in (2) has a convergent majorant series.

(ii) Calculate for each m,n ∈ Z the integral

∫ π

−π

einxe−imx dx.

Hint: consider the cases m = n and m 6= n separately.

6

(iii) Prove using (ii) that for any m ∈ Z,

∫ π

−π

∞∑

n=−∞

dneinx e−imx dx = 2πdm.

Hint: The order of integration and summation can be interchangedby Theorem 5.34 in the MAT 2 book from 2012 (Theorem 5.33 in theversion from 2009), or see Theorem 5.3.7 in the MAT4 book.

(iv) Conclude using (iii) that

dn =1

2π

∫ π

−π

f(x) e−inx dx, ∀n ∈ Z.

Problem 108 (This is a quite unusual exercise.)

Consider the function

f(x) :=1

1 + x2, x ∈ R.

(i) Show that f ∈ L2(R).

Hint: By Exercise 5.3 we know that f ∈ L1(R).

(ii) Does the function f have any vanishing moments?

(iii) Argue that there exists a function φ ∈ L2(R) such that

φ̂(γ) =1

1 + γ2, γ ∈ R. (3)

Hint: Use a result in Section 7.2

(iv) One can show that the function φ in (iii) belongs to L1(R). Does thefunction φ have any vanishing moments?

(v) Does the function φ in (iii) generates a multiresolution analysis?

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Technical University of DenmarkPage 1 out of 3 pages

Written exam. Date: June 6, 2008

Course name: Mathematics 4: Real Analysis, Course no. 01325

Part I : Problems 201 – 204 (6 questions)Part II: Problems 5 – 7 (6 questions)

The exam consists of 7 problems, all together with 12 questions. The 12 ques-tions are weighted equally. In order to obtain full credit, you are required toprovide complete arguments. The answers will be judged as a whole. Theanswers can be given in English or Danish.

Problem 201 (contained in Ex. 3.9) Consider a sequence {wk}∞k=1 ofpositive real numbers, and define the weighted ℓ1-space ℓ1w(N) by

ℓ1w(N) :=

{{xk}∞k=1 | xk ∈ C,

∞∑

k=1

|xk|wk <∞}.

(i) Show that the expression || · || given by

||{xk}∞k=1|| :=∞∑

k=1

|xk|wk

defines a norm on ℓ1w(N).

We now consider the special choice

wk := 2k, k ∈ N.

(ii) Show that ℓ1w(N) is a subspace of ℓ1(N).

The set of problems continues!

8

Problem 202 (Contained in Ex. 6.10) Consider the linear operator

U : L2(R) → L2(R), (Uf)(x) = f(2x− 2), x ∈ R.

(i) Show that U is bounded on L2(R).

(ii) Compute the adjoint operator U∗.

Problem 203 (Ex. 10.6) Consider the symmetric B-splines B2 and B3.Find an expression for the Fourier transform of the convolution B2 ∗B3, i.e.,determine the function

B̂2 ∗B3.

Problem 204 (Ex. 4.2) Let H be a Hilbert space with inner product 〈·, ·〉and associated norm || · ||. Show that for all u,v ∈ H,

||u+ v||2 + ||u− v||2 = 2

(||u||2 + ||v||2

).

Problem 5-7 Taken out, as they are not relevant for Mathematics 4 in itspresent version.

The set of problems is completed!

9


TEST!! Date: May 6, 2009

Course name: Mathematics 4, Course no. 01325

Allowed aids: All aids.

The test consists of 6 problems, all together with 12 questions. The 12 ques-tions are weighted equally. In order to obtain full credit, you are required toprovide complete arguments. The answers will be judged as a whole. Theanswers can be given in English or Danish.

Problem 205 (Ex. 1.7) Consider the function

f(x) := −xe−x, x ∈ [0,∞[.

Calculate the number

inf{f(x)∣∣ x ∈ [0,∞[}.

Can “infimum” be replaced by “minimum”?

Problem 206 (Ex. 6.15) For f ∈ L2(−π, π), the complex Fourier coeffi-cients are defined by

ck :=1

2π

∫ π

−π

f(x)e−ikx dx, k ∈ Z.

Show that the integral defining ck is well defined for f ∈ L2(−π, π), i.e., that∫ π

−π

∣∣f(x)e−ikx∣∣ dx <∞.


10

Problem 207 (Ex. 4.15) Let {vk}∞k=1 denote a sequence in a Hilbert spaceH. The purpose is to prove Lemma 4.4.4. in the book, which claims that (a)and (b) below are equivalent:

(a) {vk}∞k=1 is complete;

(b) If v ∈ H and 〈v,vk〉 = 0 for all k ∈ N, then v = 0.

Do the following:

(i) Show that if v ∈ H and 〈v,vk〉 = 0 for all k ∈ N, then 〈v,w〉 = 0 forall w ∈ span{vk}∞k=1.Hint: show first that 〈v,w〉 = 0 for all w ∈ span{vk}∞k=1.

(ii) Use (i) to show that if (a) holds, then (b) holds.

(iii) Prove that if (a) does not hold, then (b) does not hold.Hint: Let W := span{vk}∞k=1, and use Theorem 4.3.5 to argue that if(a) does not hold, then W⊥ 6= {0}.

Problem 208 (Ex. 5.17) Let p ∈ [1,∞[ and consider the mapping

T : Lp(R) → Lp(R), (Tf)(x) := f(3x+ 2).

(i) Show that T indeed maps Lp(R) into Lp(R).

(ii) Show that T is linear and bounded.

(iii) Consider the function

f(x) :=1

x2χ[1,∞[(x),

and show that f ∈ Lp(R) for all p ∈ [1,∞[.


11

Problem 209 Let ψ denote the Haar wavelet. We know that ψ can beconstructed from the multiresolution analysis generated by the function φ =χ[0,1[.

(i) Show that for f ∈ L2(R),

〈f, Tkφ〉 =∫ 1

0

f(x+ k) dx.

(ii) Consider a compactly supported function f ∈ L2(R) and the expansion(8.14) in the book. Show that the term

∑

k∈Z

〈f, Tkφ〉Tkφ

actually is a finite sum. Hint: Assume that supp f ⊆ [−N,N ] andconsider ck := 〈f, Tkφ〉 for |k| ≥ N + 1.

Problem 210 (Ex. 8.11) Let φ ∈ L2(R), and denote its Fourier transformby φ̂.

(i) Show that {Tkφ}k∈Z is an orthonormal system if and only if

〈φ, Tkφ〉 ={1 if k = 0;

0 if k 6= 0.

Let

Φ(γ) :=∑

n∈Z

|φ̂(γ + n)|2.

One can show (you are not expected to do this) that

〈φ, Tkφ〉 =∫ 1

0

Φ(γ)e2πikγ dγ.

(ii) Show that {Tkφ}k∈Z is an orthonormal system if and only if Φ(γ) = 1for γ ∈ R.

Hint: By Example 6.4.3 in the book, the numbers∫ 1

0Φ(γ)e2πikγ dγ are Fourier

coefficients for the 1-periodic function Φ.


12


Written exam. Date: June 4, 2009

Course name: Mathematics 4, Course no. 01325


The exam consists of 5 problems, all together with 10 questions. The 10 ques-tions are weighted equally. In order to obtain full credit, you are required toprovide complete arguments. The answers will be judged as a whole. Theanswers can be given in English or Danish.


f(x) = x− x2.

Determine the numbersup{f(x)

∣∣ x ∈ [0, 4]}.Problem 212 (Ex. 4.6) Consider the vector space ℓp(N) for some p ∈ [1,∞[,equipped with the usual norm

||{xk}∞k=1||p =(

∞∑

k=1

|xk|p)1/p

.

(i) Consider the vectors

x = (1, 0, 0, . . . ), y = (0, 1, 0, 0, . . . ),

and show that

||x||p = ||y||p = 1, ||x+ y||p = ||x− y||p = 21/p.

(ii) Assume that p 6= 2. Show that the norm || · ||p does not come from aninner product. Hint: use Theorem 4.1.4.

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Problem 213 Consider a Hilbert space H and let W be a closed subspaceof H. Let {ek}∞k=1 be an orthonormal basis for W . Define the operatorP : H → H by

Pv :=∞∑

k=1

〈v, ek〉ek, v ∈ H.

(i) Show that if w ∈ W and u ∈ W⊥, then

P (w + u) = w.

(ii) Show that if w ∈ W and u ∈ W⊥, then

||w + u||2 = ||w||2 + ||u||2.

(iii) Show that ||Pv|| ≤ ||v|| for all v ∈ H.Hint: By Theorem 4.3.5, any v ∈ H can be written

v = w + u for some w ∈ W, u ∈ W⊥.

Problem 214 (Ex. 5.20) Let p ∈ [1,∞[ and consider the linear mapping

T : Lp(0, 2) → Lp(0, 2), (Tf)(x) := xf(x).

One can show (you are not expected to do this) that for 0 ≤ a ≤ b,

ap∫ b

a

|f(x)|p dx ≤∫ b

a

|xf(x)|p dx ≤ bp∫ b

a

|f(x)|p dx. (4)

(i) Use (4) to argue that T indeed maps Lp(0, 2) into Lp(0, 2) and isbounded with

||T || ≤ 2.

(ii) Calculate the norm of the operator T .Hint: let f = χ[ǫ,2] for some ǫ ∈ [0, 2[ and use (4) to show that

ǫ ≤ ||Tf ||p||f ||p

≤ 2.

14

Problem 215 (Contained in Ex. 10.11) Let m ∈ N, and consider theB-spline Nm. One can show (you are not expected to do this) that thereexist constants A,B > 0 such that

A ≤∑

k∈Z

|N̂m(γ + k)|2 ≤ B, γ ∈ R.

(i) Show that the function

G(γ) :=∑

k∈Z

|N̂m(γ + k)|2

is 1-periodic.

Define the function ϕ ∈ L2(R) by its Fourier transform ϕ̂ via

ϕ̂(γ) :=1√G(γ)

N̂m(γ).

(ii) Show that

∑

k∈Z

|ϕ̂(γ + k)|2 = 1, γ ∈ R.


15

Technical University of Denmark

Page 1 out of 3 pages

Written exam. Date: June 1, 2010.

Course name: Mathematics 4, Course no. 01325.


The exam consists of 4 problems, all together with 10 questions. The 10questions are weighted equally. In order to obtain full credit, you are requiredto provide complete arguments. The answers will be judged as a whole. Theanswers can be given in English or Danish.

All references are to the book “Functions, Spaces, and Expansions, 2010”by Ole Christensen.

Problem 216 Consider the set of functions

V := {f : R → C∣∣ f(−π) = f(π) = 0}.

(i) Show that V is a vector space with respect to the usual operations ofaddition and scalar multiplication.

(ii) Give an example of a linear operator that does not map V into V.


16

Problem 217 Consider the linear mapping

T : L1(0, 2) → L1(0, 2), (Tf)(x) :=

∫ 2

0

e−x2y2 f(y) dy.

(i) Show that for any x ∈ [0, 2],

|(Tf)(x)| ≤ ||f ||L1(0,2).

(ii) Use the result in (i) to show that T indeed maps L1(0, 2) into L1(0, 2)and is bounded with

||T || ≤ 2.

Problem 218 Consider the B-spline N2 and its translate T−2N2. Define thefunction f by

f(x) = N2(x)− T−2N2(x).

The graph shows the function f . The following questions can be answeredwithout calculating the function f explicitly.

(i) Show that the Fourier transform of the function f is

f̂(γ) =(1− e4πiγ

)(1− e−2πiγ

2πiγ

)2

.

(ii) Find the number of vanishing moments for the function f .Hint: Instead of calculating the involved integrals you can use symme-try properties of the functions f(x) and xf(x).

17

Problem 219 Consider a Hilbert spaceH and let {ek}∞k=1 be an orthonormalbasis for H. Consider a bounded invertible operator T : H → H for whichthe inverse T−1 is bounded as well.

(i) Show that if v ∈ H, then∞∑

k=1

|〈v, Tek〉|2 = ||T ∗v||2.

(ii) Show that the adjoint operator T ∗ is invertible, and that

(T ∗)−1 = (T−1)∗.

Hint: Use the result in Exercise 4.17 (ii).

(iii) Show that for all v ∈ H

||v|| ≤ ||(T ∗)−1|| ||T ∗v||.

Hint: Use that v = (T ∗)−1T ∗v.

(iv) Show that for all v ∈ H

1

||T−1||2 ||v||2 ≤

∞∑

k=1

|〈v, Tek〉|2 ≤ ||T ||2 ||v||2.


18


Written exam. Date: May 31, 2011.



Duration: 2 Hours.



f(x) :=∞∑

k=2

k χ[k,k+k−3](x) = 2χ[2,2 1

8](x) + 3χ[3,3 1

27](x) + · · · , x ∈ R.

(i) Make a draft of the function f .

(ii) Is f a bounded function?

(iii) Does f ∈ L1(R)?


f(x) := exχ[0,1](x), x ∈ R.

(i) Calculate the convolution (f ∗ f)(x) for x ∈ [0, 1].


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Problem 222 Consider a complex Hilbert space H and a bounded, linear,and self-adjoint operator

T : H → H.

We say that a number λ ∈ C is an eigenvalue for T if there exists an eigen-vector, i.e., a vector v 6= 0 such that

Tv = λv.

(i) Assume that λ ∈ C is an eigenvalue for T, and let v denote a corre-sponding eigenvector with ||v|| = 1. Calculate the numbers

〈Tv,v〉 and 〈v, Tv〉.

(ii) Show that all the eigenvalues are real numbers.

Now assume that H has an orthonormal basis {ek}∞k=1 and that for eachk ∈ N there exists a number λk ∈ R such that

Tek = λkek.

(iii) Show that for any f ∈ H,

T f =∞∑

k=1

λk 〈f, ek〉 ek.

Problem 223 We will consider the modulation operator Eb for complexvalues of the parameter b. That is, for b ∈ C we let Eb act on the functionf : R → C by

(Ebf)(x) := e2πibxf(x), x ∈ R.

(i) Show that the mapping Eb is linear for any b ∈ C.

(ii) Show that for any b ∈ C, the operator Eb maps L2(0, 1) into L2(0, 1)and is bounded.

Hint: Write b = α + iβ for some α, β ∈ R, and use that every expo-

nential function is bounded on [0, 1].

(iii) Determine the values of the parameter b ∈ C for which the operatorEb : L

2(0, 1) → L2(0, 1) is self-adjoint.


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Written exam. Date: May 29, 2012.



Duration: 2 Hours.


All references are to the book “Functions, Spaces, and Expansions, 2010”by Ole Christensen.

Problem 224 Let ω : R → C be a continuous bounded function, andconsider the mapping

U : L2(R) → L2(R), (Uf)(x) = ω(x)f(x), x ∈ R.

(i) Show that U indeed maps L2(R) into L2(R).

(ii) Show that U is linear and bounded.

(iii) Compute the adjoint operator U∗.

(iv) Show that U is unitary if and only if

|ω(x)| = 1, ∀ x ∈ R.


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out of 4 pages

Problem 225 We consider the space Lp(0, π/2) for p = 1/2, i.e.,

L1/2(0, π/2) := {f :]0, π/2[→ C∣∣∫ π/2

0

|f(x)|1/2 dx <∞}.

(i) Does the function

h(x) :=1

x, x ∈]0, π/2[

belong to L1/2(0, π/2)?

(ii) Show that every bounded function f :]0, π/2[→ C belongs to L1/2(0, π/2).

One can show (you are not expected to do this) that the functions

f(x) := sin2(x), g(x) := cos2(x), x ∈]0, π/2[,

satisfy that

∫ π/2

0

|f(x)|1/2 dx = 1,

∫ π/2

0

|g(x)|1/2 dx = 1,

∫ π/2

0

|f(x) + g(x)|1/2 dx =π

2.

(iii) Does the expression

||f || :=(∫ π/2

0

|f(x)|1/2 dx)2

define a norm on Lp(0, π/2) for p = 1/2?


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out of 4 pages

Problem 226 The figure shows the graph of the function

f(x) := xe−x2

, x ∈ R.

Find the number of vanishing moments for the function f. (An argumentbased on the properties of the function f is sufficient in order to obtain fullcredit.)


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out of 4 pages

Problem 227 Let {ek}∞k=1 denote an orthonormal basis for a Hilbert spaceH, and let

vk := ek + ek+1, k ∈ N.

(i) Let v ∈ H and assume that 〈v, vk〉 = 0 for all k ∈ N. Show that then

|〈v, ek〉| = |〈v, ek+1〉|, ∀k ∈ N.

(ii) Show that span{vk}∞k=1 = H.Hint: Use (i) and Theorem 4.7.2 (iv) to show that if〈v,vk〉 = 0, k ∈ N, then v = 0.

One can show (you are not expected to do this) that for any N ∈ N

and complex numbers ck ∈ C for k = 1, . . . , N,

∣∣∣∣∣

∣∣∣∣∣e1 −N∑

k=1

ckvk

∣∣∣∣∣

∣∣∣∣∣

2

= |1− c1|2 +N−1∑

k=1

|ck + ck+1|2 + |cN |2. (5)

(iii) Show that it is impossible to choose ck ∈ C such that

e1 =∞∑

k=1

ckvk.

Hint: Use (5) to argue that we only need to consider the special se-quence ck = (−1)k−1, k ∈ N.


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Extra exercises, 01325Mathematics 4 · 2013-02-08 · Problem 106 (Extension of Problem 222, Exam...

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